Zachary Abel, Erik D. Demaine, Martin L. Demaine, Sarah Eisenstat, Jayson Lynch and Tao Schardl

**Who Needs Crossings? Hardness of Plane Graph Rigidity**
[+]

We exactly settle the complexity of graph realization, graph rigidity, and graph global rigidity as applied to three types of graphs: "globally noncrossing" graphs, which avoid crossings in all of their configurations; matchstick graphs, with unit-length edges and where only noncrossing configurations are considered; and unrestricted graphs (crossings allowed) with unit edge lengths (or in the global rigidity case, edge lengths in {1,2}). We show that all nine of these questions are complete for the class Exists-R, defined by the Existential Theory of the Reals, or its complement Forall-R; in particular, each problem is (co)NP-hard.
One of these nine results - that realization of unit-distance graphs is Exists-R-complete - was shown previously by Schaefer (2013), but the other eight are new. We strengthen several prior results. Matchstick graph realization was known to be NP-hard (Eades & Wormald 1990, or Cabello et al. 2007), but its membership in NP remained open; we show it is complete for the (possibly) larger class Exists-R. Global rigidity of graphs with edge lengths in {1,2} was known to be coNP-hard (Saxe 1979); we show it is Forall-R-complete.
The majority of the paper is devoted to proving an analog of Kempe's Universality Theorem - informally, "there is a linkage to sign your name" - for globally noncrossing linkages. In particular, we show that any polynomial curve phi(x,y)=0 can be traced by a noncrossing linkage, settling an open problem from 2004. More generally, we show that the nontrivial regions in the plane that may be traced by a noncrossing linkage are precisely the compact semialgebraic regions. Thus, no drawing power is lost by restricting to noncrossing linkages. We prove analogous results for matchstick linkages and unit-distance linkages as well.

Mikkel Abrahamsen and Mikkel Thorup

**Finding the Maximum Subset with Bounded Convex Curvature**
[+]

We describe an algorithm for solving an important geometric problem arising in computer-aided manufacturing. When machining a pocket in a solid piece of material such as steel using a rough tool in a milling machine, sharp convex corners of the pocket cannot be done properly, but have to be left for finer tools that are more expensive to use. We want to determine a tool path that maximizes the use of the rough tool. Mathematically, this boils down to the following problem. Given a simply-connected set of points P in the plane such that the boundary of P is a curvilinear polygon consisting of n line segments and circular arcs of arbitrary radii, compute the maximum subset Q of P consisting of simply-connected sets where the boundary of each set is a curve with bounded convex curvature. A closed curve has bounded convex curvature if, when traversed in counterclockwise direction, it turns to the left with curvature at most 1. There is no bound on the curvature where it turns to the right. The difference in the requirement to left- and right-curvature is a natural consequence of different conditions when machining convex and concave areas of the pocket. We devise an algorithm to compute the unique maximum such set Q. The algorithm runs in O(n log n) time and uses O(n) space.
For the correctness of our algorithm, we prove a new generalization of the Pestov-Ionin Theorem. This is needed to show that the output Q of our algorithm is indeed maximum in the sense that if Q' is any subset of P with a boundary of bounded convex curvature, then Q' is a subset of Q.

Eyal Ackerman, Balázs Keszegh and Mate Vizer

**Coloring points with respect to squares**
[+]

We consider the problem of 2-coloring geometric hypergraphs. Specifically, we show that there is a constant m such that any finite set S of points in the plane can be 2-colored such that every axis-parallel square that contains at least m points from S contains points of both colors. Our proof is constructive, that is, it provides a polynomial-time algorithm for obtaining such a 2-coloring. By affine transformations this result immediately applies also when considering homothets of a fixed parallelogram.

Pankaj Agarwal, Kyle Fox, Jiangwei Pan and Rex Ying

**Approximating Dynamic Time Warping and Edit Distance for a Pair of Point Sequences**
[+]

We present the first subquadratic algorithms for computing similarity between a pair of point sequences in R^d, for any fixed d > 1, using dynamic time warping (DTW) and edit distance, assuming that the point sequences are drawn from certain natural families of curves. In particular, our algorithms compute (1 + eps)-approximations of DTW and ED in near-linear time for point sequences drawn from k-packed or k-bounded curves, and subquadratic time for backbone sequences. Roughly speaking, a curve is k-packed if the length of its intersection with any ball of radius r is at most kr, and it is k-bounded if the sub-curve between two curve points does not go too far from the two points compared to the distance between the two points. In backbone sequences, consecutive points are spaced at approximately equal distances apart, and no two points lie very close together. Recent results suggest that a subquadratic algorithm for DTW or ED is unlikely for an arbitrary pair of point sequences even for d = 1.
The commonly used dynamic programming algorithms for these distance measures reduce the problem to computing a minimum-weight path in a grid graph. Our algorithms work by constructing a small set of rectangular regions that cover the grid vertices. The weights of vertices inside each rectangle are roughly the same, and we develop efficient procedures to compute the approximate minimum-weight paths through these rectangles.

Oswin Aichholzer, Victor Alvarez, Thomas Hackl, Alexander Pilz, Bettina Speckmann and Birgit Vogtenhuber

**An Improved Lower Bound on the Number of Triangulations**
[+]

Upper and lower bounds for the number of geometric graphs of specific types on a given set of points in the plane have been intensively studied in recent years. For most classes of geometric graphs it is now known that point sets in convex position minimize their number. However, it is still unclear which point sets minimize the number of geometric triangulations; the so-called double circles are conjectured to be the minimizing sets. In this paper we prove that any set of n points in general position in the plane has at least Omega(2.631^n) geometric triangulations. Our result improves the previously best general lower bound of Omega(2.43^n) and also covers the previously best lower bound of Omega(2.63^n) for a fixed number of extreme points. We achieve our bound by showing and combining several new results, which are of independent interest:
(1) Adding a point on the second convex layer of a given point set (of 7 or more points) at least doubles the number of triangulations.
(2) Generalized configurations of points that minimize the number of triangulations have at most n/2 points on their convex hull.
(3) We provide tight lower bounds for the number of triangulations of point sets with up to 15 points. These bounds further support the double circle conjecture.

Hugo Akitaya, Greg Aloupis, Jeff Erickson and Csaba Toth

**Recognizing Weakly Simple Polygons**
[+]

We present an O(n log n)-time algorithm that determines whether a given planar n-gon is weakly simple. This improves upon an O(n^2 log n)-time algorithm by [Chang, Erickson, and Xu, SODA, 2015]. Weakly simple polygons are required as input for several geometric algorithms. As such, how to recognize simple or weakly simple polygons is a fundamental question.

Alexandr Andoni and Ilya Razenshteyn

**Tight Lower Bounds for Data-Dependent Locality-Sensitive Hashing**
[+]

We prove a tight lower bound for the exponent rho for
data-dependent Locality-Sensitive Hashing schemes, recently used to
design efficient solutions for the c-approximate nearest neighbor
search. In particular, our lower bound matches the bound of rho<=
1/(2c-1)+o(1) for the l_1 space, obtained via the recent
algorithm from [Andoni-Razenshteyn, STOC'15].
In recent years it emerged that data-dependent hashing is strictly
superior to the classical Locality-Sensitive Hashing, when
the hash function is data-independent. In the latter setting, the best
exponent has been already known: for the l_1 space, the tight
bound is rho=1/c, with the upper bound from [Indyk-Motwani,
STOC'98] and the matching lower bound from [O'Donnell-Wu-Zhou,
ITCS'11].
We prove that, even if the hashing is data-dependent, it must hold
that rho>=1/(2c-1)-o(1). To prove the result, we need to
formalize the exact notion of data-dependent hashing that also
captures the complexity of the hash functions (in addition to their
collision properties). Without restricting such complexity, we would
allow for obviously infeasible solutions such as the Voronoi diagram
of a dataset. To preclude such solutions, we require our hash functions
to be succinct. This condition is satisfied by all the known algorithmic results.

