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Shakhar Smorodinsky

Publications and source records attributed to Shakhar Smorodinsky.

At least 37 records · Page 2Linked to original sources
arXiv2018

A new lower bound on Hadwiger-Debrunner numbers in the plane

A family of sets $F$ is said to satisfy the $(p,q)$ property if among any $p$ sets in $F$, some $q$ have a non-empty intersection. Hadwiger and Debrunner (1957) conjectured that for any $p \geq q \geq d+1$ there exists $c=c_d(p,q)$, such that any family of compact convex sets in $\mathbb{R}^d$ that satisfies the $(p,q)$ property, can be pierced by at most $c$ points. In a celebrated result from 1992, Alon and Kleitman proved the conjecture. However, obtaining sharp bounds on $c_d(p,q)$, called `the Hadwiger-Debrunner numbers', is still a major open problem in discrete and computational geometry. The best currently known lower bound on the Hadwiger-Debrunner numbers in the plane is $c_2(p,q) = Ω( \frac{p}{q}\log(\frac{p}{q}))$ while the best known upper bound is $O(p^{(1.5+δ)(1+\frac{1}{q-2})})$. In this paper we improve the lower bound significantly by showing that $c_2(p,q) \geq p^{1+Ω(1/q)}$. Furthermore, the bound is obtained by a family of lines, and is tight for all families that have a bounded VC-dimension. Unlike previous bounds on the Hadwiger-Debrunner numbers which mainly used the weak epsilon-net theorem, our bound stems from a surprising connection of the $(p,q)$ problem to an old problem of Erdős on points in general position in the plane. We use a novel construction for the Erdős' problem, obtained recently by Balogh and Solymosi using the hypergraph container method, to get the lower bound on $c_2(p,3)$. We then generalize the bound to $c_2(p,q)$ for any $q \geq 3$.

math.CO↗
arXiv2017

From a $(p,2)$-Theorem to a Tight $(p,q)$-Theorem

A family $F$ of sets is said to satisfy the $(p,q)$-property if among any $p$ sets of $F$ some $q$ intersect. The celebrated $(p,q)$-theorem of Alon and Kleitman asserts that any family of compact convex sets in $\mathbb{R}^d$ that satisfies the $(p,q)$-property for some $q \geq d+1$, can be pierced by a fixed number $f_d(p,q)$ of points. The minimum such piercing number is denoted by $HD_d(p,q)$. Already in 1957, Hadwiger and Debrunner showed that whenever $q>\frac{d-1}{d}p+1$ the piercing number is $HD_d(p,q)=p-q+1$; no exact values of $HD_d(p,q)$ were found ever since. While for an arbitrary family of compact convex sets in $\mathbb{R}^d$, $d \geq 2$, a $(p,2)$-property does not imply a bounded piercing number, such bounds were proved for numerous specific families. The best-studied among them is axis-parallel rectangles in the plane. Wegner and (independently) Dol'nikov used a $(p,2)$-theorem for axis-parallel rectangles to show that $HD_{\mathrm{rect}}(p,q)=p-q+1$ holds for all $q>\sqrt{2p}$. These are the only values of $q$ for which $HD_{\mathrm{rect}}(p,q)$ is known exactly. In this paper we present a general method which allows using a $(p,2)$-theorem as a bootstrapping to obtain a tight $(p,q)$-theorem, for families with Helly number 2, even without assuming that the sets in the family are convex or compact. To demonstrate the strength of this method, we obtain a significant improvement of an over 50 year old result by Wegner and Dol'nikov. Namely, we show that $HD_{\mathrm{d-box}}(p,q)=p-q+1$ holds for all $q > c' \log^{d-1} p$, and in particular, $HD_{\mathrm{rect}}(p,q)=p-q+1$ holds for all $q \geq 7 \log_2 p$ (compared to $q \geq \sqrt{2p}$ of Wegner and Dol'nikov). In addition, for several classes of families, we present improved $(p,2)$-theorems, some of which can be used as a bootstrapping to obtain tight $(p,q)$-theorems.

math.CO↗
arXiv2017

Hypergraphical Clustering Games of Mis-Coordination

We introduce and motivate the study of hypergraphical clustering games of mis-coordination. For two specific variants we prove the existence of a pure Nash equilibrium and provide bounds on the price of anarchy as a function of the cardinality of the action set and the size of the hyperedges.

