Dániel Gerbner, Balázs Keszegh, Kartal Nagy, Balázs Patkós, Gábor Wiener
In the game theoretical approach of the basic problem in Combinatorial Search an adversary thinks of a defective element $d$ of an $n$-element pool $X$, and the questioner needs to find $x$ by asking questions of type is $d\in Q$? for certain subsets $Q$ of $X$. We study cooperative versions of this problem, where there are multiple questioners, but not all of them learn the answer to the queries. We consider various models that differ in how it is decided who gets to ask the next query, who obtains the answer to the query, and who needs to know the defective element by the end of the process.
Dániel Gerbner, Balázs Keszegh, Nathan Lemons, Dömötör Pálvölgyi, Cory Palmer, Balázs Patkós
A family $\cF \subseteq 2^{[n]}$ saturates the monotone decreasing property $\cP$ if $\cF$ satisfies $\cP$ and one cannot add any set to $\cF$ such that property $\cP$ is still satisfied by the resulting family. We address the problem of finding the minimum size of a family saturating the $k$-Sperner property and the minimum size of a family that saturates the Sperner property and that consists only of $l$-sets and $(l+1)$-sets.
Balázs Keszegh, Dömötör Pálvölgyi
We prove that the number of tangencies between the members of two families, each of which consists of $n$ pairwise disjoint curves, can be as large as $Ω(n^{4/3})$. We show that from a conjecture about forbidden $0$-$1$ matrices it would follow that this bound is sharp for doubly-grounded families. We also show that if the curves are required to be $x$-monotone, then the maximum number of tangencies is $Θ(n\log n)$, which improves a result by Pach, Suk, and Treml. Finally, we also improve the best known bound on the number of tangencies between the members of a family of at most $t$-intersecting curves.
Rinat Ben-Avraham, Matthias Henze, Rafel Jaume, Balázs Keszegh, Orit E. Raz, Micha Sharir, Igor Tubis
We consider the RMS distance (sum of squared distances between pairs of points) under translation between two point sets in the plane, in two different setups. In the partial-matching setup, each point in the smaller set is matched to a distinct point in the bigger set. Although the problem is not known to be polynomial, we establish several structural properties of the underlying subdivision of the plane and derive improved bounds on its complexity. These results lead to the best known algorithm for finding a translation for which the partial-matching RMS distance between the point sets is minimized. In addition, we show how to compute a local minimum of the partial-matching RMS distance under translation, in polynomial time. In the Hausdorff setup, each point is paired to its nearest neighbor in the other set. We develop algorithms for finding a local minimum of the Hausdorff RMS distance in nearly linear time on the line, and in nearly quadratic time in the plane. These improve substantially the worst-case behavior of the popular ICP heuristics for solving this problem.
Balázs Keszegh, Balázs Patkós, Xuding Zhu
A coloring $c$ of the vertices of a graph $G$ is nonrepetitive if there exists no path $v_1v_2\ldots v_{2l}$ for which $c(v_i)=c(v_{l+i})$ for all $1\le i\le l$. Given graphs $G$ and $H$ with $|V(H)|=k$, the lexicographic product $G[H]$ is the graph obtained by substituting every vertex of $G$ by a copy of $H$, and every edge of $G$ by a copy of $K_{k,k}$. %Our main results are the following. We prove that for a sufficiently long path $P$, a nonrepetitive coloring of $P[K_k]$ needs at least $3k+\lfloor k/2\rfloor$ colors. If $k>2$ then we need exactly $2k+1$ colors to nonrepetitively color $P[E_k]$, where $E_k$ is the empty graph on $k$ vertices. If we further require that every copy of $E_k$ be rainbow-colored and the path $P$ is sufficiently long, then the smallest number of colors needed for $P[E_k]$ is at least $3k+1$ and at most $3k+\lceil k/2\rceil$. Finally, we define fractional nonrepetitive colorings of graphs and consider the connections between this notion and the above results.
Dániel Gerbner, Balázs Keszegh, Dániel T. Nagy, Kartal Nagy, Dömötör Pálvölgyi, Balázs Patkós, Gábor Wiener
We study the query complexity on slices of Boolean functions. Among other results we show that there exists a Boolean function for which we need to query all but 7 input bits to compute its value, even if we know beforehand that the number of 0's and 1's in the input are the same, i.e., when our input is from the middle slice. This answers a question of Byramji. Our proof is non-constructive, but we also propose a concrete candidate function that might have the above property. Our results are related to certain natural discrepancy type questions that, somewhat surprisingly, have not been studied before.
Eyal Ackerman, Balázs Keszegh
We prove a quasi-linear upper bound on the size of $K_{t,t}$-free polygon visibility graphs. For visibility graphs of star-shaped and monotone polygons we show a linear bound. In the more general setting of $n$ points on a simple closed curve and visibility pseudo-segments, we provide an $O(n \log n)$ upper bound and an $Ω(nα(n))$ lower bound.