Boris Aronov, Otfried Cheong, Michael Gene Dobbins and Xavier Goaoc

**The Number of Holes in the Union of Translates of a Convex Set in Three Dimensions**
[+]

We show that the union of translates of a convex body in three dimensional space can have a cubic number holes in the worst case, where a hole in a set is a connected component of its compliment. This refutes a 20-year-old conjecture. As a consequence, we also obtain improved lower bounds on the complexity of motion planning problems and of Voronoi diagrams with convex distance functions.

Sunil Arya, Guilherme D. Da Fonseca and David Mount

**On the Combinatorial Complexity of Approximating Polytopes**
[+]

Approximating convex bodies succinctly by convex polytopes is a fundamental problem in discrete geometry. A convex body $K$ of diameter $diam(K)$ is given in Euclidean $d$-dimensional space, where $d$ is a constant. Given an error parameter $eps > 0$, the objective is to determine a polytope of minimum combinatorial complexity whose Hausdorff distance from $K$ is at most $eps diam(K)$. By combinatorial complexity we mean the total number of faces of all dimensions of the polytope. A well-known result by Dudley implies that $O(1/eps^{(d-1)/2})$ facets suffice, and a dual result by Bronshteyn and Ivanov similarly bounds the number of vertices, but neither result bounds the total combinatorial complexity. We show that there exists an approximating polytope whose total combinatorial complexity is $O-tilde(1/eps^{(d-1)/2})$, where $O-tilde$ conceals a polylogarithmic factor in $1/eps$. This is an improvement upon the best known bound, which is roughly $O(1/eps^{d-2})$.
Our result is based on a novel combination of both new and old ideas. First, we employ Macbeath regions, a classical structure from the theory of convexity. The construction of our approximating polytope employs a new stratified placement of these regions. Second, in order to analyze the combinatorial complexity of the approximating polytope, we present a tight analysis of a width-based variant of Barany and Larman's economical cap covering, which may be of independent interest. Finally, we use a deterministic variation of the witness-collector technique (developed recently by Devillers et al.) in the context of our stratified construction.

Bhaskar Bagchi, Benjamin A. Burton, Basudeb Datta, Nitin Singh and Jonathan Spreer

**Efficient algorithms to decide tightness**
[+]

Tightness is a generalisation of the notion of convexity: a space is tight if and only if it is "as convex as possible", given its topological constraints. For a simplicial complex, deciding tightness has a straightforward exponential time algorithm, but more efficient methods to decide tightness are only known in the trivial setting of triangulated surfaces.
In this article, we present a new polynomial time procedure to decide tightness for triangulations of 3-manifolds -- a problem which previously was thought to be hard. In addition, for the more difficult problem of deciding tightness of 4-dimensional combinatorial manifolds, we describe an algorithm that is fixed parameter tractable in the treewidth of the 1-skeletons of the vertex links. Finally, we show that simpler treewidth parameters are not viable: for all non-trivial inputs, we show that the treewidths of both the 1-skeleton and the dual graph must grow too quickly for a standard treewidth-based algorithm to remain tractable.

Kevin Balas, Adrian Dumitrescu and Csaba Toth

**Anchored Rectangle and Square Packings**
[+]

For points p_1,...,p_n in the unit square [0,1]^2, an anchored rectangle packing consists of interior-disjoint axis-aligned empty rectangles r_1,...,r_n in [0,1]^2 such that point p_i is a corner of the rectangle r_i (that is, r_i is anchored at p_i) for
i=1,...,n. We show that for every set of n points in [0,1]^2, there is an anchored rectangle packing of area at least 7/12-O(1/n), and for every n, there are point sets for which the area of every anchored rectangle packing is at most 2/3. The maximum area of an anchored square packing is always at least 5/32 and sometimes at most 7/27.
The above constructive lower bounds immediately yield constant-factor
approximations, of 7/12 -epsilon for rectangles and 5/32 for squares, for computing anchored packings of maximum area in
O(n log n) time. We prove that a simple greedy strategy achieves a 9/47-approximation for anchored square packings, and 1/3 for lower-left anchored square packings. Reductions
to maximum weight independent set (MWIS) yield a QPTAS and a PTAS for anchored rectangle and square packings in n^{O(1/epsilon)}
and exp(poly(log (n/epsilon))) time, respectively.

Sayan Bandyapadhyay and Kasturi Varadarajan

**On Variants of k-means Clustering**
[+]

Clustering problems often arise in fields like data mining and machine learning. Clustering usually refers to the task of partitioning a collection of objects into groups with similar elements, with respect to a similarity (or dissimilarity) measure. Among the clustering problems, k-means clustering in particular has received much attention from researchers. Despite the fact that k-means is a well studied problem, its status in the plane is still open. In particular, it is unknown whether it admits a PTAS in the plane. The best known approximation bound achievable in polynomial time is 9+epsilon.
In this paper, we consider the following variant of k-means. Given a set C of points in R^d and a real f > 0, find a finite set F of points in R^d that minimizes the quantity f*|F|+sum_{p in C} min_{q in F} {||p-q||}^2. For any fixed dimension d, we design a PTAS for this problem that is based on local search. We also give a ``bi-criterion'' local search algorithm for k-means which uses (1+epsilon)k centers and yields a solution whose cost is at most (1+epsilon) times the cost of an optimal k-means solution. The algorithm runs in polynomial time for any fixed dimension.
The contribution of this paper is two-fold. On the one hand, we are able to handle the square of distances in an elegant manner, obtaining a near-optimal approximation bound. This leads us towards a better understanding of the k-means problem. On the other hand, our analysis of local search might also be useful for other geometric problems. This is important considering that little is known about the local search method for geometric approximation.

Luis Barba, Stefan Langerman, Sarah R. Allen and John Iacono

**Incremental Voronoi diagrams**
[+]

We study the amortized number of combinatorial changes (edge insertions and removals) needed to update the graph structure of the Voronoi diagram VD(S) (and several variants thereof) of a set S of n sites in the plane as sites are added to the set.
To that effect, we define a general update operation for planar graphs that can be used to model the incremental construction of several variants of Voronoi diagrams as well as the incremental construction of an intersection of halfspaces in R^3.
We show that the amortized number of edge insertions and removals needed to add a new site to the Voronoi diagram is O(n^(1/2)).
A matching Omega(n^(1/2)) combinatorial lower bound is shown, even in the case where the graph representing the Voronoi diagram is a tree.
This contrasts with the O(log(n)) upper bound of Aronov et al. [Aronov et al., in proc. of LATIN, 2006] for farthest-point Voronoi diagrams in the special case where points are inserted in clockwise order along their convex hull.
We then present a semi-dynamic data structure that maintains the Voronoi diagram of a set S of n sites in convex position.
This data structure supports the insertion of a new site p (and hence the addition of its Voronoi cell) and finds the asymptotically minimal number K of edge insertions and removals needed to obtain the diagram of S U (p) from the diagram of S, in time O(K polylog n) worst case, which is O(n^(1/2) polylog n) amortized by the aforementioned combinatorial result.
The most distinctive feature of this data structure is that the graph of the Voronoi diagram is maintained explicitly at all times and can be retrieved and traversed in the natural way; this contrasts with other known data structures supporting nearest neighbor queries.
Our data structure supports general search operations on the current Voronoi diagram, which can, for example, be used to perform point location queries in the cells of the current Voronoi diagram in O(log n) time, or to determine whether two given sites are neighbors in the Delaunay triangulation.

Yair Bartal and Lee-Ad Gottlieb

**Dimension reduction techniques for l_p (1<p<2), with applications**
[+]

For Euclidean space (l_2), there exists the powerful dimension reduction transform of Johnson and Lindenstrauss [Conf. in modern analysis and probability, AMS 1984], with a host of known applications. Here, we consider the problem of dimension
reduction for all l_p spaces 1<p<2. Although strong lower bounds are known for dimension reduction in l_1, Ostrovsky and Rabani [JACM 2002] successfully circumvented these by presenting an l_1 embedding that maintains fidelity in only a bounded distance range, with applications to clustering and nearest neighbor search. However, their embedding techniques are specific to l_1 and do not naturally extend to other norms. In this paper, we apply a range of advanced techniques and produce bounded range dimension reduction embeddings for all of 1<p<2, thereby demonstrating that the approach initiated
by Ostrovsky and Rabani for l_1 can be extended to a much more general framework. We also obtain improved bounds in terms of the intrinsic dimensionality. As a result we achieve improved
bounds for proximity problems including snowflake embeddings and clustering.

