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Paweł Rzążewski

Publications and source records attributed to Paweł Rzążewski.

At least 37 records · Page 2Linked to original sources
arXiv2025

On Computational Aspects of Ordered Matching Problems

Ordered matchings, defined as graphs with linearly ordered vertices, where each vertex is connected to exactly one edge, play a crucial role in the area of ordered graphs and their homomorphisms. Therefore, we consider related problems from the complexity point of view and determine their corresponding computational and parameterized complexities. We show that the subgraph of ordered matchings problem is NP-complete and we prove that the problem of finding ordered homomorphisms between ordered matchings is NP-complete as well, implying NP-completeness of more generic problems. In parameterized complexity setting, we consider a natural choice of parameter - a number of vertices of the image ordered graph. We show that in contrast to the complexity context, finding homomorphisms if the image ordered graph is an ordered matching, this problem parameterized by the number of vertices of the image ordered graph is FPT, which is known to be W[1]-hard for the general problem. We also determine that the problem of core for ordered matchings is solvable in polynomial time which is again in contrast to the NP-completeness of the general problem. We provide several algorithms and generalize some of these problems into ordered graphs with colored edges.

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arXiv2025

On Computational Aspects of Cores of Ordered Graphs

An ordered graph is a graph enhanced with a linear order on the vertex set. An ordered graph is a core if it does not have an order-preserving homomorphism to a proper subgraph. We say that $H$ is the core of $G$ if (i) $H$ is a core, (ii) $H$ is a subgraph of $G$, and (iii) $G$ admits an order-preserving homomorphism to $H$. We study complexity aspects of several problems related to the cores of ordered graphs. Interestingly, they exhibit a different behavior than their unordered counterparts. We show that the retraction problem, i.e., deciding whether a given graph admits an ordered-preserving homomorphism to its specific subgraph, can be solved in polynomial time. On the other hand, it is \NP-hard to decide whether a given ordered graph is a core. In fact, we show that it is even \NP-hard to distinguish graphs $G$ whose core is largest possible (i.e., if $G$ is a core) from those, whose core is the smallest possible, i.e., its size is equal to the ordered chromatic number of $G$. The problem is even \wone-hard with respect to the latter parameter.

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arXiv2025

On coarse tree decompositions and coarse balanced separators

It is known that there is a linear dependence between the treewidth of a graph and its balanced separator number: the smallest integer $k$ such that for every weighing of the vertices, the graph admits a balanced separator of size at most $k$. We investigate whether this connection can be lifted to the setting of coarse graph theory, where both the bags of the considered tree decompositions and the considered separators should be coverable by a bounded number of bounded-radius balls. As the first result, we prove that if an $n$-vertex graph $G$ admits balanced separators coverable by $k$ balls of radius $r$, then $G$ also admits tree decompositions ${\cal T}_1$ and ${\cal T}_2$ such that: - in ${\cal T}_1$, every bag can be covered by $O(k\log n)$ balls of radius $r$; and - in ${\cal T}_2$, every bag can be covered by $O(k^2\log k)$ balls of radius $r(\log k+\log\log n+O(1))$. As the second result, we show that if we additionally assume that $G$ has doubling dimension at most $m$, then the functional equivalence between the existence of small balanced separators and of tree decompositions of small width can be fully lifted to the coarse setting. Precisely, we prove that for a positive integer $r$ and a graph $G$ of doubling dimension at most $m$, the following conditions are equivalent, with constants $k_1,k_2,k_3,k_4,Δ_3,Δ_4$ depending on each other and on $m$: - $G$ admits balanced separators consisting of $k_1$ balls of radius $r$; - $G$ has a tree decomposition with bags coverable by $k_2$ balls of radius $r$; - $G$ has a tree-partition of maximum degree $\leq Δ_3$ with bags coverable by $k_3$ balls of radius $r$; - $G$ is quasi-isometric to a graph of maximum degree $\leq Δ_4$ and tree-partition width $\leq k_4$.