Vladimir Bošković, Balázs Keszegh
Recently, the saturation problem of $0$-$1$ matrices gained a lot of attention. This problem can be regarded as a saturation problem of ordered bipartite graphs. Motivated by this, we initiate the study of the saturation problem of ordered and cyclically ordered graphs. We prove that dichotomy holds also in these two cases, i.e., for a (cyclically) ordered graph its saturation function is either bounded or linear. We also determine the order of magnitude for large classes of (cyclically) ordered graphs, giving infinite many examples exhibiting both possible behaviours, answering a problem of Pálvölgyi. In particular, in the ordered case we define a natural subclass of ordered matchings, the class of linked matchings, and we start their systematic study, concentrating on linked matchings with at most three links and prove that many of them have bounded saturation function. In both the ordered and cyclically ordered case we also consider the semisaturation problem, where dichotomy holds as well and we can even fully characterize the graphs that have bounded semisaturation function.
Dániel Gerbner, Balázs Keszegh, Dániel Lenger, Dániel T. Nagy, Dömötör Pálvölgyi, Balázs Patkós, Máté Vizer, Gábor Wiener
We study $\mathrm{exa}_k(n,F)$, the largest number of edges in an $n$-vertex graph $G$ that contains exactly $k$ copies of a given subgraph $F$. The case $k=0$ is the Turán number $\mathrm{ex}(n,F)$ that is among the most studied parameters in extremal graph theory. We show that for any $F$ and $k$, $\mathrm{exa}_k(n,F)=(1+o(1))\mathrm{ex}(n,F))$ and determine the exact values of $\mathrm{exa}_k(n,K_3)$ and $\mathrm{exa}_1(n,K_r)$ for $n$ large enough. We also explore a connection to the following well-known problem in search theory. We are given a graph of order $n$ that consists of an unknown copy of $F$ and some isolated vertices. We can ask pairs of vertices as queries, and the answer tells us whether there is an edge between those vertices. Our goal is to describe the graph using as few queries as possible. Aigner and Triesch in 1990 showed that the number of queries needed is at least $\binom{n}{2}-\mathrm{exa}_1(n,F)$. Among other results we show that the number of queries that were answered NO is at least $\binom{n}{2}-\mathrm{exa}_1(n,F)$.
Dániel Gerbner, Balázs Keszegh, Abhishek Methuku, Dániel T. Nagy, Balázs Patkós, Casey Tompkins, Chuanqi Xiao
We are given a set $A$ of buyers, a set $B$ of houses, and for each buyer a preference list, i.e., an ordering of the houses. A house allocation is an injective mapping $τ$ from $A$ to $B$, and $τ$ is strictly better than another house allocation $τ'\neq τ$ if for every buyer $i$, $τ'(i)$ does not come before $τ(i)$ in the preference list of $i$. A house allocation is Pareto optimal if there is no strictly better house allocation. Let $s(τ)$ be the image of $τ$ (i.e., the set of houses sold in the house allocation $τ$). We are interested in the largest possible cardinality $f(m)$ of the family of sets $s(τ)$ for Pareto optimal mappings $τ$ taken over all sets of preference lists of $m$ buyers. We improve the earlier upper bound on $f(m)$ given by Asinowski, Keszegh and Miltzow by making a connection between this problem and some problems in extremal set theory.
Chaya Keller, Balázs Keszegh, Dömötör Pálvölgyi
Consider the hypergraph whose vertex set is a family of $n$ lines in general position in the plane, and whose hyperedges are induced by intersections with a family of pseudo-discs. We prove that the number of $t$-hyperedges is bounded by $O_t(n^2)$ and that the total number of hyperedges is bounded by $O(n^3)$. Both bounds are tight.
Eyal Ackerman, Balázs Keszegh, Dömötör Pálvölgyi
What is the minimum number of colors that always suffice to color every planar set of points such that any disk that contains enough points contains two points of different colors? It is known that the answer to this question is either three or four. We show that three colors always suffice if the condition must be satisfied only by disks that contain a fixed point. Our result also holds, and is even tight, when instead of disks we consider their topological generalization, namely pseudo-disks, with a non-empty intersection. Our solution uses the equivalence that a hypergraph can be realized by stabbed pseudo-disks if and only if it is ABAB-free. These hypergraphs are defined in a purely abstract, combinatorial way and our proof that they are 3-chromatic is also combinatorial.
Balázs Keszegh, Dömötör Pálvölgyi
In this note we improve our upper bound given earlier by showing that every 9-fold covering of a point set in the space by finitely many translates of an octant decomposes into two coverings, and our lower bound by a construction for a 4-fold covering that does not decompose into two coverings. The same bounds also hold for coverings of points in $\R^2$ by finitely many homothets or translates of a triangle. We also prove that certain dynamic interval coloring problems are equivalent to the above question.