Piotr Berman, Meiram Murzabulatov and Sofya Raskhodnikova

**Testing Convexity of Figures Under the Uniform Distribution**
[+]

We consider the following basic geometric problem: Given epsilon in (0,1/2), a 2-dimensional figure that consists of a black object and a white background is epsilon-far from convex if it differs in at least an epsilon fraction of the area from every figure where the black object is convex. How many uniform and independent samples from a figure that is epsilon-far from convex are needed to detect a violation of convexity with probability at least 2/3? This question arises in the context of designing property testers for convexity.
Specifically, a (1-sided error) tester for convexity gets samples from the figure, labeled by their color; it always accepts if the black object is convex; it rejects with probability at least 2/3 if the figure is epsilon-far from convex.
We show that Theta(epsilon^{-4/3}) uniform samples are necessary and sufficient for detecting a violation of convexity in an epsilon-far figure and, equivalently, for testing convexity of figures with 1-sided error. Our testing algorithm runs in time O(epsilon^{-4/3}) and thus beats the Omega(epsilon^{-3/2}) sample lower bound for learning convex figures under the uniform distribution from the work of Schmeltz (Data Structures and Efficient Algorithms,1992). It shows that, with uniform samples, we can check if a set is approximately convex much faster than we can find an approximate representation of a convex set.

V.S.P. Vijay Bhattiprolu and Sariel Har-Peled

**Separating a Voronoi Diagram via Local Search**
[+]

Given a set P of n points in R^d , we show how to insert a set Z of O(n^(1-1/d)) additional points,
such that P can be broken into two sets P1 and P2 , of roughly equal size, such that in the Voronoi
diagram V(P u Z), the cells of P1 do not touch the cells of P2 ; that is, Z separates P1 from P2
in the Voronoi diagram (and also in the dual Delaunay triangulation). In addition, given such
a partition (P1 , P2 ) of P , we present an approximation algorithm to compute a minimum size
separator realizing this partition. We also present a simple local search algorithm that is a PTAS
for approximating the optimal Voronoi partition.

Therese Biedl, Giuseppe Liotta and Fabrizio Montecchiani

**On Visibility Representations of Non-planar Graphs**
[+]

A rectangle visibility representation (RVR) of a graph consists of an assignment of axis-aligned
rectangles to vertices such that for every edge there exists a horizontal or vertical line of sight
between the rectangles assigned to its endpoints. Testing whether a graph has an RVR is known
to be NP-hard. In this paper, we study the problem of finding an RVR under the assumption
that an embedding in the plane of the input graph is fixed and we are looking for an RVR that
reflects this embedding. We show that in this case the problem can be solved in polynomial
time for general embedded graphs and in linear time for 1-plane graphs (i.e., embedded graphs
having at most one crossing per edge). The linear time algorithm uses a precise list of forbidden
configurations, which extends the set known for straight-line drawings of 1-plane graphs. These
forbidden configurations can be tested for in linear time, and so in linear time we can test whether
a 1-plane graph has an RVR and either compute such a representation or report a negative witness.
Finally, we discuss some extensions of our study to the case when the embedding is not fixed but
the RVR can have at most one crossing per edge.

Mikhail Bogdanov, Monique Teillaud and Gert Vegter

**Delaunay triangulations on orientable surfaces of low genus**
[+]

Earlier work on Delaunay triangulation of point sets on the 2D flat
torus, which is locally isometric to the Euclidean plane, was based on
lifting the point set to a locally isometric 9-sheeted covering space
of the torus. Under mild conditions the Delaunay triangulation of the
lifted point set, consisting of 9 copies of the input set, projects to
the Delaunay triangulation of the input set.
We improve and generalize this work. First we present a new
construction based on an 8-sheeted covering space, which shows that
eight copies suffice for the standard flat torus. Then we generalize
this construction to the context of compact orientable surfaces of
higher genus, which are locally isometric to the hyperbolic plane. We
investigate more thoroughly the Bolza surface, homeomorphic to a
sphere with two handles, both because it is the hyperbolic surface
with lowest genus, and because triangulations on the Bolza surface
have applications in various fields such as neuromathematics and
cosmological models. While the general properties (existence results
of appropriate covering spaces) show similarities with the results for
the flat case, explicit constructions and their proofs are much more
complex, even in the case of the apparently simple Bolza surface. One
of the main reasons is the fact that two hyperbolic translations do
not commute in general.
To the best of our knowledge, the results in this paper are the first
ones of this kind. The interest of our contribution lies not only in
the results, but most of all in the construction of covering spaces
itself and the study of their properties.

Cecilia Bohler, Rolf Klein and Chih-Hung Liu

**An Efficient Randomized Algorithm for Higher-Order Abstract Voronoi Diagrams**
[+]

Given a set of n sites in the plane,
the order-k Voronoi diagram is a planar subdivision such that all points in a region share the same k nearest sites.
The order-k Voronoi diagram arises for the k-nearest-neighbor problem, and there has been a lot of work for point sites in the Euclidean metric.
In this paper, we study order-k Voronoi diagrams defined by an abstract bisecting curve system that satisfies several practical axioms,
and thus our study covers many concrete order-k Voronoi diagrams.
We propose a randomized incremental construction algorithm that runs in O(k(n-k) log^2 n +n log^3 n) steps, where O(k(n-k)) is the number of faces in the worst case.
Due to those axioms, this result applies to disjoint line segments in the L_p norm,
convex polygons of constant size, points in the Karlsruhe metric, and so on. In fact,
this kind of run time with a polylog factor to the number of faces was only achieved for point sites in the L_1 or Euclidean metric before.

Glencora Borradaile, David Eppstein, Christian Wulff-Nilsen and Amir Nayyeri

**All-Pairs Minimum Cuts in Near-Linear Time for Surface-Embedded Graphs**
[+]

For an undirected n-vertex graph G with non-negative edge-weights, we consider the following type of query: given two vertices s and t in G, what is the weight of a minimum st-cut in G? We solve this problem in preprocessing time O(n log^3 n) for graphs of bounded genus, giving the first sub-quadratic time algorithm for this class of graphs. Our result also improves by a logarithmic factor a previous algorithm by Borradaile, Sankowski and Wulff-Nilsen (FOCS 2010) that applied only to planar graphs. Our algorithm constructs a Gomory-Hu tree for the given graph, providing a data structure with space O(n) that can answer minimum-cut queries in constant time. The dependence on the genus of the input graph in our preprocessing time is 2^{O(g^2)}.

Glencora Borradaile, Erin Wolf Chambers, Kyle Fox and Amir Nayyeri

**Minimum cycle and homology bases of surface embedded graphs**
[+]

We study the problems of finding a minimum cycle basis (a minimum weight set of cycles that form a basis for the cycle space) and a minimum homology basis (a minimum weight set of cycles that generates the 1-dimensional (Z_2)-homology classes) of an undirected graph embedded on an orientable surface of genus g. The problems are closely related, because the minimum cycle basis of a graph contains its minimum homology basis, and the minimum homology basis of the 1-skeleton of any graph is exactly its minimum cycle basis.
For the minimum cycle basis problem, we give a deterministic O(n^omega + 2^2g n^2)-time algorithm. The best known existing algorithms for surface embedded graphs are those for general sparse graphs: an O(n^omega) time Monte Carlo algorithm [Amaldi et. al., ESA '09] and a deterministic O(n^3) time algorithm [Mehlhorn and Michail, TALG '09]. For the minimum homology basis problem, we give an O(g^3 n log n)-time algorithm, improving on existing algorithms for many values of g and n.