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arXiv2025

Induced matching treewidth and tree-independence number, revisited

We study two graph parameters defined via tree decompositions: tree-independence number and induced matching treewidth. Both parameters are defined similarly as treewidth, but with respect to different measures of a tree decomposition $\mathcal{T}$ of a graph $G$: for tree-independence number, the measure is the maximum size of an independent set in $G$ included in some bag of $\mathcal{T}$, while for the induced matching treewidth, the measure is the maximum size of an induced matching in $G$ such that some bag of $\mathcal{T}$ contains at least one endpoint of every edge of the matching. While the induced matching treewidth of any graph is bounded from above by its tree-independence number, the family of complete bipartite graphs shows that small induced matching treewidth does not imply small tree-independence number. On the other hand, Abrishami, Briański, Czyżewska, McCarty, Milanič, Rzążewski, and Walczak~[SIAM Journal on Discrete Mathematics, 2025] showed that, if a fixed biclique $K_{t,t}$ is excluded as an induced subgraph, then the tree-independence number is bounded from above by some function of the induced matching treewidth. The function resulting from their proof is exponential even for fixed $t$, as it relies on multiple applications of Ramsey's theorem. In this note we show, using the Kövári-Sós-Turán theorem, that for any class of $K_{t,t}$-free graphs, the two parameters are in fact polynomially related.

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arXiv2025

Hitting all longest paths in $H$-free graphs and $H$-graphs

The \textit{longest path transversal number} of a connected graph $G$, denoted by $lpt(G)$, is the minimum size of a set of vertices of $G$ that intersects all longest paths in $G$. We present constant upper bounds for the longest path transversal number of \textit{hereditary classes of graphs}, that is, classes of graphs closed under taking induced subgraphs. Our first main result is a structural theorem that allows us to \textit{refine} a given longest path transversal in a graph using domination properties. This has several consequences: First, it implies that for every $t \in \{5,6\}$, every connected $P_t$-free graph $G$ satisfies $lpt(G) \leq t-2$. Second, it shows that every $(\textit{bull}, \textit{chair})$-free graph $G$ satisfies $lpt(G) \leq 5$. Third, it implies that for every $t \in \mathbb{N}$, every connected chordal graph $G$ with no induced subgraph isomorphic to $K_t \mat \overline{K_t}$ satisfies $lpt(G) \leq t-1$, where $K_t \mat \overline{K_t}$ is the graph obtained from a $t$-clique and an independent set of size $t$ by adding a perfect matching between them. Our second main result provides an upper bound for the longest path transversal number in \textit{$H$-intersection graphs}. For a given graph $H$, a graph $G$ is called an \textit{$H$-graph} if there exists a subdivision $H'$ of $H$ such that $G$ is the intersection graph of a family of vertex subsets of $H'$ that each induce connected subgraphs. The concept of $H$-graphs, introduced by Biró, Hujter, and Tuza, naturally captures interval graphs, circular-arc graphs, and chordal graphs, among others. Our result shows that for every connected graph $H$ with at least two vertices, there exists an integer $k = k(H)$ such that every connected $H$-graph $G$ satisfies $lpt(G) \leq k$.

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arXiv2025

List coloring ordered graphs with forbidden induced subgraphs

In the List $k$-Coloring problem we are given a graph whose every vertex is equipped with a list, which is a subset of $\{1,\ldots,k\}$. We need to decide if $G$ admits a proper coloring, where every vertex receives a color from its list. The complexity of the problem in classes defined by forbidding induced subgraphs is a widely studied topic in algorithmic graph theory. Recently, Hajebi, Li, and Spirkl [SIAM J. Discr. Math. 38 (2024)] initiated the study of List $3$-Coloring in ordered graphs, i.e., graphs with fixed linear ordering of vertices. Forbidding ordered induced subgraphs allows us to investigate the boundary of tractability more closely. We continue this direction of research, focusing mostly on the case of List $4$-Coloring. We present several algorithmic and hardness results, which altogether provide an almost complete dichotomy for classes defined by forbidding one fixed ordered graph: our investigations leave one minimal open case.