math.CO↗
arXiv2017

On interference among moving sensors and related problems

We show that for any set of $n$ points moving along "simple" trajectories (i.e., each coordinate is described with a polynomial of bounded degree) in $\Re^d$ and any parameter $2 \le k \le n$, one can select a fixed non-empty subset of the points of size $O(k \log k)$, such that the Voronoi diagram of this subset is "balanced" at any given time (i.e., it contains $O(n/k)$ points per cell). We also show that the bound $O(k \log k)$ is near optimal even for the one dimensional case in which points move linearly in time. As applications, we show that one can assign communication radii to the sensors of a network of $n$ moving sensors so that at any given time their interference is $O(\sqrt{n\log n})$. We also show some results in kinetic approximate range counting and kinetic discrepancy. In order to obtain these results, we extend well-known results from $\varepsilon$-net theory to kinetic environments.

cs.CG↗
arXiv2017

Bounding a global red-blue proportion using local conditions

We study the following local-to-global phenomenon: Let $B$ and $R$ be two finite sets of (blue and red) points in the Euclidean plane $\mathbb{R}^2$. Suppose that in each "neighborhood" of a red point, the number of blue points is at least as large as the number of red points. We show that in this case the total number of blue points is at least one fifth of the total number of red points. We also show that this bound is optimal and we generalize the result to arbitrary dimension and arbitrary norm using results from Minkowski arrangements.

cs.CG↗
arXiv2017

Conflict-Free Coloring of Intersection Graphs of Geometric Objects

In FOCS'2002, Even et al. introduced and studied the notion of conflict-free colorings of geometrically defined hypergraphs. They motivated it by frequency assignment problems in cellular networks. This notion has been extensively studied since then. A conflict-free coloring of a graph is a coloring of its vertices such that the neighborhood (pointed or closed) of each vertex contains a vertex whose color differs from the colors of all other vertices in that neighborhood. In this paper we study conflict-colorings of intersection graphs of geometric objects. We show that any intersection graph of n pseudo-discs in the plane admits a conflict-free coloring with O(\log n) colors, with respect to both closed and pointed neighborhoods. We also show that the latter bound is asymptotically sharp. Using our methods, we also obtain a strengthening of the two main results of Even et al. which we believe is of independent interest. In particular, in view of the original motivation to study such colorings, this strengthening suggests further applications to frequency assignment in wireless networks. Finally, we present bounds on the number of colors needed for conflict-free colorings of other classes of intersection graphs, including intersection graphs of axis-parallel rectangles and of ρ-fat objects in the plane.

math.CO↗
arXiv2017

On Piercing Numbers of Families Satisfying the $(p,q)_r$ Property

The Hadwiger-Debrunner number $HD_d(p,q)$ is the minimal size of a piercing set that can always be guaranteed for a family of compact convex sets in $\mathbb{R}^d$ that satisfies the $(p,q)$ property. Hadwiger and Debrunner showed that $HD_d(p,q) \geq p-q+1$ for all $q$, and equality is attained for $q > \frac{d-1}{d}p +1$. Almost tight upper bounds for $HD_d(p,q)$ for a `sufficiently large' $q$ were obtained recently using an enhancement of the celebrated Alon-Kleitman theorem, but no sharp upper bounds for a general $q$ are known. In [L. Montejano and P. Soberón, Piercing numbers for balanced and unbalanced families, Disc. Comput. Geom., 45(2) (2011), pp. 358-364], Montejano and Soberón defined a refinement of the $(p,q)$ property: $F$ satisfies the $(p,q)_r$ property if among any $p$ elements of $F$, at least $r$ of the $q$-tuples intersect. They showed that $HD_d(p,q)_r \leq p-q+1$ holds for all $r>{{p}\choose{q}}-{{p+1-d}\choose{q+1-d}}$; however, this is far from being tight. In this paper we present improved asymptotic upper bounds on $HD_d(p,q)_r$ which hold when only a tiny portion of the $q$-tuples intersect. In particular, we show that for $p,q$ sufficiently large, $HD_d(p,q)_r \leq p-q+1$ holds with $r = \frac{1}{p^{\frac{q}{2d}}}{{p}\choose{q}}$. Our bound misses the known lower bound for the same piercing number by a factor of less than $pq^d$. Our results use Kalai's Upper Bound Theorem for convex sets, along with the Hadwiger-Debrunner theorem and the recent improved upper bound on $HD_d(p,q)$ mentioned above.