Benjamin A. Burton, Arnaud De Mesmay and Uli Wagner

**Finding non-orientable surfaces in 3-manifolds**
[+]

We investigate the complexity of finding an embedded non-orientable surface of Euler genus g in a triangulated 3-manifold. This problem occurs both as a natural question in low-dimensional topology, and as a first non-trivial instance of embeddability of complexes into 3-manifolds.
We prove that the problem is NP-hard, thus adding to the relatively few hardness results that are currently known in 3-manifold topology. In addition, we show that the problem lies in NP when the Euler genus g is odd, and we give an explicit algorithm in this case.

Mathieu Carrière and Steve Oudot

**Structure and Stability of the 1-Dimensional Mapper**
[+]

Given a continuous function f:X->R and a cover I of its image
by intervals, the Mapper is the nerve of a refinement of the pullback
cover f^{-1}(I). Despite its success in applications, little is
known about the structure and stability of this construction from a
theoretical point of view. As a pixelized version of the Reeb graph
of f, it is expected to capture a subset of its features (branches,
holes), depending on how the interval cover is positioned with respect
to the critical values of the function. Its stability should also
depend on this positioning. We propose a theoretical framework
relating the structure of the Mapper to that of the Reeb graph,
making it possible to predict which features will be present and which
will be absent in the Mapper given the function and the cover, and for
each feature, to quantify its degree of (in-)stability. Using this
framework, we can derive guarantees on the structure of the Mapper, on
its stability, and on its convergence to the Reeb graph as the
granularity of the cover I goes to zero.

Alfonso Cevallos, Friedrich Eisenbrand and Rico Zenklusen

**Max-sum diversity via convex programming**
[+]

Diversity maximization is an important concept in information retrieval, computational geometry and operations research. Usually, it is a variant of the following problem: Given a ground set, constraints, and a function f that measures diversity of a subset, the task is to select a feasible subset S such that f(S) is maximized. The sum-dispersion function f(S) which is the sum of the pairwise distances in S, is in this context a prominent diversification measure. The corresponding diversity maximization is the "max-sum" or "sum-sum" diversification. Many recent results deal with the design of constant-factor approximation algorithms of diversification problems involving sum-dispersion function under a matroid constraint.
In this paper, we present a PTAS for the max-sum diversity problem under a matroid constraint for distances d(.,.) of negative type. Distances of negative type are, for example, metric distances stemming from the l_2 and l_1 norms, as well as the cosine or spherical, or Jaccard distance which are popular similarity metrics in web and image search.
Our algorithm is based on techniques developed in geometric algorithms like metric embeddings and convex optimization. We show that one can compute a fractional solution of the usually non-convex relaxation of the problem which yields an upper bound on the optimum integer solution. Starting from this fractional solution, we employ a deterministic rounding approach which only incurs a small loss in terms of objective, thus leading to a PTAS. This technique can be applied to other previously studied variants of the max-sum dispersion function, including combinations of diversity with linear-score maximization, improving over the previous constant-factor approximation algorithms.

Timothy M. Chan

**Dynamic Streaming Algorithms for Epsilon-Kernels**
[+]

Introduced by Agarwal, Har-Peled, and Varadarajan [J. ACM, 2004], an epsilon-kernel of a point set is a coreset that can be used to approximate the width, minimum enclosing cylinder, minimum bounding box, and solve various related geometric optimization problems. Such coresets form one of the most important tools in the design of linear-time approximation algorithms in computational geometry, as well as efficient insertion-only streaming algorithms and dynamic (non-streaming) data structures. In this paper, we continue the theme and explore dynamic streaming algorithms (in the so-called turnstile model).
Andoni and Nguyen [SODA 2012] described a dynamic streaming algorithm for maintaining a (1+epsilon)-approximation of the width using O(polylog U) space and update time for a point set in [U]^d for any constant dimension d and any constant epsilon>0. Their sketch, based on a "polynomial method", does not explicitly maintain an epsilon-kernel. We extend their method to maintain an epsilon-kernel, and at the same time reduce some of logarithmic factors. As an application, we obtain the first randomized dynamic streaming algorithm for the width problem (and related geometric optimization problems) that supports k outliers, using poly(k, log U) space and time.

Timothy M. Chan and Simon Pratt

**Two Approaches to Building Time-Windowed Geometric Data Structures**
[+]

Given a set of geometric objects each associated with a time value, we wish to determine whether a given property is true for a subset of those objects whose time values fall within a query time window. We call such problems time-windowed decision problems, and they have been the subject of much recent attention, for instance studied by Bokal, Cabello, and Eppstein [SoCG 2015]. In this paper, we present new approaches to this class of problems that are conceptually simpler than Bokal et al.'s, and also lead to faster algorithms. For instance, we present algorithms for preprocessing for the time-windowed 2D diameter decision problem in O(n log n) time and the time-windowed 2D convex hull area decision problem in O(n alpha(n) log n) time (where alpha is the inverse Ackermann function), improving Bokal et al.'s O(n log^2 n) and O(n log n loglog n) solutions respectively.
Our first approach is to reduce time-windowed decision problems to a generalized range successor problem, which we solve using a novel way to search range trees. Our other approach is to use dynamic data structures directly, taking advantage of a new observation that the total number of combinatorial changes to a planar convex hull is near linear for any FIFO update sequence, in which deletions occur in the same order as insertions. We also apply these approaches to obtain the first O(n polylog n) algorithms for the time-windowed 3D diameter decision and 2D orthogonal segment intersection detection problems.

Hsien-Chih Chang and Jeff Erickson

**Untangling Planar Curves**
[+]

Any generic closed curve in the plane can be transformed into a simple closed curve by a finite sequence of local transformations called homotopy moves. We prove that simplifying a planar closed curve with n self-crossings requires Theta(n^{3/2}) homotopy moves in the worst case. Our algorithm improves the best previous upper bound O(n^2), which is already implicit in the classical work of Steinitz; the matching lower bound follows from the construction of closed curves with large *defect*, a topological invariant of generic closed curves introduced by Aicardi and Arnold. This lower bound also implies that Omega(n^{3/2}) degree-1 reductions, series-parallel reductions, and Delta-Y transformations are required to reduce any planar graph with treewidth Omega(sqrt{n}) to a single edge, matching known upper bounds for rectangular and cylindrical grid graphs. Finally, we prove that Omega(n^2) homotopy moves are required in the worst case to transform one non-contractible closed curve on the torus to another; this lower bound is tight if the curve is homotopic to a simple closed curve.

Markus Chimani and Petr Hlineny

**Inserting Multiple Edges into a Planar Graph**
[+]

Let G be a connected planar (but not yet embedded) graph and F a set of additional edges not in G. The multiple edge insertion problem (MEI) asks for a drawing of G+F with the minimum number of pairwise edge crossings, such that the subdrawing of G is plane. An optimal solution to this problem is known to approximate the crossing number of the graph G+F.
Finding an exact solution to MEI is NP-hard for general F, but linear time solvable for the special case of |F|=1 [Gutwenger et al, SODA 2001/Algorithmica] and polynomial time solvable when all of F are incident to a new vertex [Chimani et al, SODA 2009]. The complexity for general F but with constant k=|F| was open, but algorithms both with relative and absolute approximation guarantees have been presented [Chuzhoy et al, SODA 2011], [Chimani-Hlineny, ICALP 2011]. We show that the problem is fixed parameter tractable (FPT) in k for biconnected G, or if the cut vertices of G have bounded degrees. We give the first exact algorithm for this problem; it requires only O(|V(G)|) time for any constant k.