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arXiv2025

Maximum Weight Independent Set in Graphs with no Long Claws in Quasi-Polynomial Time

We show that the Maximum Weight Independent Set problem (MWIS) can be solved in quasi-polynomial time on $H$-free graphs (graphs excluding a fixed graph $H$ as an induced subgraph) for every $H$ whose every connected component is a path or a subdivided claw (i.e., a tree with at most three leaves). This completes the dichotomy of the complexity of MWIS in $\mathcal{F}$-free graphs for any finite set $\mathcal{F}$ of graphs into NP-hard cases and cases solvable in quasi-polynomial time, and corroborates the conjecture that the cases not known to be NP-hard are actually polynomial-time solvable. The key graph-theoretic ingredient in our result is as follows. Fix an integer $t \geq 1$. Let $S_{t,t,t}$ be the graph created from three paths on $t$ edges by identifying one endpoint of each path into a single vertex. We show that, given a graph $G$, one can in polynomial time find either an induced $S_{t,t,t}$ in $G$, or a balanced separator consisting of $\mathcal{O}(\log |V(G)|)$ vertex neighborhoods in $G$, or an extended strip decomposition of $G$ (a decomposition almost as useful for recursion for MWIS as a partition into connected components) with each particle of weight multiplicatively smaller than the weight of $G$. This is a strengthening of a result of Majewski, Masařík, Novotná, Okrasa, Pilipczuk, Rzążewski, and Sokołowski [ICALP 2022] which provided such an extended strip decomposition only after the deletion of $\mathcal{O}(\log |V(G)|)$ vertex neighborhoods. To reach the final result, we employ an involved branching strategy that relies on the structural lemma presented above.

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arXiv2025

Parameterized Complexity of Directed Traveling Salesman Problem

The Directed Traveling Salesman Problem (DTSP) is a variant of the classical Traveling Salesman Problem in which the edges in the graph are directed and a vertex and edge can be visited multiple times. The goal is to find a directed closed walk of minimum length (or total weight) that visits every vertex of the given graph at least once. In a yet more general version, Directed Waypoint Routing Problem (DWRP), some vertices are marked as terminals and we are only required to visit all terminals. Furthermore, each edge has its capacity bounding the number of times this edge can be used by a solution. While both problems (and many other variants of TSP) were extensively investigated, mostly from the approximation point of view, there are surprisingly few results concerning the parameterized complexity. Our starting point is the result of Marx et al. [APPROX/RANDOM 2016] who proved that DTSP is W[1]-hard parameterized by distance to pathwidth 3. In this paper we aim to initiate the systematic complexity study of variants of DTSP with respect to various, mostly structural, parameters. We show that DWRP is FPT parameterized by the solution size, the feedback edge number, and the vertex integrity of the underlying undirected graph. Furthermore, the problem is XP parameterized by treewidth. On the complexity side, we show that the problem is W[1]-hard parameterized by the distance to constant treedepth.

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arXiv2025

Tabular intermediate logics comparison

Tabular intermediate logics are intermediate logics characterized by finite posets treated as Kripke frames. For a poset $\mathbb{P}$, let $L(\mathbb{P})$ denote the corresponding tabular intermediate logic. We investigate the complexity of the following decision problem $\mathsf{LogContain}$: given two finite posets $\mathbb P$ and $\mathbb Q$, decide whether $L(\mathbb P) \subseteq L(\mathbb Q)$. By Jankov's and de Jongh's theorem, the problem $\mathsf{LogContain}$ is related to the problem $\mathsf{SPMorph}$: given two finite posets $\mathbb P$ and $\mathbb Q$, decide whether there exists a surjective $p$-morphism from $\mathbb P$ onto $\mathbb Q$. Both problems belong to the complexity class NP. We present two contributions. First, we describe a construction which, starting with a graph $\mathbb{G}$, gives a poset $\mathsf{Pos}(\mathbb{G})$ such that there is a surjective locally surjective homomorphism (the graph-theoretic analog of a $p$-morphism) from $\mathbb{G}$ onto $\mathbb{H}$ if and only if there is a surjective $p$-morphism from $\mathsf{Pos}(\mathbb{G})$ onto $\mathsf{Pos}(\mathbb{H})$. This allows us to translate some hardness results from graph theory and obtain that several restricted versions of the problems $\mathsf{LogContain}$ and $\mathsf{SPMorph}$ are NP-complete. Among other results, we present a 18-element poset $\mathbb{Q}$ such that the problem to decide, for a given poset $\mathbb{P}$, whether $L(\mathbb{P})\subseteq L(\mathbb{Q})$ is NP-complete. Second, we describe a polynomial-time algorithm that decides $\mathsf{LogContain}$ and $\mathsf{SPMorph}$ for posets $\mathbb{T}$ and $\mathbb{Q}$, when $\mathbb{T}$ is a tree.

