arXiv subjects

Pavel Dvořák

Publications and source records attributed to Pavel Dvořák.

18 recordsLinked to original sources

arXiv2025

Semi-Robust Communication Complexity of Maximum Matching

We study the one-way two-party communication complexity of Maximum Matching in the semi-robust setting where the edges of a maximum matching are randomly partitioned between Alice and Bob, but all remaining edges of the input graph are adversarially partitioned between the two parties. We show that the simple protocol where Alice solely communicates a lexicographically-first maximum matching of their edges to Bob is surprisingly powerful: We prove that it yields a $3/4$-approximation in expectation and that our analysis is tight. The semi-robust setting is at least as hard as the fully robust setting. In this setting, all edges of the input graph are randomly partitioned between Alice and Bob, and the state-of-the-art result is a fairly involved $5/6$-approximation protocol that is based on the computation of edge-degree constrained subgraphs [Azarmehr, Behnezhad, ICALP'23]. Our protocol also immediately yields a $3/4$-approximation in the fully robust setting. One may wonder whether an improved analysis of our protocol in the fully robust setting is possible: While we cannot rule this out, we give an instance where our protocol only achieves a $0.832 < 5/6 = 0.83$-approximation. Hence, while our simple protocol performs surprisingly well, it cannot be used to improve over the state-of-the-art in the fully robust setting.

cs.DS↗

arXiv2025

Bounds on Functionality and Symmetric Difference -- Two Intriguing Graph Parameters

Functionality ($\mathrm{fun}$) is a graph parameter that generalizes graph degeneracy defined by Alecu et al. [JCTB, 2021]. They research the relation of functionality to many other graphs parameters (tree-width, clique-width, VC-dimension, etc.). Extending their research, we completely characterize the functionality of random graph $G(n,p)$ for all possible $p$. We provide matching (up to a constant factor) lower and upper bound for a large range of $p$. It follows from our bounds for $G(n,p)$, that the maximum functionality (roughly $\sqrt{n}$) is achieved for $p \approx 1/\sqrt{n}$. We complement this by showing that every graph $G$ on $n$ vertices have $\mathrm{fun}(G) \le O(\sqrt{ n \ln n})$ and we give a nearly matching $Ω(\sqrt{n})$-lower bound provided by incident graphs of projective planes. Previously known lower bounds for functionality were only logarithmic in the number of vertices. Further, we study a related graph parameter symmetric difference ($\mathrm{sd}$), the minimum of $|N(u) ~Δ~ N(v)|$ over all pairs of vertices of the ``worst possible'' induced subgraph. It was observed by Alecu et al. that $\mathrm{fun}(G) \le \mathrm{sd}(G)+1$ for every graph $G$. They asked whether the functionality of interval graphs is bounded. Recently, Dallard et al. [RiM, 2024] answered this positively and they constructed an interval graph $G$ with $\mathrm{sd}(G) = Θ(\sqrt[4]{n})$ (even though they did not mention the explicit bound), i.e., they separate the functionality and symmetric difference of interval graphs. We show that $\mathrm{sd}$ of interval graphs is at most $O(\sqrt[3]{n})$ and we provide a different example of an interval graph $G$ with $\mathrm{sd}(G) = Θ(\sqrt[4]{n})$. Further, we show that $\mathrm{sd}$ of circular arc graphs is $Θ(\sqrt{n})$.

math.CO↗

arXiv2024

A Quantum Pigeonhole Principle and Two Semidefinite Relaxations of Communication Complexity

We study semidefinite relaxations of $Π_1$ combinatorial statements. By relaxing the pigeonhole principle, we obtain a new "quantum" pigeonhole principle which is a stronger statement. By relaxing statements of the form "the communication complexity of $f$ is $> k$", we obtain new communication models, which we call "$γ_2$ communication" and "quantum-lab protocols". We prove, via an argument from proof complexity, that any natural model obtained by such a relaxation must solve all Karchmer--Wigderson games efficiently. However, the argument is not constructive, so we work to explicitly construct such protocols in these two models.

