arXiv subjects

Dömötör Pálvölgyi

Publications and source records attributed to Dömötör Pálvölgyi.

At least 37 records · Page 2Linked to original sources

arXiv2024

Spanners in Planar Domains via Steiner Spanners and non-Steiner Tree Covers

We study spanners in planar domains, including polygonal domains, polyhedral terrain, and planar metrics. Previous work showed that for any constant $ε\in (0,1)$, one could construct a $(2+ε)$-spanner with $O(n\log(n))$ edges (SICOMP 2019), and there is a lower bound of $Ω(n^2)$ edges for any $(2-ε)$-spanner (SoCG 2015). The main open question is whether a linear number of edges suffices and the stretch can be reduced to $2$. We resolve this problem by showing that for stretch $2$, one needs $Ω(n\log n)$ edges, and for stretch $2+ε$ for any fixed $ε\in (0,1)$, $O(n)$ edges are sufficient. Our lower bound is the first super-linear lower bound for stretch $2$. En route to achieve our result, we introduce the problem of constructing non-Steiner tree covers for metrics, which is a natural variant of the well-known Steiner point removal problem for trees (SODA 2001). Given a tree and a set of terminals in the tree, our goal is to construct a collection of a small number of dominating trees such that for every two points, at least one tree in the collection preserves their distance within a small stretch factor. Here, we identify an unexpected threshold phenomenon around $2$ where a sharp transition from $n$ trees to $Θ(\log n)$ trees and then to $O(1)$ trees happens. Specifically, (i) for stretch $ 2-ε$, one needs $Ω(n)$ trees; (ii) for stretch $2$, $Θ(\log n)$ tree is necessary and sufficient; and (iii) for stretch $2+ε$, a constant number of trees suffice. Furthermore, our lower bound technique for the non-Steiner tree covers of stretch $2$ has further applications in proving lower bounds for two related constructions in tree metrics: reliable spanners and locality-sensitive orderings. Our lower bound for locality-sensitive orderings matches the best upper bound (STOC 2022).

cs.CG↗

arXiv2024

Odd-Sunflowers

Extending the notion of sunflowers, we call a family of at least two sets an odd-sunflower if every element of the underlying set is contained in an odd number of sets or in none of them. It follows from the Erd\H os--Szemerédi conjecture, recently proved by Naslund and Sawin, that there is a constant $μ<2$ such that every family of subsets of an $n$-element set that contains no odd-sunflower consists of at most $μ^n$ sets. We construct such families of size at least $1.5021^n$. We also characterize minimal odd-sunflowers of triples.

math.CO↗

arXiv2024

Partitioned Matching Games for International Kidney Exchange

We introduce partitioned matching games as a suitable model for international kidney exchange programmes, where in each round the total number of available kidney transplants needs to be distributed amongst the participating countries in a "fair" way. A partitioned matching game $(N,v)$ is defined on a graph $G=(V,E)$ with an edge weighting $w$ and a partition $V=V_1 \cup \dots \cup V_n$. The player set is $N = \{1, \dots, n\}$, and player $p \in N$ owns the vertices in $V_p$. The value $v(S)$ of a coalition $S \subseteq N$ is the maximum weight of a matching in the subgraph of $G$ induced by the vertices owned by the players in $S$. If $|V_p|=1$ for all $p\in N$, then we obtain the classical matching game. Let $c=\max\{|V_p| \; |\; 1\leq p\leq n\}$ be the width of $(N,v)$. We prove that checking core non-emptiness is polynomial-time solvable if $c\leq 2$ but co-NP-hard if $c\leq 3$. We do this via pinpointing a relationship with the known class of $b$-matching games and completing the complexity classification on testing core non-emptiness for $b$-matching games. With respect to our application, we prove a number of complexity results on choosing, out of possibly many optimal solutions, one that leads to a kidney transplant distribution that is as close as possible to some prescribed fair distribution.