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arXiv2025

On Approximate MMS Allocations on Restricted Graph Classes

We study the problem of fair division of a set of indivisible goods with connectivity constraints. Specifically, we assume that the goods are represented as vertices of a connected graph, and sets of goods allocated to the agents are connected subgraphs of this graph. We focus on the widely-studied maximin share criterion of fairness. It has been shown that an allocation satisfying this criterion may not exist even without connectivity constraints, i.e., if the graph of goods is complete. In view of this, it is natural to seek approximate allocations that guarantee each agent a connected bundle of goods with value at least a constant fraction of the maximin share value to the agent. It is known that for some classes of graphs, such as complete graphs, cycles, and $d$-claw-free graphs for any fixed $d$, such approximate allocations indeed exist. However, it is an open problem whether they exist for the class of all graphs. In this paper, we continue the systematic study of the existence of approximate allocations on restricted graph classes. In particular, we show that such allocations exist for several well-studied classes, including block graphs, cacti, complete multipartite graphs, and split graphs.

cs.DM↗
arXiv2025

Kernelization for list $H$-coloring for graphs with small vertex cover

For a fixed graph $H$, in the List $H$-Coloring problem, we are given a graph $G$ along with list $L(v) \subseteq V(H)$ for every $v \in V(G)$, and we have to determine if there exists a list homomorphism $φ$ from $(G,L)$ to $H$, i.e., an edge preserving mapping $φ: V(G)\to V(H)$ that satisfies $φ(v)\in L(v)$ for every $v\in V(G)$. Note that if $H$ is the complete graph on $q$ vertices, the problem is equivalent to List $q$-Coloring. We investigate the kernelization properties of List $H$-Coloring parameterized by the vertex cover number of $G$: given an instance $(G,L)$ and a vertex cover of $G$ of size $k$, can we reduce $(G,L)$ to an equivalent instance $(G',L')$ of List $H$-Coloring where the size of $G'$ is bounded by a low-degree polynomial $p(k)$ in $k$? This question has been investigated previously by Jansen and Pieterse [Algorithmica 2019], who provided an upper bound, which turns out to be optimal if $H$ is a complete graph, i.e., for List $q$-Coloring. This result was one of the first applications of the method of kernelization via bounded-degree polynomials. We define two new integral graph invariants, $c^*(H)$ and $d^*(H)$, with $d^*(H) \leq c^*(H) \leq d^*(H)+1$, and show that for every graph $H$, List $H$-Coloring -- has a kernel with $\mathcal{O}(k^{c^*(H)})$ vertices, -- admits no kernel of size $\mathcal{O}(k^{d^*(H)-\varepsilon})$ for any $\varepsilon > 0$, unless the polynomial hierarchy collapses. -- Furthermore, if $c^*(H) > d^*(H)$, then there is a kernel with $\mathcal{O}(k^{c^*(H)-\varepsilon})$ vertices where $\varepsilon \geq 2^{1-c^*(H)}$. Additionally, we show that for some classes of graphs, including powers of cycles and graphs $H$ where $Δ(H) \leq c^*(H)$ (which in particular includes cliques), the bound $d^*(H)$ is tight, using the polynomial method. We conjecture that this holds in general.

math.CO↗
arXiv2025

Finding large $k$-colorable induced subgraphs in (bull, chair)-free and (bull,E)-free graphs