math.CO↗
arXiv2016

Improved bounds on the Hadwiger-Debrunner numbers

Let $HD_d(p,q)$ denote the minimal size of a transversal that can always be guaranteed for a family of compact convex sets in $\mathbb{R}^d$ which satisfy the $(p,q)$-property ($p \geq q \geq d+1$). In a celebrated proof of the Hadwiger-Debrunner conjecture, Alon and Kleitman proved that $HD_d(p,q)$ exists for all $p \geq q \geq d+1$. Specifically, they prove that $HD_d(p,d+1)$ is $\tilde{O}(p^{d^2+d})$. We present several improved bounds: (i) For any $q \geq d+1$, $HD_d(p,q) = \tilde{O}(p^{d \left(\frac{q-1}{q-d}\right)})$. (ii) For $q \geq \log p$, $HD_d(p,q) = \tilde{O}(p+(p/q)^d)$. (iii) For every $ε> 0$ there exists a $p_0 = p_0(ε)$ such that for every $p \geq p_0$ and for every $q \geq p^{\frac{d-1}{d}+ε}$ we have: $p-q+1 \leq HD_d(p,q) \leq p-q+2$. The latter is the first near tight estimate of $HD_d(p,q)$ for an extended range of values of $(p,q)$ since the 1957 Hadwiger-Debrunner theorem. We also prove a $(p,2)$-theorem for families in $\mathbb{R}^2$ with union complexity below a specific quadratic bound. Based on this, we introduce a polynomial time constant factor approximation algorithm for MAX-CLIQUE of intersection graphs of convex sets satisfying this property.

math.CO↗
arXiv2016

On Vertex Rankings of Graphs and its Relatives

A vertex ranking of a graph is an assignment of ranks (or colors) to the vertices of the graph, in such a way that any simple path connecting two vertices of equal rank, must contain a vertex of a higher rank. In this paper we study a relaxation of this notion, in which the requirement above should only hold for paths of some bounded length $l$ for some fixed $l$. For instance, already the case $l=2$ exhibit quite a different behavior than proper coloring. We prove upper and lower bounds on the minimum number of ranks required for several graph families, such as trees, planar graphs, graphs excluding a fixed minor and degenerate graphs.

math.CO↗
arXiv2015

Distinct distances between points and lines

We show that for $m$ points and $n$ lines in the real plane, the number of distinct distances between the points and the lines is $Ω(m^{1/5}n^{3/5})$, as long as $m^{1/2}\le n\le m^2$. We also prove that for any $m$ points in the plane, not all on a line, the number of distances between these points and the lines that they span is $Ω(m^{4/3})$. The problem of bounding the number of distinct point-line distances can be reduced to the problem of bounding the number of tangent pairs among a finite set of lines and a finite set of circles in the plane, and we believe that this latter question is of independent interest. In the same vein, we show that $n$ circles in the plane determine at most $O(n^{3/2})$ points where two or more circles are tangent, improving the previously best known bound of $O(n^{3/2}\log n)$. Finally, we study three-dimensional versions of the distinct point-line distances problem, namely, distinct point-line distances and distinct point-plane distances. The problems studied in this paper are all new, and the bounds that we derive for them, albeit most likely not tight, are non-trivial to prove. We hope that our work will motivate further studies of these and related problems.

math.MG↗
arXiv2015

A short proof of the first selection lemma and weak $\frac{1}{r}$-nets for moving points

(i) We provide a short and simple proof of the first selection lemma. (ii) We also prove a selection lemma of a new type in $\Re^d$. For example, when $d=2$ assuming $n$ is large enough we prove that for any set $P$ of $n$ points in general position there are $Ω(n^4)$ pairs of segments spanned by $P$ all of which intersect in some fixed triangle spanned by $P$. (iii) Finally, we extend the weak $\frac{1}{r}$-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 $\frac{1}{r}$-net of cardinality $O(r^{\frac{d(d+1)}{2}}\log^{d}r)$ whose points are moving with coordinates that are rational functions with bounded description complexity. Moreover, each member of $N$ has one polynomial coordinate.

cs.DM↗
arXiv2014

Epsilon-Nets for Halfspaces Revisited

Given a set $P$ of $n$ points in $\mathbb{R}^3$, we show that, for any $\varepsilon >0$, there exists an $\varepsilon$-net of $P$ for halfspace ranges, of size $O(1/\varepsilon)$. We give five proofs of this result, which are arguably simpler than previous proofs \cite{msw-hnlls-90, cv-iaags-07, pr-nepen-08}. We also consider several related variants of this result, including the case of points and pseudo-disks in the plane.