cs.GT↗

arXiv2023

Note on polychromatic coloring of hereditary hypergraph families

We exhibit a 5-uniform hypergraph that has no polychromatic 3-coloring, but all its restricted subhypergraphs with edges of size at least 3 are 2-colorable. This disproves a bold conjecture of Keszegh and the author, and can be considered as the first step to understand polychromatic colorings of hereditary hypergraph families better since the seminal work of Berge. We also show that our method cannot give hypergraphs of arbitrary high uniformity, and mention some connections to panchromatic colorings.

math.CO↗

arXiv2023

Projective and external saturation problem for posets

We introduce two variants of the poset saturation problem. For a poset $P$ and the Boolean lattice $\mathcal{B}_n$, a family $\mathcal{F}$ of sets, not necessarily from $\mathcal{B}_n$, is \textit{projective $P$-saturated} if (i) it does not contain any strong copies of $P$, (ii) for any $G\in \mathcal{B}_n\setminus \mathcal{F}$, the family $\mathcal{F}\cup \{G\}$ contains a strong copy of $P$, and (iii) for any two different $F,F'\in\mathcal{F}$ we have $F\cap[n]\neq F'\cap [n]$. Ordinary strongly $P$-saturated families, i.e., subfamilies $\mathcal{F}$ required to be from $\mathcal{B}_n$ satisfying (i) and (ii), automatically satisfy (iii) as they lie within $\mathcal{B}_n$. We study what phenomena are valid both for the ordinary saturation number $\mathrm{sat}^*(n,P)$ and the projective saturation number, the size of the smallest projective $P$-saturated family. Note that the projective saturation number might differ for a poset and its dual. We also introduce an even more relaxed and symmetric version of poset saturation, \textit{external saturation}. We conjecture that all finite posets have bounded external saturation number, and prove this in some special cases.

math.CO↗

arXiv2022

On graphs that contain exactly k copies of a subgraph, and a related problem in search theory

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 $\binom{n}{2}-\mathrm{exa}_1(n,F)$.

math.CO↗

arXiv2022

On the number of hyperedges in the hypergraph of lines and pseudo-discs

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.

math.CO↗

arXiv2022

Exchange properties of finite set-systems

In a recent breakthrough, Adiprasito, Avvakumov, and Karasev constructed a triangulation of the $n$-dimensional real projective space with a subexponential number of vertices. They reduced the problem to finding a small downward closed set-system $\cal F$ covering an $n$-element ground set which satisfies the following condition: For any two disjoint members $A, B\in\cal F$, there exist $a\in A$ and $b\in B$ such that either $B\cup\{a\}\in\cal F$ and $A\cup\{b\}\setminus\{a\}\in\cal F$, or $A\cup\{b\}\in\cal F$ and $B\cup\{a\}\setminus\{b\}\in\cal F$. Denoting by $f(n)$ the smallest cardinality of such a family $\cal F$, they proved that $f(n)<2^{O(\sqrt{n}\log n)}$, and they asked for a nontrivial lower bound. It turns out that the construction of Adiprasito et al. is not far from optimal; we show that $2^{(1.42+o(1))\sqrt{n}}\le f(n)\le 2^{(1+o(1))\sqrt{2n\log n}}$. We also study a variant of the above problem, where the condition is strengthened by also requiring that for any two disjoint members $A, B\in\cal F$ with $|A|>|B|$, there exists $a\in A$ such that $B\cup\{a\}\in\cal F$. In this case, we prove that the size of the smallest $\cal F$ satisfying this stronger condition lies between $2^{Ω(\sqrt{n}\log n)}$ and $2^{O(n\log\log n/\log n)}$.