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arXiv2024

Exponential Separation Between Powers of Regular and General Resolution Over Parities

Proving super-polynomial lower bounds on the size of proofs of unsatisfiability of Boolean formulas using resolution over parities is an outstanding problem that has received a lot of attention after its introduction by Raz and Tzamaret [Ann. Pure Appl. Log.'08]. Very recently, Efremenko, Garlík and Itsykson [ECCC'23] proved the first exponential lower bounds on the size of ResLin proofs that were additionally restricted to be bottom-regular. We show that there are formulas for which such regular ResLin proofs of unsatisfiability continue to have exponential size even though there exists short proofs of their unsatisfiability in ordinary, non-regular resolution. This is the first super-polynomial separation between the power of general ResLin and and that of regular ResLin for any natural notion of regularity. Our argument, while building upon the work of Efremenko et al., uses additional ideas from the literature on lifting theorems.

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arXiv2023

Bears with Hats and Independence Polynomials

Consider the following hat guessing game. A bear sits on each vertex of a graph $G$, and a demon puts on each bear a hat colored by one of $h$ colors. Each bear sees only the hat colors of his neighbors. Based on this information only, each bear has to guess $g$ colors and he guesses correctly if his hat color is included in his guesses. The bears win if at least one bear guesses correctly for any hat arrangement. We introduce a new parameter - fractional hat chromatic number $\hatμ$, arising from the hat guessing game. The parameter $\hatμ$ is related to the hat chromatic number which has been studied before. We present a surprising connection between the hat guessing game and the independence polynomial of graphs. This connection allows us to compute the fractional hat chromatic number of chordal graphs in polynomial time, to bound fractional hat chromatic number by a function of maximum degree of $G$, and to compute the exact value of $\hatμ$ of cliques, paths, and cycles.

math.CO↗

arXiv2023

Improved Weighted Matching in the Sliding Window Model

We consider the Maximum-weight Matching (MWM) problem in the streaming sliding window model of computation. In this model, the input consists of a sequence of weighted edges on a given vertex set $V$ of size $n$. The objective is to maintain an approximation of a maximum-weight matching in the graph spanned by the $L$ most recent edges, for some integer $L$, using as little space as possible. Prior to our work, the state-of-the-art results were a $(3.5+\varepsilon)$-approximation algorithm for MWM by Biabani et al. [ISAAC'21] and a $(3+\varepsilon)$-approximation for (unweighted) Maximum Matching (MM) by Crouch et al. [ESA'13]. Both algorithms use space $\tilde{O}(n)$. We give the following results: 1. We give a $(2+\varepsilon)$-approximation algorithm for MWM with space $\tilde{O}(\sqrt{nL})$. Under the reasonable assumption that the graphs spanned by the edges in each sliding window are simple, our algorithm uses space $\tilde{O}(n \sqrt{n})$. 2. In the $\tilde{O}(n)$ space regime, we give a $(3+\varepsilon)$-approximation algorithm for MWM, thereby closing the gap between the best-known approximation ratio for MWM and MM. Similar to Biabani et al.'s MWM algorithm, both our algorithms execute multiple instances of the $(2+\varepsilon)$-approximation $\tilde{O}(n)$-space streaming algorithm for MWM by Paz and Schwartzman [SODA'17] on different portions of the stream. Our improvements are obtained by selecting these substreams differently. Furthermore, our $(2+\varepsilon)$-approximation algorithm runs the Paz-Schwartzman algorithm in reverse direction over some parts of the stream, and in forward direction over other parts, which allows for an improved approximation guarantee at the cost of increased space requirements.