cs.CG↗
arXiv2013

On Totally Positive Matrices and Geometric Incidences

A matrix is called totally positive if every minor of it is positive. Such matrices are well studied and have numerous applications in Mathematics and Computer Science. We study how many times the value of a minor can repeat in a totally positive matrix and show interesting connections with incidence problems in combinatorial geometry. We prove that the maximum possible number of repeated $d \times d$-minors in a $d \times n$ totally-positive matrix is $O(n^{d-\frac{d}{d+1}})$. For the case $d=2$ we also show that our bound is optimal. We consider some special families of totally postive matrices to show non-trivial lower bounds on the number of repeated minors. In doing so, we arrive at a new interesting problem: How many unit-area and axis-parallel rectangles can be spanned by two points in a set of $n$ points in the plane? This problem seems to be interesting in its own right especially since it seem to have a flavor of additive combinatorics and relate to interesting incidence problems where considering only the topology of the curves involved is not enough. We prove an upper bound of $O(n^{\frac{4}{3}})$ and provide a lower bound of $n^{1+\frac{1}{O(\log\log n)}}$.

math.CO↗
arXiv2012

Hitting Sets Online and Unique-Max Coloring

We consider the problem of hitting sets online. The hypergraph (i.e., range-space consisting of points and ranges) is known in advance, and the ranges to be stabbed are input one-by-one in an online fashion. The online algorithm must stab each range upon arrival. An online algorithm may add points to the hitting set but may not remove already chosen points. The goal is to use the smallest number of points. The best known competitive ratio for hitting sets online by Alon et al. \cite{alon2009online} is $O(\log n \cdot \log m)$ for general hypergraphs, where $n$ and $m$ denote the number of points and the number of ranges, respectively. We consider hypergraphs in which the union of two intersecting ranges is also a range. Our main result for such hypergraphs is as follows. The competitive ratio of the online hitting set problem is at most the unique-max number and at least this number minus one.

cs.DS↗
arXiv2012

Conflict-free coloring with respect to a subset of intervals

Given a hypergraph H = (V, E), a coloring of its vertices is said to be conflict-free if for every hyperedge S \in E there is at least one vertex in S whose color is distinct from the colors of all other vertices in S. The discrete interval hypergraph Hn is the hypergraph with vertex set {1,...,n} and hyperedge set the family of all subsets of consecutive integers in {1,...,n}. We provide a polynomial time algorithm for conflict-free coloring any subhypergraph of Hn, we show that the algorithm has approximation ratio 2, and we prove that our analysis is tight, i.e., there is a subhypergraph for which the algorithm computes a solution which uses twice the number of colors of the optimal solution. We also show that the problem of deciding whether a given subhypergraph of Hn can be colored with at most k colors has a quasipolynomial time algorithm.

math.CO↗
arXiv2012

Conflict-Free Coloring and its Applications

Let $H=(V,E)$ be a hypergraph. A {\em conflict-free} coloring of $H$ is an assignment of colors to $V$ such that in each hyperedge $e \in E$ there is at least one uniquely-colored vertex. This notion is an extension of the classical graph coloring. Such colorings arise in the context of frequency assignment to cellular antennae, in battery consumption aspects of sensor networks, in RFID protocols and several other fields, and has been the focus of many recent research papers. In this paper, we survey this notion and its combinatorial and algorithmic aspects.

math.CO↗
arXiv2010

The potential to improve the choice: list conflict-free coloring for geometric hypergraphs

Given a geometric hypergraph (or a range-space) $H=(V,\cal E)$, a coloring of its vertices is said to be conflict-free if for every hyperedge $S \in \cal E$ there is at least one vertex in $S$ whose color is distinct from the colors of all other vertices in $S$. The study of this notion is motivated by frequency assignment problems in wireless networks. We study the list-coloring (or choice) version of this notion. In this version, each vertex is associated with a set of (admissible) colors and it is allowed to be colored only with colors from its set. List coloring arises naturally in the context of wireless networks. Our main result is a list coloring algorithm based on a new potential method. The algorithm produces a stronger unique-maximum coloring, in which colors are positive integers and the maximum color in every hyperedge occurs uniquely. As a corollary, we provide asymptotically sharp bounds on the size of the lists required to assure the existence of such unique-maximum colorings for many geometric hypergraphs (e.g., discs or pseudo-discs in the plane or points with respect to discs). Moreover, we provide an algorithm, such that, given a family of lists with the appropriate sizes, computes such a coloring from these lists.

math.CO↗