math.CO↗

arXiv2022

Induced and non-induced poset saturation problems

A subfamily $\mathcal{G}\subseteq \mathcal{F}\subseteq 2^{[n]}$ of sets is a non-induced (weak) copy of a poset $P$ in $\mathcal{F}$ if there exists a bijection $i:P\rightarrow \mathcal{G}$ such that $p\le_P q$ implies $i(p)\subseteq i(q)$. In the case where in addition $p\le_P q$ holds if and only if $i(p)\subseteq i(q)$, then $\mathcal{G}$ is an induced (strong) copy of $P$ in $\mathcal{F}$. We consider the minimum number $sat(n,P)$ [resp.\ $sat^*(n,P)$] of sets that a family $\mathcal{F}\subseteq 2^{[n]}$ can have without containing a non-induced [induced] copy of $P$ and being maximal with respect to this property, i.e., the addition of any $G\in 2^{[n]}\setminus \mathcal{F}$ creates a non-induced [induced] copy of $P$. We prove for any finite poset $P$ that $sat(n,P)\le 2^{|P|-2}$, a bound independent of the size $n$ of the ground set. For induced copies of $P$, there is a dichotomy: for any poset $P$ either $sat^*(n,P)\le K_P$ for some constant depending only on $P$ or $sat^*(n,P)\ge \log_2 n$. We classify several posets according to this dichotomy, and also show better upper and lower bounds on $sat(n,P)$ and $sat^*(n,P)$ for specific classes of posets. Our main new tool is a special ordering of the sets based on the colexicographic order. It turns out that if $P$ is given, processing the sets in this order and adding the sets greedily into our family whenever this does not ruin non-induced [induced] $P$-freeness, we tend to get a small size non-induced [induced] $P$-saturating family.

math.CO↗

arXiv2022

On Communication Complexity of Fixed Point Computation

Brouwer's fixed point theorem states that any continuous function from a compact convex space to itself has a fixed point. Roughgarden and Weinstein (FOCS 2016) initiated the study of fixed point computation in the two-player communication model, where each player gets a function from $[0,1]^n$ to $[0,1]^n$, and their goal is to find an approximate fixed point of the composition of the two functions. They left it as an open question to show a lower bound of $2^{Ω(n)}$ for the (randomized) communication complexity of this problem, in the range of parameters which make it a total search problem. We answer this question affirmatively. Additionally, we introduce two natural fixed point problems in the two-player communication model. $\bullet$ Each player is given a function from $[0,1]^n$ to $[0,1]^{n/2}$, and their goal is to find an approximate fixed point of the concatenation of the functions. $\bullet$ Each player is given a function from $[0,1]^n$ to $[0,1]^{n}$, and their goal is to find an approximate fixed point of the interpolation of the functions. We show a randomized communication complexity lower bound of $2^{Ω(n)}$ for these problems (for some constant approximation factor). Finally, we initiate the study of finding a panchromatic simplex in a Sperner-coloring of a triangulation (guaranteed by Sperner's lemma) in the two-player communication model: A triangulation $T$ of the $d$-simplex is publicly known and one player is given a set $S_A\subset T$ and a coloring function from $S_A$ to $\{0,\ldots ,d/2\}$, and the other player is given a set $S_B\subset T$ and a coloring function from $S_B$ to $\{d/2+1,\ldots ,d\}$, such that $S_A\dot\cup S_B=T$, and their goal is to find a panchromatic simplex. We show a randomized communication complexity lower bound of $|T|^{Ω(1)}$ for the aforementioned problem as well (when $d$ is large).

cs.CC↗

arXiv2022

The number of tangencies between two families of curves

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.

math.CO↗

arXiv2022

Almost-monochromatic sets and the chromatic number of the plane

In a colouring of $\mathbb{R}^d$ a pair $(S,s_0)$ with $S\subseteq \mathbb{R}^d$ and with $s_0\in S$ is \emph{almost monochromatic} if $S\setminus \{s_0\}$ is monochromatic but $S$ is not. We consider questions about finding almost monochromatic similar copies of pairs $(S,s_0)$ in colourings of $\mathbb{R}^d$, $\mathbb{Z}^d$, and in $\mathbb{Q}$ under some restrictions on the colouring. Among other results, we characterise those $(S,s_0)$ with $S\subseteq \mathbb{Z}$ for which every finite colouring of $\mathbb{R}$ without an infinite monochromatic arithmetic progression contains an almost monochromatic similar copy of $(S,s_0)$. We also show that if $S\subseteq \mathbb{Z}^d$ and $s_0$ is outside of the convex hull of $S\setminus \{s_0\}$, then every finite colouring of $\mathbb{R}^d$ without a similar monochromatic copy of $\mathbb{Z}^d$ contains an almost monochromatic similar copy of $(S,s_0)$. Further, we propose an approach of finding almost-monochromatic sets that might lead to a non-computer assisted proof of $χ(\R^2)\geq 5$.