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arXiv2022

Parameterized Inapproximability of Independent Set in $H$-Free Graphs

We study the Independent Set (IS) problem in $H$-free graphs, i.e., graphs excluding some fixed graph $H$ as an induced subgraph. We prove several inapproximability results both for polynomial-time and parameterized algorithms. Halldórsson [SODA 1995] showed that for every $δ>0$ IS has a polynomial-time $(\frac{d-1}{2}+δ)$-approximation in $K_{1,d}$-free graphs. We extend this result by showing that $K_{a,b}$-free graphs admit a polynomial-time $O(α(G)^{1-1/a})$-approximation, where $α(G)$ is the size of a maximum independent set in $G$. Furthermore, we complement the result of Halldórsson by showing that for some $γ=Θ(d/\log d),$ there is no polynomial-time $γ$-approximation for these graphs, unless NP = ZPP. Bonnet et al. [IPEC 2018] showed that IS parameterized by the size $k$ of the independent set is W[1]-hard on graphs which do not contain (1) a cycle of constant length at least $4$, (2) the star $K_{1,4}$, and (3) any tree with two vertices of degree at least $3$ at constant distance. We strengthen this result by proving three inapproximability results under different complexity assumptions for almost the same class of graphs (we weaken condition (2) that $G$ does not contain $K_{1,5}$). First, under the ETH, there is no $f(k)\cdot n^{o(k/\log k)}$ algorithm for any computable function $f$. Then, under the deterministic Gap-ETH, there is a constant $δ>0$ such that no $δ$-approximation can be computed in $f(k) \cdot n^{O(1)}$ time. Also, under the stronger randomized Gap-ETH there is no such approximation algorithm with runtime $f(k)\cdot n^{o(\sqrt{k})}$. Finally, we consider the parameterization by the excluded graph $H$, and show that under the ETH, IS has no $n^{o(α(H))}$ algorithm in $H$-free graphs and under Gap-ETH there is no $d/k^{o(1)}$-approximation for $K_{1,d}$-free graphs with runtime $f(d,k) n^{O(1)}$.

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arXiv2022

List Locally Surjective Homomorphisms in Hereditary Graph Classes

A locally surjective homomorphism from a graph $G$ to a graph $H$ is an edge-preserving mapping from $V(G)$ to $V(H)$ that is surjective in the neighborhood of each vertex in $G$. In the list locally surjective homomorphism problem, denoted by LLSHom($H$), the graph $H$ is fixed and the instance consists of a graph $G$ whose every vertex is equipped with a subset of $V(H)$, called list. We ask for the existence of a locally surjective homomorphism from $G$ to $H$, where every vertex of $G$ is mapped to a vertex from its list. In this paper, we study the complexity of the LLSHom($H$) problem in $F$-free graphs, i.e., graphs that exclude a fixed graph $F$ as an induced subgraph. We aim to understand for which pairs $(H,F)$ the problem can be solved in subexponential time. We show that for all graphs $H$, for which the problem is NP-hard in general graphs, it cannot be solved in subexponential time in $F$-free graphs unless $F$ is a bounded-degree forest or the ETH fails. The initial study reveals that a natural subfamily of bounded-degree forests $F$ that might lead to some tractability results is the family $\mathcal S$ consisting of forests whose every component has at most three leaves. In this case, we exhibit the following dichotomy theorem: besides the cases that are polynomial-time solvable in general graphs, the graphs $H \in \{P_3,C_4\}$ are the only connected ones that allow for a subexponential-time algorithm in $F$-free graphs for every $F \in \mathcal S$ (unless the ETH fails).

cs.DS↗

arXiv2021

Data Structures Lower Bounds and Popular Conjectures

In this paper, we investigate the relative power of several conjectures that attracted recently lot of interest. We establish a connection between the Network Coding Conjecture (NCC) of Li and Li and several data structure like problems such as non-adaptive function inversion of Hellman and the well-studied problem of polynomial evaluation and interpolation. In turn these data structure problems imply super-linear circuit lower bounds for explicit functions such as integer sorting and multi-point polynomial evaluation.