Aruni Choudhary, Michael Kerber and Sharath Raghvendra

**Polynomial-sized Topological Approximations using the Permutahedron**
[+]

Classical methods to model topological properties of point clouds, such as the Vietoris-Rips complex, suffer from the combinatorial explosion of complex sizes. We propose a novel technique to approximate a multi-scale filtration
of the Rips complex with improved bounds for size: precisely, for n points in R^d, we obtain a O(d)-approximation with at most
n2^{O(d log k)} simplices of dimension k or lower. In conjunction with dimension reduction techniques, our approach yields a O(polylog (n))-approximation of size n^{O(1)} for Rips filtrations on arbitrary metric spaces. This result stems from high-dimensional lattice geometry and exploits properties of the permutahedral lattice, a well-studied structure in discrete geometry.
Building on the same geometric concept, we also present a lower bound result on the size of an approximate filtration: we construct a point set for which every (1+epsilon)-approximation of the Cech filtration has to contain n^{Omega(log log n)} features, provided that epsilon < 1/(log^{1+c}n) for c in (0,1).

Mark de Berg, Joachim Gudmundsson and Mehran Mehr

**Faster Algorithms for Computing Plurality Points**
[+]

Let V be a set of n points in R^d, which we call voters, where d is a fixed constant. A point p in R^d is preferred over another point p' in R^d by a voter v in V if dist(v,p) < dist(v,p'). A point p is called a plurality point if it is preferred by at least as many voters as any other point p'.
We present an algorithm that decides in O(n log n) time whether V admits a plurality point in the L_2 norm and, if so, finds the (unique) plurality point. We also give efficient algorithms to compute the smallest subset W of V such that V - W admits a plurality point, and to compute a so-called minimum-radius plurality ball.
Finally, we consider the problem in the personalized L_1 norm, where each point v in V has a preference vector <w_1(v), ...,w_d(v)> and the distance from v to any point p in R^d is given by sum_{i=1}^d w_i(v) cdot |x_i(v)-x_i(p)|. For this case we can compute in O(n^(d-1)) time the set of all plurality points of V. When all preference vectors are equal, the running time improves to O(n).

Olivier Devillers, Menelaos Karavelas and Monique Teillaud

**Qualitative Symbolic Perturbation**
[+]

In a classical Symbolic Perturbation scheme, degeneracies are handled by substituting some polynomials in epsilon for the inputs of a predicate. Instead of a single perturbation, we propose to use a sequence of (simpler) perturbations. Moreover, we look at their effects geometrically instead of algebraically; this allows us to tackle cases that were not tractable with the classical algebraic approach.

Hu Ding, Jing Gao and Jinhui Xu

**Finding Global Optimum for Truth Discovery: Entropy Based Geometric Variance**
[+]

Truth Discovery is an important problem arising in data analytics related fields such as data mining, database, and big data. It concerns about finding the most trustworthy information from a dataset acquired from a number of unreliable sources. Due to its importance, the problem has been extensively studied in recent years and a number techniques have already been proposed. However, all of them are of heuristic nature and do not have any quality guarantee. In this paper, we formulate the problem as a high dimensional geometric optimization problem, called Entropy based Geometric Variance. Relying on a number of novel geometric techniques (such as Log-Partition and Modified Simplex Lemma), we further discover new insights to this problem. We show, for the first time, that the truth discovery problem can be solved with guaranteed quality of solution. Particularly, we show that it is possible to achieve a (1+eps)-approximation within nearly linear time under some reasonable assumptions. We expect that our algorithm will be useful for other data related applications.

Dominic Dotterrer, Tali Kaufman and Uli Wagner

**On Expansion and Topological Overlap**
[+]

We give a detailed and easily accessible proof of Gromov's Topological Overlap Theorem. Let X be a finite simplicial complex or, more generally, a finite polyhedral cell complex of dimension d. Informally, the theorem states that if X has sufficiently strong higher-dimensional expansion properties (which generalize edge expansion of graphs and are defined in terms of cellular cochains of X) then X has the following topological overlap property: for every continuous map X rightarrow R^d there exists a point p in R^d whose preimage intersects a positive fraction mu > 0 of the d-cells of X. More generally, the conclusion holds if R^d is replaced by any d-dimensional piecewise-linear (PL) manifold M, with a constant mu that depends only on d and on the expansion properties of X, but not on M.

Adrian Dumitrescu and Minghui Jiang

**On the number of maximum empty boxes amidst $n$ points**
[+]

We revisit the following problem (along with its higher dimensional variant):
Given a set S of n points inside an axis-parallel rectangle U in the
plane, find a maximum-area axis-parallel sub-rectangle that is contained in U but contains no points of S.
(I) We prove that the number of maximum-area empty rectangles amidst n points in the plane is O(n log n 2^alpha(n)), where alpha(n) is the
extremely slowly growing inverse of Ackermann's function. The previous best bound, O(n^2), is due to Naamad, Lee, and Hsu (1984).
(II) For any d at least 3,
we prove that the number of maximum-volume empty boxes amidst n points in R^d is always O(n^d) and sometimes Omega(n^floor(d/2)).
This is the first superlinear lower bound derived for this problem.
(III) We discuss some algorithmic aspects regarding the search for a maximum empty box
in R^3. In particular, we present an algorithm that finds a (1-epsilon)-approximation
of the maximum empty box amidst n points in
O(epsilon^{-2} n^{5/3} log^2{n}) time.

Stefan Felsner, Alexander Igamberdiev, Philipp Kindermann, Boris Klemz, Tamara Mchedlidze and Manfred Scheucher

**Strongly Monotone Drawings of Planar Graphs**
[+]

A straight-line drawing of a graph is a monotone drawing if for each pair of vertices there is a path which is monotonically increasing in some direction, and it is called a strongly monotone drawing if the direction of monotonicity is given by the direction of the line segment connecting the two vertices.
We present algorithms to compute crossing-free strongly monotone drawings for some classes of planar graphs; namely, 3-connected planar graphs, outerplanar graphs, and 2-trees. The drawings of 3-connected planar graphs are based on primal-dual circle packings. Our drawings of outerplanar graphs depend on a new algorithm that constructs strongly monotone drawings of trees which are also convex. For irreducible trees, these drawings are strictly convex.

Martin Fink, John Hershberger, Nirman Kumar and Subhash Suri

**Hyperplane Separability and Convexity of Probabilistic Point Sets**
[+]

We describe an O(n^d) time algorithm for computing the exact probability that two d-dimensional probabilistic point sets are linearly separable, for any fixed d >= 2. A probabilistic point in d-space is the usual point, but with an associated (independent) probability of existence. We also show that the d-dimensional separability problem is equivalent to a (d+1)-dimensional convex hull membership problem, which asks for the probability that a query point lies inside the convex hull of n probabilistic points. Using this reduction, we improve the current best bound for the convex hull membership by a factor of n [Agarwal et al., ESA, 2014]. In addition, our algorithms can handle "input degeneracies" in which more than k+1 points may lie on a k-dimensional subspace, thus resolving an open problem in [Agarwal et al., ESA, 2014]. Finally, we prove lower bounds for the separability problem via a reduction from the k-SUM problem, which shows in particular that our O(n^2) algorithms for 2-dimensional separability and 3-dimensional convex hull membership are nearly optimal.

Fedor Fomin, Sudeshna Kolay, Daniel Lokshtanov, Fahad Panolan and Saket Saurabh

**Subexponential algorithms for rectilinear Steiner tree and arborescence problems**
[+]

A rectilinear Steiner tree for a set T of points in the plane is a tree
which connects T using horizontal and vertical lines. In the Rectilinear Steiner Tree problem, input is a set T of n points in the Euclidean plane (R^2) and the goal is to find an rectilinear Steiner tree for T of smallest possible total length. A rectilinear Steiner arborescence for a set T of points and root r in T is a rectilinear Steiner tree S for T such that the path in S from r to any point t in T is a shortest path. In the Rectilinear Steiner Arborescense problem the input is a set T of n points in R^2, and a root r in T, the task is to find an rectilinear Steiner arborescence for T, rooted at r of smallest possible total length. In this paper, we give the first subexponential time algorithms for both problems. Our algorithms are deterministic and run in 2^{O(sqrt{n}log n)} time.