Dániel Gerbner, Balázs Keszegh, Abhishek Methuku, Balázs Patkós, Máté Vizer
Erdős and Moser raised the question of determining the maximum number of maximal cliques or equivalently, the maximum number of maximal independent sets in a graph on $n$ vertices. Since then there has been a lot of research along these lines. A $k$-dominating independent set is an independent set $D$ such that every vertex not contained in $D$ has at least $k$ neighbours in $D$. Let $mi_k(n)$ denote the maximum number of $k$-dominating independent sets in a graph on $n$ vertices, and let $ζ_k:=\lim_{n \rightarrow \infty} \sqrt[n]{mi_k(n)}$. Nagy initiated the study of $mi_k(n)$. In this article we disprove a conjecture of Nagy and prove that for any even $k$ we have $$1.489 \approx \sqrt[9]{36} \le ζ^k_k.$$ We also prove that for any $k \ge 3$ we have $$ζ_k^{k} \le 2.053^{\frac{1}{1.053+1/k}}< 1.98,$$ improving the upper bound of Nagy.
Eyal Ackerman, Balázs Keszegh, Máté Vizer
We consider the problem of $2$-coloring geometric hypergraphs. Specifically, we show that there is a constant $m$ such that any finite set of points in the plane $\mathcal{S} \subset {\mathbb R}^2$ can be $2$-colored such that every axis-parallel square that contains at least $m$ points from $\mathcal{S}$ 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 $2$-coloring points with respect to homothets of a fixed parallelogram.
Dániel Gerbner, Balázs Keszegh, Cory Palmer, Dömötör Pálvölgyi
We call a topological ordering of a weighted directed acyclic graph non-negative if the sum of weights on the vertices in any prefix of the ordering is non-negative. We investigate two processes for constructing non-negative topological orderings of weighted directed acyclic graphs. The first process is called a mark sequence and the second is a generalization called a mark-unmark sequence. We answer a question of Erickson by showing that every non-negative topological ordering that can be realized by a mark-unmark sequence can also be realized by a mark sequence. We also investigate the question of whether a given weighted directed acyclic graph has a non-negative topological ordering. We show that even in the simple case when every vertex is a source or a sink the question is NP-complete.
János Barát, Zoltán L. Blázsik, Balázs Keszegh, Zeyu Zheng
We investigate the extremal properties of saturated partial plane embeddings of maximal planar graphs. For a planar graph $G$, the plane-saturation number $\mathrm{sat}_{\mathcal{P}}(G)$ denotes the minimum number of edges in a plane subgraph of $G$ such that the addition of any edge either violates planarity or results in a graph that is not a subgraph of $G$. We focus on maximal planar graphs and establish an upper bound on $\mathrm{sat}_{\mathcal{P}}(G)$ by showing there exists a universal constant $ε> 0$ such that $\mathrm{sat}_{\mathcal{P}}(G) < (3-ε)v(G)$ for any maximal planar graph $G$ with $v(G) \geq 16$. This answers a question posed by Clifton and Simon. Additionally, we derive lower bound results and demonstrate that for maximal planar graphs with sufficiently large number of vertices, the minimum ratio $\mathrm{sat}_{\mathcal{P}}(G)/e(G)$ lies within the interval $(1/16, 1/9 + o(1)]$.
Yan Alves Radtke, Balázs Keszegh, Robert Lauff
We consider the faces in pseudoline arrangements in which the pseudolines are colored with two colors. Björner, Las Vergnas, Sturmfels, White, and Ziegler conjecture the existence of a two-colored triangle in such arrangements. We consider variants of this problem. We show that in any non-trivial two-coloring of a pseudoline arrangement there exists a two-colored triangle or quadrangle. We also investigate the existence of a bichromatic triangle assuming certain structures on the coloring. Previously, several authors investigated the chromatic number and independence number of hypergraphs whose vertices correspond to the pseudolines of an arrangement and the hyperedges correspond to the faces of the arrangement. We show that the maximum of the independence numbers of such hypergraphs is $\lceil \frac{2}{3}n-1\rceil$. We also prove that if we only consider the triangular faces then this maximum becomes $n-Θ(\log n)$.
Gábor Damásdi, Balázs Keszegh, Dömötör Pálvölgyi, Karamjeet Singh
The study of geometric hypergraphs gave rise to the notion of $ABAB$-free hypergraphs. A hypergraph $\mathcal{H}$ is called $ABAB$-free if there is an ordering of its vertices such that there are no hyperedges $A,B$ and vertices $v_1,v_2,v_3,v_4$ in this order satisfying $v_1,v_3\in A\setminus B$ and $v_2,v_4\in B\setminus A$. In this paper, we prove that it is NP-complete to decide if a hypergraph is $ABAB$-free. We show a number of analogous results for hypergraphs with similar forbidden patterns, such as $ABABA$-free hypergraphs. As an application, we show that deciding whether a hypergraph is realizable as the incidence hypergraph of points and pseudodisks is also NP-complete.
Balázs Keszegh
We prove lower and upper bounds for the chromatic number of certain hypergraphs defined by geometric regions. This problem has close relations to conflict-free colorings. One of the most interesting type of regions to consider for this problem is that of the axis-parallel rectangles. We completely solve the problem for a special case of them, for bottomless rectangles. We also give an almost complete answer for half-planes and pose several open problems. Moreover we give efficient coloring algorithms.