cs.CC↗

arXiv2020

Automorphisms of the Cube $n^d$

Consider a hypergraph $H_n^d$ where the vertices are points of the $d$-dimensional combinatorial cube $n^d$ and the edges are all sets of $n$ points such that they are in one line. We study the structure of the group of automorphisms of $H_n^d$, i.e., permutations of points of $n^d$ preserving the edges. In this paper we provide a complete characterization. Moreover, we consider the Colored Cube Isomorphism problem of deciding whether for two colorings of the vertices of $H_n^d$ there exists an automorphism of $H_n^d$ preserving the colors. We show that this problem is ${\sf GI}$-complete.

cs.DM↗

arXiv2020

Barrington Plays Cards: The Complexity of Card-based Protocols

In this paper we study the computational complexity of functions that have efficient card-based protocols. Card-based protocols were proposed by den Boer [EUROCRYPT '89] as a means for secure two-party computation. Our contribution is two-fold: We classify a large class of protocols with respect to the computational complexity of functions they compute, and we propose other encodings of inputs which require fewer cards than the usual 2-card representation.

cs.CC↗

arXiv2020

Lower Bounds for Semi-adaptive Data Structures via Corruption

In a dynamic data structure problem we wish to maintain an encoding of some data in memory, in such a way that we may efficiently carry out a sequence of queries and updates to the data. A long-standing open problem in this area is to prove an unconditional polynomial lower bound of a trade-off between the update time and the query time of an adaptive dynamic data structure computing some explicit function. Ko and Weinstein provided such lower bound for a restricted class of {\em semi-adaptive\} data structures, which compute the Disjointness function. There, the data are subsets $x_1,\dots,x_k$ and $y$ of $\{1,\dots,n\}$, the updates can modify $y$ (by inserting and removing elements), and the queries are an index $i \in \{1,\dots,k\}$ (query $i$ should answer whether $x_i$ and $y$ are disjoint, i.e., it should compute the Disjointness function applied to $(x_i, y)$). The semi-adaptiveness places a restriction in how the data structure can be accessed in order to answer a query. We generalize the lower bound of Ko and Weinstein to work not just for the Disjointness, but for any function having high complexity under the smooth corruption bound.

cs.CC↗

arXiv2020

Target Set Selection in Dense Graph Classes

In this paper, we study the Target Set Selection problem from a parameterized complexity perspective. Here for a given graph and a threshold for each vertex, the task is to find a set of vertices (called a target set) which activates the whole graph during the following iterative process. A vertex outside the active set becomes active if the number of so far activated vertices in its neighborhood is at least its threshold. We give two parameterized algorithms for a special case where each vertex has the threshold set to the half of its neighbors (the so-called Majority Target Set Selection problem) for parameterizations by the neighborhood diversity and the twin cover number of the input graph. We complement these results from the negative side. We give a hardness proof for the Majority Target Set Selection problem when parameterized by (a restriction of) the modular-width - a natural generalization of both previous structural parameters. We also show the Target Set Selection problem parameterized by the neighborhood diversity or by the twin cover number is W[1]-hard when there is no restriction on the thresholds.

cs.DS↗

arXiv2020

Parameterized Approximation Schemes for Steiner Trees with Small Number of Steiner Vertices

We study the Steiner Tree problem, in which a set of terminal vertices needs to be connected in the cheapest possible way in an edge-weighted graph. This problem has been extensively studied from the viewpoint of approximation and also parametrization. In particular, on one hand Steiner Tree is known to be APX-hard, and W[2]-hard on the other, if parameterized by the number of non-terminals (Steiner vertices) in the optimum solution. In contrast to this we give an efficient parameterized approximation scheme (EPAS), which circumvents both hardness results. Moreover, our methods imply the existence of a polynomial size approximate kernelization scheme (PSAKS) for the considered parameter. We further study the parameterized approximability of other variants of Steiner Tree, such as Directed Steiner Tree and Steiner Forest. For neither of these an EPAS is likely to exist for the studied parameter: for Steiner Forest an easy observation shows that the problem is APX-hard, even if the input graph contains no Steiner vertices. For Directed Steiner Tree we prove that approximating within any function of the studied parameter is W[1]-hard. Nevertheless, we show that an EPAS exists for Unweighted Directed Steiner Tree, but a PSAKS does not. We also prove that there is an EPAS and a PSAKS for Steiner Forest if in addition to the number of Steiner vertices, the number of connected components of an optimal solution is considered to be a parameter.