Bernd Gärtner, Johannes Lengler and May Szedlak

**Random Sampling with Removal**
[+]

Random sampling is a classical tool in constrained optimization.
Under favorable conditions, the optimal solution subject to a small
subset of randomly chosen constraints violates only a small subset of
the remaining constraints. Here we study the following variant that we
call random sampling with removal: suppose that after sampling the
subset, we remove a fixed number of constraints from the sample,
according to an arbitrary rule. Is it still true that the optimal
solution of the reduced sample violates only a small subset of the constraints?
The question naturally comes up in situations where the solution
subject to the sampled constraints is used as an approximate solution to
the original problem. In this case, it makes sense to improve cost and
volatility of the sample solution by removing some of the constraints
that appear most restricting. At the same time, the approximation
quality (measured in terms of violated constraints) should remain high.
We study random sampling with removal in a generalized, completely abstract
setting where we assign to each subset R of the constraints an arbitrary
set V(R) of constraints disjoint from R; in applications, V(R)
corresponds to the constraints violated by the optimal solution subject
to only the constraints in R. Furthermore, our results are
parametrized by the dimension d, i.e., we assume that every set R has a
subset B of size at most d with the same set of violated constraints. This
is the first time this generalized setting is studied.
In this setting, we prove matching upper and lower bounds for the
expected number of constraints violated by a random sample, after the
removal of k elements. For a large range of values of k, the new upper
bounds improve the previously best bounds for LP-type problems, which
moreover had only been known in special cases.
We show that this bound on special LP-type problems, can be derived in
the much more general setting of violator
spaces, and with very elementary proofs.

Markus Geyer, Michael Hoffmann, Michael Kaufmann, Vincent Kusters and Csaba Toth

**The Planar Tree Packing Theorem**
[+]

Packing graphs is a combinatorial problem where several given graphs are being mapped into a common host graph such that every edge is used at most once. In the planar tree packing problem we are given two trees T1 and T2 on n vertices and have to find a planar graph on n vertices that is the edge-disjoint union of T1 and T2. A clear exception that must be made is the star which cannot be packed together with any other tree. But according to a conjecture of García et al. from 1997 this is the only exception, and all other pairs of trees admit a planar packing. Previous results addressed various special cases, such as a tree and a spider tree, a tree and a caterpillar, two trees of diameter four, two isomorphic trees, and trees of maximum degree three. Here we settle the conjecture in the affirmative and prove its general form, thus making it the planar tree packing theorem. The proof is constructive and provides a polynomial time algorithm to obtain a packing for two given nonstar trees.

Petr Hlineny and Marek Derňár

**Crossing Number is Hard for Kernelization**
[+]

The graph crossing number problem, cr(G)<=k, asks for a drawing of a graph G in the plane with at most k edge crossings. Although this problem is in general notoriously difficult, it is fixed-parameter tractable for the parameter k [Grohe, STOC 2001]. This suggests a closely related question of whether this problem has a polynomial kernel, meaning whether every instance of cr(G)<=k can be in polynomial time reduced to an equivalent instance of size polynomial in k (and independent of |G|). We answer this question in the negative. Along the proof we show that the tile crossing number problem of twisted planar tiles is NP-hard, which has been an open problem for some time, too, and then employ the complexity technique of cross-composition. Our result holds already for the special case of graphs obtained from planar graphs by adding one edge.

Alfredo Hubard, Vojtěch Kaluža, Arnaud de Mesmay and Martin Tancer

**Shortest path embeddings of graphs on surfaces**
[+]

The classical theorem of Fáry states that every planar graph can be represented by an embedding in which every edge is represented by a straight line segment. We consider generalizations of Fáry's theorem to surfaces equipped with Riemannian metrics. In this setting, we require that every edge is drawn as a shortest path between its two endpoints and we call an embedding with this property a shortest path embedding. The main question addressed in this paper is whether given a closed surface S, there exists a Riemannian metric for which every topologically embeddable graph admits a shortest path embedding. This question is also motivated by various problems regarding crossing numbers on surfaces.
We observe that the round metrics on the sphere and the projective plane have this property. We provide flat metrics on the torus and the Klein bottle which also have this property.
Then we show that for the unit square flat metric on the Klein bottle there exists a graph without shortest path embeddings. We show, moreover, that for large g, there exist graphs G embeddable into the orientable surface of genus g, such that with large probability a random hyperbolic metric does not admit a shortest path embedding of G, where the probability measure is proportional to the Weil-Petersson volume on moduli space.
Finally, we construct a hyperbolic metric on every orientable surface S of genus g, such that every graph embeddable into S can be embedded so that every edge is a concatenation of at most O(g) shortest paths.

Piotr Indyk, Robert Kleinberg, Sepideh Mahabadi and Yang Yuan

**Simultaneous Nearest Neighbor Search**
[+]

Motivated by applications in computer vision and databases, we introduce and study the Simultaneous Nearest Neighbor Search (SNN) problem. Given a set of data points, the goal of SNN is to design a data structure that, given a *collection* of queries, finds a *collection* of close points that are *compatible* with each other. Formally, we are given k query points Q=q_1,...,q_k, and a compatibility graph G with vertices in Q, and the goal is to return data points p_1,...,p_k that minimize (i) the weighted sum of the distances from q_i to p_i and (ii) the weighted sum, over all edges (i,j) in the compatibility graph G, of the distances between p_i and p_j. The problem has several applications in computer vision and databases, where one wants to return a set of *consistent* answers to multiple related queries. Furthermore, it generalizes several well-studied computational problems, including Nearest Neighbor Search, Aggregate Nearest Neighbor Search and the 0-extension problem.
In this paper we propose and analyze the following general two-step method for designing efficient data structures for SNN. In the first step, for each query point q_i we find its (approximate) nearest neighbor point p'_i; this can be done efficiently using existing approximate nearest neighbor structures. In the second step, we solve an off-line optimization problem over sets q_1,...,q_k and p'_1,...,p'_k; this can be done efficiently given that k is much smaller than n. Even though p'_1,...,p'_k might not constitute the optimal answers to queries q_1,...,q_k, we show that, for the unweighted case, the resulting algorithm satisfies a O(log k/log log k)-approximation guarantee. Furthermore, we show that the approximation factor can be in fact reduced to a constant for compatibility graphs frequently occurring in practice, e.g., 2D grids, 3D grids or planar graphs.
Finally, we validate our theoretical results by preliminary experiments. In particular, we show that the *empirical approximation factor* provided by the above approach is very close to 1.

Iyad Kanj, Ljubomir Perkovic and Duru Turkoglu

**Degree Four Plane Spanners: Simpler and Better**
[+]

Let P be a set of n points embedded in the plane, and let
C be the complete Euclidean graph whose point-set is P.
Each edge in C between two points p, q is realized as the line
segment [pq], and is assigned a weight equal to the Euclidean
distance |pq|. In this paper, we show how to construct in
O(nlg{n}) time a plane spanner of C of maximum degree at
most 4 and of stretch factor at most 20. This improves a long
sequence of results on the construction of bounded degree plane spanners of C. Our result matches the smallest known upper bound of 4 by Bonichon et al. on the maximum degree while
significantly improving their stretch factor upper bound from 156.82 to 20. The construction of our spanner is based on Delaunay triangulations defined with respect to the equilateral-triangle distance, and uses a different approach than that used by Bonichon et al. Our approach leads to a simple and intuitive construction of a well-structured spanner, and reveals useful structural properties of the Delaunay triangulations defined with respect to the equilateral-triangle distance.

Akitoshi Kawamura, Sonoko Moriyama, Yota Otachi and Janos Pach

**A lower bound on opaque sets**
[+]

It is proved that the total length of any set of countably many rectifiable curves, whose union meets all straight lines that intersect the unit square U, is at least 2.00002. This is the first improvement on the lower bound of 2 by Jones in 1964. A similar bound is proved for all convex sets U other than a triangle.