We study the Max Partial $k$-Coloring problem, where we are given a vertex-weighted graph, and we ask for a maximum-weight induced subgraph that admits a proper $k$-coloring. For $k=1$ this problem coincides with Maximum Weight Independent Set, and for $k=2$ the problem is equivalent (by complementation) to Minimum Odd Cycle Transversal. Furthermore, it generalizes $k$-Coloring. We show that Max Partial $k$-Coloring on $n$-vertex instances with clique number $ω$ can be solved in time * $n^{\mathcal{O}(kω)}$ if the input graph excludes the bull and the chair as an induced subgraph, * $n^{\mathcal{O}(kω\log n)}$ if the input graph excludes the bull and E as an induced subgraph. This implies that $k$-Coloring can be solved in polynomial time in the former class, and in quasipolynomial-time in the latter one.

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arXiv2024

Excluding a clique or a biclique in graphs of bounded induced matching treewidth

For a tree decomposition $\mathcal{T}$ of a graph $G$, let $μ(\mathcal{T})$ denote the maximum size of an induced matching in $G$ with the property that some bag of $\mathcal{T}$ contains at least one endpoint of every edge of the matching. The induced matching treewidth of a graph $G$ is the minimum value of $μ(\mathcal{T})$ over all tree decompositions $\mathcal{T}$ of $G$. Classes of graphs with bounded induced matching treewidth admit polynomial-time algorithms for a number of problems, including INDEPENDENT SET, $k$-COLORING, ODD CYCLE TRANSVERSAL, and FEEDBACK VERTEX SET. In this paper, we focus on combinatorial properties of such classes. First, we show that graphs with bounded induced matching treewidth that exclude a fixed biclique as an induced subgraph have bounded tree-independence number, which is another well-studied parameter defined in terms of tree decompositions. This sufficient condition about excluding a biclique is also necessary, as bicliques have unbounded tree-independence number. Second, we show that graphs with bounded induced matching treewidth that exclude a fixed clique have bounded chromatic number, that is, classes of graphs with bounded induced matching treewidth are $χ$-bounded. The two results confirm two conjectures due to Lima et al. [ESA 2024].

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arXiv2024

An $11/6$-Approximation Algorithm for Vertex Cover on String Graphs

We present a 1.8334-approximation algorithm for Vertex Cover on string graphs given with a representation, which takes polynomial time in the size of the representation; the exact approximation factor is $11/6$. Recently, the barrier of 2 was broken by Lokshtanov et al. [SoGC '24] with a 1.9999-approximation algorithm. Thus we increase by three orders of magnitude the distance of the approximation ratio to the trivial bound of 2. Our algorithm is very simple. The intricacies reside in its analysis, where we mainly establish that string graphs without odd cycles of length at most 11 are 8-colorable. Previously, Chudnovsky, Scott, and Seymour [JCTB '21] showed that string graphs without odd cycles of length at most 7 are 80-colorable, and string graphs without odd cycles of length at most 5 have bounded chromatic number.

cs.DS↗
arXiv2024

Tree decompositions meet induced matchings: beyond Max Weight Independent Set

For a tree decomposition $\mathcal{T}$ of a graph $G$, by $μ(\mathcal{T})$ we denote the size of a largest induced matching in $G$ all of whose edges intersect one bag of $\mathcal{T}$. Induced matching treewidth of a graph $G$ is the minimum value of $μ(\mathcal{T})$ over all tree decompositions $\mathcal{T}$ of $G$. Yolov [SODA 2018] proved that Max Weight Independent Set can be solved in polynomial time for graphs of bounded induced matching treewidth. In this paper we explore what other problems are tractable in such classes of graphs. As our main result, we give a polynomial-time algorithm for Min Weight Feedback Vertex Set. We also provide some positive results concerning packing induced subgraphs, which in particular imply a PTAS for the problem of finding a largest induced subgraph of bounded treewidth. These results suggest that in graphs of bounded induced matching treewidth, one could find in polynomial time a maximum-weight induced subgraph of bounded treewidth satisfying a given CMSO$_2$ formula. We conjecture that such a result indeed holds and prove it for graphs of bounded tree-independence number, which form a rich and important family of subclasses of graphs of bounded induced matching treewidth. We complement these algorithmic results with a number of complexity and structural results concerning induced matching treewidth.