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arXiv2020

Solving Integer Linear Programs with a Small Number of Global Variables and Constraints

Integer Linear Programming (ILP) has a broad range of applications in various areas of artificial intelligence. Yet in spite of recent advances, we still lack a thorough understanding of which structural restrictions make ILP tractable. Here we study ILP instances consisting of a small number of "global" variables and/or constraints such that the remaining part of the instance consists of small and otherwise independent components; this is captured in terms of a structural measure we call fracture backdoors which generalizes, for instance, the well-studied class of N -fold ILP instances. Our main contributions can be divided into three parts. First, we formally develop fracture backdoors and obtain exact and approximation algorithms for computing these. Second, we exploit these backdoors to develop several new parameterized algorithms for ILP; the performance of these algorithms will naturally scale based on the number of global variables or constraints in the instance. Finally, we complement the developed algorithms with matching lower bounds. Altogether, our results paint a near-complete complexity landscape of ILP with respect to fracture backdoors.

cs.DM↗

arXiv2019

On Induced Online Ramsey Number of Paths, Cycles, and Trees

An online Ramsey game is a game between Builder and Painter, alternating in turns. They are given a graph $H$ and a graph $G$ of an infinite set of independent vertices. In each round Builder draws an edge and Painter colors it either red or blue. Builder wins if after some finite round there is a monochromatic copy of the graph $H$, otherwise Painter wins. The online Ramsey number $\widetilde{r}(H)$ is the minimum number of rounds such that Builder can force a monochromatic copy of $H$ in $G$. This is an analogy to the size-Ramsey number $\overline{r}(H)$ defined as the minimum number such that there exists graph $G$ with $\overline{r}(H)$ edges where for any edge two-coloring $G$ contains a monochromatic copy of $H$. In this paper, we introduce the concept of induced online Ramsey numbers: the induced online Ramsey number $\widetilde{r}_{ind}(H)$ is the minimum number of rounds Builder can force an induced monochromatic copy of $H$ in $G$. We prove asymptotically tight bounds on the induced online Ramsey numbers of paths, cycles and two families of trees. Moreover, we provide a result analogous to Conlon [On-line Ramsey Numbers, SIAM J. Discr. Math. 2009], showing that there is an infinite family of trees $T_1,T_2,\dots$, $|T_i|<|T_{i+1}|$ for $i\ge1$, such that \[ \lim_{i\to\infty} \frac{\widetilde{r}(T_i)}{\overline{r}(T_i)} = 0. \]

cs.DM↗

arXiv2016

Anti-Path Cover on Sparse Graph Classes

We show that it is possible to use Bondy-Chvatal closure to design an FPT algorithm that decides whether or not it is possible to cover vertices of an input graph by at most k vertex disjoint paths in the complement of the input graph. More precisely, we show that if a graph has tree-width at most w and its complement is closed under Bondy-Chvatal closure, then it is possible to bound neighborhood diversity of the complement by a function of w only. A simpler proof where tree-depth is used instead of tree-width is also presented.

cs.DS↗

arXiv2015

Parameterized complexity of length-bounded cuts and multi-cuts

We show that the Minimal Length-Bounded L-But problem can be computed in linear time with respect to L and the tree-width of the input graph as parameters. In this problem the task is to find a set of edges of a graph such that after removal of this set, the shortest path between two prescribed vertices is at least L long. We derive an FPT algorithm for a more general multi-commodity length bounded cut problem when parameterized by the number of terminals also. For the former problem we show a W[1]-hardness result when the parameterization is done by the path-width only (instead of the tree-width) and that this problem does not admit polynomial kernel when parameterized by tree-width and L. We also derive an FPT algorithm for the Minimal Length-Bounded Cut problem when parameterized by the tree-depth. Thus showing an interesting paradigm for this problem and parameters tree-depth and path-width.

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