Marc Khoury and Jonathan Richard Shewchuk

**Fixed Points of the Restricted Delaunay Triangulation Operator**
[+]

The restricted Delaunay triangulation can be conceived as an operator that takes as input a k-manifold (typically smooth) embedded in R^d and a set of points sampled with sufficient density on that manifold, and produces as output a k-dimensional triangulation of the manifold, the input points serving as its vertices. What happens if we feed that triangulation back into the operator, replacing the original manifold, while retaining the same set of input points? If k = 2 and the sample points are sufficiently dense, we obtain another triangulation of the manifold. Iterating this process, we soon reach an iteration for which the input and output triangulations are the same. We call this triangulation a fixed point of the restricted Delaunay triangulation operator.
With this observation, and a new test for distinguishing ``critical points'' near the manifold from those near its medial axis, we develop a provably good surface reconstruction algorithm for R^3 with unusually modest sampling requirements. We develop a similar algorithm for constructing a simplicial complex that models a 2-manifold embedded in a high-dimensional space R^d, also with modest sampling requirements (especially compared to algorithms that depend on sliver exudation). The latter algorithm builds a non-manifold representation similar to the flow complex, but made solely of Delaunay simplices. The algorithm avoids the curse of dimensionality: its running time is polynomial, not exponential, in d.

Heuna Kim and Günter Rote

**Congruence Testing of Point Sets in 4-space**
[+]

We give a deterministic O(n log n)-time algorithm to decide if two n-point sets in 4-dimensional Euclidean space are the same up to rotations and translations.
It has been conjectured that O(n log n) algorithms should exist for any fixed dimension. The best algorithms in d-space so far are a deterministic algorithm by Brass and Knauer [Int. J. Comput. Geom. Appl., 2000] and a randomized Monte Carlo algorithm by Akutsu [Comp. Geom., 1998]. They take time O(n^2 log n) and O(n^(3/2) log n) respectively in 4-space.
Our algorithm exploits many geometric structures and properties of 4-dimensional space.

Irina Kostitsyna, Maarten Löffler, Valentin Polishchuk and Frank Staals

**On the complexity of minimum-link path problems**
[+]

We revisit the minimum-link path problem: Given a polyhedral domain and two points in it, connect the points by a polygonal path with minimum number of edges. We consider settings where the min-link path's vertices or edges can be restricted to lie on the boundary of the domain, or can be in its interior. Our results include bit complexity bounds, a novel general hardness construction, and a polynomial-time approximation scheme. We fully characterize the situation in 2D, and provide first results in dimensions 3 and higher for several versions of the problem.
Concretely, our results resolve several open problems. We prove that computing the minimum-link diffuse reflection path, motivated by ray tracing in computer graphics, is NP-hard, even for two-dimensional polygonal domains with holes. This has remained an open problem [Ghosh et al. 2012] despite a large body of work on the topic. We also resolve the open problem from [Mitchell et al. 1992] mentioned in the handbook [Goodman and O'Rourke, 2004] (see Chapter 27.5, Open problem 3) and The Open Problems Project [Demaine et al. TOPP] (see Problem 22): "What is the complexity of the minimum-link path problem in 3-space?" Our results imply that the problem is NP-hard even on terrains (and hence, due to discreteness of the answer, there is no FPTAS unless P=NP), but admits a PTAS.

Stefan Langerman and Andrew Winslow

**A Quasilinear-Time Algorithm for Tiling the Plane Isohedrally with a Polyomino**
[+]

A plane tiling consisting of congruent copies of a shape is isohedral provided that for any pair of copies, there exists a symmetry of the tiling mapping one copy to the other.
We give a O(n*log^2(n))-time algorithm for deciding if a polyomino with n edges can tile the plane isohedrally. This improves on the O(n^{18})-time algorithm of Keating and Vince and generalizes recent work by Brlek, Provençal, Fédou, and the second author.

Isaac Mabillard and Uli Wagner

**Eliminating Higher-Multiplicity Intersections, II. The Deleted Product Criterion in the r-Metastable Range**
[+]

Motivated by Tverberg-type problems in topological combinatorics and by classical results about embeddings (maps without double points), we study the question whether a finite simplicial complex K can be mapped into R^d without higher-multiplicity intersections.
We focus on conditions for the existence of almost r-embeddings, i.e., maps f: K -> R^d such that the intersection of f(sigma_1), ..., f(sigma_r) is empty whenever sigma_1,...,sigma_r are pairwise disjoint simplices of K.
Generalizing the classical Haefliger-Weber embeddability criterion, we show that a well-known necessary deleted product condition for the existence of almost r-embeddings is sufficient in a suitable r-metastable range of dimensions: If r d > (r+1) dim K + 2 then there exists an almost r-embedding K-> R^d if and only if there exists an equivariant map of the r-fold deleted product of K to the sphere S^(d(r-1)-1).
This significantly extends one of the main results of our previous paper (which treated the special case where d=rk and dim K=(r-1)k, for some k> 2), and settles an open question raised there.

Dániel Marx and Tillmann Miltzow

**Peeling and Nibbling the Cactus: Subexponential-Time Algorithms for Counting Triangulations and Related Problems**
[+]

Given a set of n points S in the plane, a triangulation T of
S is a maximal set of non-crossing segments with endpoints in S.
We present an algorithm that computes the number of triangulations
on a given set of n points in time n^{ (11+ o(1)) sqrt{n} }, significantly improving the previous best running time of
O(2^n n^2) by Alvarez and Seidel [SoCG 2013]. Our main tool is
identifying separators of size O(sqrt{n}) of a triangulation in a
canonical way. The definition of the separators are based on the
decomposition of the triangulation into nested layers (``cactus
graphs''). Based on the above algorithm, we develop a simple and
formal framework to count other non-crossing straight-line graphs in
n^{O(sqrt{n})} time. We demonstrate the usefulness of the framework by applying it to counting non-crossing Hamilton cycles, spanning trees, perfect matchings, $3$-colorable triangulations, connected graphs, cycle decompositions, quadrangulations, $3$-regular graphs, and more.

Elizabeth Munch and Bei Wang

**Convergence between Categorical Representations of Reeb Space and Mapper**
[+]

The Reeb space, which generalizes the notion of a Reeb graph, is one of the few tools in topological data analysis and visualization suitable for the study of multivariate scientific datasets.
First introduced by Edelsbrunner et al., it compresses the components of the level sets of a multivariate mapping and obtains a summary representation of their relationships.
A related construction called mapper, and a special case of the mapper construction called the Joint Contour Net have been shown to be effective in visual analytics.
Mapper and JCN are intuitively regarded as discrete approximations of the Reeb space, however without formal proofs or approximation guarantees.
An open question has been proposed by Dey et al. as to whether the mapper construction converges to the Reeb space in the limit.
In this paper, we are interested in developing the theoretical understanding of the relationship between the Reeb space and its discrete approximations to support its use in practical data analysis. Using tools from category theory, we formally prove the convergence between the Reeb space and mapper in terms of an interleaving distance between their categorical representations.
Given a sequence of refined discretizations, we prove that these approximations converge to the Reeb space in the interleaving distance; this also helps to quantify the approximation quality of the discretization at a fixed resolution.

Nabil Mustafa, Andrey Kupavskiy and Janos Pach

**New Lower Bounds for epsilon-nets**
[+]

Following groundbreaking work by Haussler and Welzl (1987), the use of small epsilon-nets has become a standard technique for solving algorithmic and extremal problems in geometry and learning theory. Two significant recent developments are: (i) an upper bound on the size of the smallest epsilon-nets for set systems, as a function of their so-called shallow-cell complexity (Chan, Grant, Konemann, and Sharpe); and (ii) the construction of a set system whose members can be obtained by intersecting a point set in R^4 by a family of half-spaces such that the size of any epsilon-net for them is at least (1/(9*epsilon)) log (1/epsilon) (Pach and Tardos).
The present paper completes both of these avenues of research. We (i) give a lower bound, matching the result of Chan et al., and (ii) generalize the construction of Pach and Tardos to half-spaces in R^d, for any d >= 4, to show that the general upper bound of Haussler and Welzl for the size of the smallest epsilon-nets is tight.