cs.DS↗
arXiv2024

Fundamental Problems on Bounded-Treewidth Graphs: The Real Source of Hardness

It is known for many algorithmic problems that if a tree decomposition of width $t$ is given in the input, then the problem can be solved with exponential dependence on $t$. A line of research by Lokshtanov, Marx, and Saurabh [SODA 2011] produced lower bounds showing that in many cases known algorithms achieve the best possible exponential dependence on $t$, assuming the SETH. The main message of our paper is showing that the same lower bounds can be obtained in a more restricted setting: a graph consisting of a block of $t$ vertices connected to components of constant size already has the same hardness as a general tree decomposition of width $t$. Formally, a $(σ,δ)$-hub is a set $Q$ of vertices such that every component of $Q$ has size at most $σ$ and is adjacent to at most $δ$ vertices of $Q$. $\bullet$ For every $ε> 0$, there are $σ,δ> 0$ such that Independent Set/Vertex Cover cannot be solved in time $(2-ε)^p\cdot n$, even if a $(σ,δ)$-hub of size $p$ is given in the input, assuming the SETH. This matches the earlier tight lower bounds parameterized by the width of the tree decomposition. Similar tight bounds are obtained for Odd Cycle Transversal, Max Cut, $q$-Coloring, and edge/vertex deletions versions of $q$-Coloring. $\bullet$ For every $ε>0$, there are $σ,δ> 0$ such that Triangle-Partition cannot be solved in time $(2-ε)^p\cdot n$, even if a $(σ,δ)$-hub of size $p$ is given in the input, assuming the Set Cover Conjecture (SCC). In fact, we prove that this statement is equivalent to the SCC, thus it is unlikely that this could be proved assuming the SETH. $\bullet$ For Dominating Set, we can prove a non-tight lower bound ruling out $(2-ε)^p\cdot n^{O(1)}$ algorithms, assuming either the SETH or the SCC, but this does not match the $3^p\cdot n^{O(1)}$ upper bound.

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arXiv2024

List homomorphisms by deleting edges and vertices: tight complexity bounds for bounded-treewidth graphs

The goal of this paper is to investigate a family of optimization problems arising from list homomorphisms, and to understand what the best possible algorithms are if we restrict the problem to bounded-treewidth graphs. For a fixed $H$, the input of the optimization problem LHomVD($H$) is a graph $G$ with lists $L(v)$, and the task is to find a set $X$ of vertices having minimum size such that $(G-X,L)$ has a list homomorphism to $H$. We define analogously the edge-deletion variant LHomED($H$). This expressive family of problems includes members that are essentially equivalent to fundamental problems such as Vertex Cover, Max Cut, Odd Cycle Transversal, and Edge/Vertex Multiway Cut. For both variants, we first characterize those graphs $H$ that make the problem polynomial-time solvable and show that the problem is NP-hard for every other fixed $H$. Second, as our main result, we determine for every graph $H$ for which the problem is NP-hard, the smallest possible constant $c_H$ such that the problem can be solved in time $c^t_H\cdot n^{O(1)}$ if a tree decomposition of $G$ having width $t$ is given in the input.Let $i(H)$ be the maximum size of a set of vertices in $H$ that have pairwise incomparable neighborhoods. For the vertex-deletion variant LHomVD($H$), we show that the smallest possible constant is $i(H)+1$ for every $H$. The situation is more complex for the edge-deletion version. For every $H$, one can solve LHomED($H$) in time $i(H)^t\cdot n^{O(1)}$ if a tree decomposition of width $t$ is given. However, the existence of a specific type of decomposition of $H$ shows that there are graphs $H$ where LHomED($H$) can be solved significantly more efficiently and the best possible constant can be arbitrarily smaller than $i(H)$. Nevertheless, we determine this best possible constant and (assuming the SETH) prove tight bounds for every fixed $H$.

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