math.CO↗

arXiv2022

Query complexity and the polynomial Freiman-Ruzsa conjecture

We prove a query complexity variant of the weak polynomial Freiman-Ruzsa conjecture in the following form. For any $ε> 0$, a set $A \subset \mathbb{Z}^d$ with doubling $K$ has a subset of size at least $K^{-\frac{4}ε}|A|$ with coordinate query complexity at most $ε\log_2 |A|$. We apply this structural result to give a simple proof of the "few products, many sums" phenomenon for integer sets. The resulting bounds are explicit and improve on the seminal result of Bourgain and Chang.

math.NT↗

arXiv2021

An improved constant factor for the unit distance problem

We prove that the number of unit distances among $n$ planar points is at most $1.94\cdot n^{4/3}$, improving on the previous best bound of $8n^{4/3}$. We also give better upper and lower bounds for several small values of $n$. We also prove some variants of the crossing lemma and improve some constant factors.

math.CO↗

arXiv2021

Turán problems for Edge-ordered graphs

In this paper we initiate a systematic study of the Turán problem for edge-ordered graphs. A simple graph is called $\textit{edge-ordered}$, if its edges are linearly ordered. An isomorphism between edge-ordered graphs must respect the edge-order. A subgraph of an edge-ordered graph is itself an edge-ordered graph with the induced edge-order. We say that an edge-ordered graph $G$ $\textit{avoids}$ another edge-ordered graph $H$, if no subgraph of $G$ is isomorphic to $H$. The $\textit{Turán number}$ of an edge-ordered graph $H$ is the maximum number of edges in an edge-ordered graph on $n$ vertices that avoids $H$. We study this problem in general, and establish an Erdős-Stone-Simonovits-type theorem for edge-ordered graphs -- we discover that the relevant parameter for the Turán number of an edge-ordered graph is its $\textit{order chromatic number}$. We establish several important properties of this parameter. We also study Turán numbers of edge-ordered paths, star forests and the cycle of length four. We make strong connections to Davenport-Schinzel theory, the theory of forbidden submatrices, and show an application in Discrete Geometry.

math.CO↗

arXiv2021

Unlabeled Compression Schemes Exceeding the VC-dimension

In this note we disprove a conjecture of Kuzmin and Warmuth claiming that every family whose VC-dimension is at most d admits an unlabeled compression scheme to a sample of size at most d. We also study the unlabeled compression schemes of the joins of some families and conjecture that these give a larger gap between the VC-dimension and the size of the smallest unlabeled compression scheme for them.

math.CO↗

arXiv2021

At most $3.55^n$ stable matchings

We improve the upper bound for the maximum possible number of stable matchings among $n$ jobs and $n$ applicants from $131072^n+O(1)$ to $3.55^n+O(1)$. To establish this bound, we state a novel formulation of a certain entropy bound that is easy to apply and may be of independent interest in counting other combinatorial objects

math.CO↗

arXiv2021

Grid Drawings of Graphs with Constant Edge-Vertex Resolution

We study the algorithmic problem of computing drawings of graphs in which $(i)$ each vertex is a disk with fixed radius $ρ$, $(ii)$ each edge is a straight-line segment connecting the centers of the two disks representing its end-vertices, $(iii)$ no two disks intersect, and $(iv)$ the distance between an edge segment and the center of a non-incident disk, called \emph{edge-vertex resolution}, is at least $ρ$. We call such drawings \emph{disk-link drawings}. In this paper we focus on the case of constant edge-vertex resolution, namely $ρ=\frac{1}{2}$ (i.e., disks of unit diameter). We prove that star graphs, which trivially admit straight-line drawings in linear area, require quadratic area in any such disk-link drawing. On the positive side, we present constructive techniques that yield improved upper bounds for the area requirements of disk-link drawings for several (planar and nonplanar) graph classes, including bounded bandwidth, complete, and planar graphs. In particular, the presented bounds for complete and planar graphs are asymptotically tight.

cs.DS↗