Amir Nayyeri and Hanzhong Xu

**On computing the Frechet distance between surfaces**
[+]

We describe two (1+epsilon)-approximation algorithms for computing the Frechet distance between two homeomorphic piecewise linear surfaces R and S of genus zero and total complexity n, with Frechet distance delta.
(1) A 2^{O((n + ( (Area(R)+Area(S))/(epsilon.delta)^2 )^2 )}$ time algorithm if R and S are composed of fat triangles (triangles with angles larger than a constant).
(2) An O(D/(epsilon.delta)^2) n + 2^{O(D^4/(epsilon^4.delta^2))} time algorithm if R and S are polyhedral terrains over [0,1]^2 with slope at most D.
Although, the Frechet distance between curves has been studied extensively, very little is known for surfaces. Our results are the first algorithms (both for surfaces and terrains) that are guaranteed to terminate in finite time. Our latter result, in particular, implies a linear time algorithm for terrains of constant maximum slope and constant Frechet distance.

Eunjin Oh, Luis Barba and Hee-Kap Ahn

**The farthest-point geodesic Voronoi diagram of points on the boundary of a simple polygon**
[+]

Given a set of sites (points) in a simple polygon, the farthest-point geodesic Voronoi diagram partitions the polygon into cells, at most one cell per site, such that every point in a cell has the same farthest site with respect to the geodesic metric. We present an O((n+m)loglogn)-time algorithm to compute the farthest-point geodesic Voronoi diagram for m sites lying on the boundary of a simple n-gon.

Benjamin Raichel and C. Seshadhri

**Avoiding the Global Sort: A Faster Contour Tree Algorithm**
[+]

We revisit the classical problem of computing the contour tree of a scalar field f:M to R, where M is a triangulated simplicial mesh in R^d. The contour tree is a fundamental topological structure that tracks the evolution of level sets of f and has numerous applications in data analysis and visualization.
All existing algorithms begin with a global sort of at least all critical values of f, which can require (roughly) Omega(n log n) time. Existing lower bounds show that there are pathological instances where this sort is required. We present the first algorithm whose time complexity depends on the contour tree structure, and avoids the global sort for non-pathological inputs. If C denotes the set of critical points in M, the running time is roughly O(sum_{v in C} log l_v), where l_v is the depth of v in the contour tree. This matches all existing upper bounds, but is a significant asymptotic improvement when the contour tree is short and fat. Specifically, our approach ensures that any comparison made is between nodes that are either adjacent in M or in the same descending path in the contour tree, allowing us to argue strong optimality properties of our algorithm.
Our algorithm requires several novel ideas: partitioning M in well-behaved portions, a local growing procedure to iteratively build contour trees, and the use of heavy path decompositions for the time complexity analysis.

Orit E. Raz

**Configurations of lines in 3-space and rigidity of planar structures**
[+]

Let L be a sequence (l_1,l_2,...,l_n) of n lines in C^3. We define the *intersection graph* G_L=([n],E) of L, where [n]:={1,..., n}, and with {i,j} in E if and only if ineq j and the corresponding lines l_i and l_j intersect, or are parallel (or coincide). For a graph G=([n],E), we say that a sequence L is a *realization* of G if Gsubset G_L. One of the main results of this paper is to provide a combinatorial characterization of graphs G=([n],E) that have the following property:
For every *generic* realization L of G, that consists of $n$ pairwise distinct lines, we have G_L=K_n,
in which case the lines of L are either all concurrent or all coplanar.
The general statements that we obtain about lines, apart from their independent interest, turns out to be closely related to the notion of graph rigidity. The connection is established due to the so-called Elekes--Sharir framework, which allows us to transform the problem into an incidence problem involving lines in three dimensions. By exploiting the geometry of contacts between lines in 3D, we can obtain alternative, simpler, and more precise characterizations of the rigidity of graphs.

Alexandre Rok and Shakhar Smorodinsky

**Weak 1/r-nets for moving points**
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In this paper, we extend the weak 1/r-net theorem to a kinetic setting where the underlying set of points is moving polynomially with bounded description complexity. We establish that
one can find a kinetic analog N of a weak 1/r-net of cardinality O(r^(d(d+1)/2)log^d r) whose points are moving with coordinates that are rational functions with bounded description complexity. Moreover, each member of N has one polynomial coordinate.

Ingo van Duijn, Edvin Berglin, Peyman Afshani and Jesper Sindahl Nielsen

**Applications of incidence bounds in point covering problems**
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In the Line Cover problem a set of n points is given and the task is to cover the points using either the minimum number of lines or at most k lines. In Curve Cover, a generalization of Line Cover, the task is to cover the points using curves with d degrees of freedom. Another generalization is the Hyperplane Cover problem where points in d-dimensional space are to be covered by hyperplanes. All these problems have kernels of polynomial size, where the parameter is the minimum number of lines, curves, or hyperplanes needed.
First we give a non-parameterized algorithm for both problems in O*(2^n) (where the O*(.) notation hides polynomial factors of n) time and polynomial space, beating a previous exponential-space result. Combining this with incidence bounds similar to the famous Szemeredi-Trotter bound, we present a Curve Cover algorithm with running time O*((Ck/log k)^((d-1)k)), where C is some constant. Our result improves the previous best times O*((k/1.35)^k) for Line Cover (where d=2), O*(k^(dk)) for general Curve Cover, as well as a few other bounds for covering points by parabolas or conics. We also present an algorithm for Hyperplane Cover in R^3 with running time O*((Ck^2/log^(1/5) k)^k), improving on the previous time of O*((k^2/1.3)^k).

Arthur van Goethem, Marc Van Kreveld, Maarten Löffler, Bettina Speckmann and Frank Staals

**Grouping Time-varying Data for Interactive Exploration**
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We present algorithms and data structures that support the interactive analysis
of the grouping structure of one-, two-, or higher-dimensional time-varying
data while varying all defining parameters. Grouping structures characterise
important patterns in the temporal evaluation of sets of time-varying data. We
follow Buchin et al. [JoCG 2015] who define groups using three
parameters: group-size, group-duration, and inter-entity distance. We give
upper and lower bounds on the number of maximal groups over all parameter
values, and show how to compute them efficiently. Furthermore, we describe
data structures that can report changes in the set of maximal groups in an
output-sensitive manner. Our results hold in R^d for fixed d.

Jie Xue, Yuan Li and Ravi Janardan

**On the separability of stochastic geometric objects, with applications**
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In this paper, we study the linear separability problem for stochastic geometric objects under the well-known unipoint/multipoint uncertainty models. Let S=S_R U S_B be a given set of stochastic bichromatic points, and define n = min{|S_R|, |S_B|} and N = max{|S_R|, |S_B|}. We show that the separable-probability (SP) of S can be computed in O(nN^{d-1}) time for d >= 3 and O(min{nN log N, N^2}) time for d=2, while the expected separation-margin (ESM) of S can be computed in O(nN^d) time for d >= 2. In addition, we give an Omega(nN^{d-1}) witness-based lower bound for computing SP, which implies the optimality of our algorithm among all those in this category. Also, a hardness result for computing ESM is given to show the difficulty of further improving our algorithm. As an extension, we generalize the same problems from points to general geometric objects, i.e., polytopes and/or balls, and extend our algorithms to solve the generalized SP and ESM problems in O(nN^d) and O(nN^{d+1}) time, respectively. Finally, we present some applications of our algorithms to stochastic convex-hull related problems.

Juyoung Yon, Sang Won Bae, Siu-Wing Cheng, Otfried Cheong and Bryan Wilkinson

**Approximating Convex Shapes with respect to Symmetric Difference under Homotheties**
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The symmetric difference is a robust operator for measuring the
error of approximating one shape by another. Given two convex
shapes P and C, we study the problem of minimizing the volume of
their symmetric difference under all possible scalings and
translations of C. We prove that the problem can be solved by
convex programming. We also present a combinatorial algorithm
for convex polygons in the plane that runs in O((m+n) log^3(m+n))
expected time, where n and m denote the number of vertices of P
and C, respectively.