arXiv subjects

Dániel Marx

Publications and source records attributed to Dániel Marx.

At least 19 recordsLinked to original sources

arXiv2026

Independence and Domination on Bounded-Treewidth Graphs: Integer, Rational, and Irrational Distances

The distance-d variants of Independent Set and Dominating Set problems have been extensively studied from different algorithmic viewpoints. In particular, the complexity of these problems are well understood on bounded-treewidth graphs [Katsikarelis, Lampis, and Paschos, Discret. Appl. Math 2022][Borradaile and Le, IPEC 2016]: given a tree decomposition of width t, the two problems can be solved in time $d^t \cdot n^{O(1)}$ and $(2d + 1)t \cdot n^{O(1)}$, respectively. Furthermore, assuming the Strong Exponential-Time Hypothesis (SETH), the base constants are best possible in these running times: they cannot be improved to $d-ε$ and $2d+1-ε$, respectively, for any $ε > 0$. We investigate continuous versions of these problems in a setting introduced by Megiddo and Tamir [SICOMP 1983], where every edge is modeled by a unit-length interval of points. In the δ-Dispersion problem, the task is to find a maximum number of points (possibly inside edges) that are pairwise at distance at least δ from each other. Similarly, in the δ-Covering problem, the task is to find a minimum number of points (possibly inside edges) such that every point of the graph (including those inside edges) is at distance at most δ from the selected point set. We provide a comprehensive understanding of these two problems on bounded-treewidth graphs.

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arXiv2026

Pattern-Sparse Tree Decompositions in $H$-Minor-Free Graphs

Given an $H$-minor-free graph $G$ and an integer $k$, our main technical contribution is sampling in randomized polynomial time an induced subgraph $G'$ of $G$ and a tree decomposition of $G'$ of width $\widetilde{O}(k)$ such that for every $Z\subseteq V(G)$ of size $k$, with probability at least $\left(2^{\widetilde{O}(\sqrt{k})}|V(G)|^{O(1)}\right)^{-1}$, we have $Z \subseteq V(G')$ and every bag of the tree decomposition contains at most $\widetilde{O}(\sqrt{k})$ vertices of $Z$. Having such a tree decomposition allows us to solve a wide range of problems in (randomized) time $2^{\widetilde{O}(\sqrt{k})}n^{O(1)}$ where the solution is a pattern $Z$ of size $k$, e.g., Directed $k$-Path, $H$-Packing, etc. In particular, our result recovers all the algorithmic applications of the pattern-covering result of Fomin et al. [SIAM J. Computing 2022] (which requires the pattern to be connected) and the planar subgraph-finding algorithms of Nederlof [STOC 2020]. Furthermore, for $K_{h,3}$-free graphs (which include bounded-genus graphs) and for a fixed constant $d$, we signficantly strengthen the result by ensuring that not only $Z$ has intersection $\widetilde{O}(\sqrt{k})$ with each bag, but even the distance-$d$ neighborhood $N^d_{G}[Z]$ as well. This extension makes it possible to handle a wider range of problems where the neighborhood of the pattern also plays a role in the solution, such as partial domination problems and problems involving distance constraints.

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arXiv2026

Approximation Schemes for Subset TSP and Steiner Tree on Geometric Intersection Graphs

We give approximation schemes for Subset TSP and Steiner Tree on unit disk graphs, and more generally, on intersection graphs of similarly sized connected fat (not necessarily convex) polygons in the plane. As a first step towards this goal, we prove spanner-type results: finding an induced subgraph of bounded size that is $(1+\varepsilon)$-equivalent to the original instance in the sense that the optimum value increases only by a factor of at most $(1+\varepsilon)$ when the solution can use only the edges in this subgraph. - For Subset TSP, our algorithms find a $(1+\varepsilon)$-equivalent induced subgraph of size $\mathrm{poly}(1/\varepsilon)\cdot\mathrm{OPT}$ in polynomial time, and use it to find a $(1+\varepsilon)$-approximate solution in time $2^{\mathrm{poly}(1/\varepsilon)}\cdot n^{O(1)}$. - For Steiner Tree, our algorithms find a $(1+\varepsilon)$-equivalent induced subgraph of size $2^{\mathrm{poly}(1/\varepsilon)}\cdot\mathrm{OPT}$ in time $2^{\mathrm{poly}(1/\varepsilon)}\cdot n^{O(1)}$, and use it to find a $(1+\varepsilon)$-approximate solution in time $2^{2^{\mathrm{poly}(1/\varepsilon)}}\cdot n^{O(1)}$. - An improved algorithm finds a $(1+\varepsilon)$-approximate solution for Steiner Tree in time $2^{\mathrm{poly}(1/\varepsilon)}\cdot n^{O(1)}$. An easy reduction shows that approximation schemes for unit disks imply approximation schemes for planar graphs. Thus our results are far-reaching generalizations of analogous results of Klein [STOC'06] and Borradaile, Klein, and Mathieu [ACM TALG'09] for Subset TSP and Steiner Tree in planar graphs. We show that our results are best possible in the sense that dropping any of (i) similarly sized, (ii) connected, or (iii) fat makes both problems APX-hard.

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arXiv2026

The Price of Being Partial: Complexity of Partial Generalized Dominating Set on Bounded-Treewidth Graphs

For fixed sets $σ, ρ$ of non-negative integers, the $(σ, ρ)$-domination framework introduced by Telle [Nord. J. Comput. 1994] captures many classical graph problems. For a graph $G$, a $(σ,ρ)$-set is a set $S$ of vertices such that for every $v\in V(G)$, we have (1) if $v \in S$, then $|N(v) \cap S| \in σ$, and (2) if $v \notin S$, then $|N(v) \cap S| \in ρ$. We initiate the study of a natural partial variant $(σ,ρ)$-MinParDomSet of the problem, in which the constraints given by $σ, ρ$ need not be fulfilled for all vertices, but we want to find a set of size at most $k$ that maximizes the number of vertices that are satisfied in the sense that they satisfy (1) or (2) above. Our goal is to understand whether $(σ,ρ)$-MinParDomSet can be solved in the same running time as the nonpartial version, or whether it is strictly harder. Formally, we consider nonempty finite or simple cofinite sets $σ$ and $ρ$ (simple cofinite sets are of the form $\mathbb{Z}_{\geq c}$), and we try to determine the smallest constant $c_{σ,ρ}$ such that there is a $c_{σ,ρ}^{tw}\cdot n^{O(1)}$ time algorithm for the problem if a tree decomposition of width $tw$ is given. We obtain matching upper and lower bounds on $c_{σ,ρ}$ for every such fixed $σ$ and $ρ$ under the Primal Pathwidth Strong Exponential Time Hypothesis, and establish whether the partial problem is harder than the nonpartial variant. For some sets $σ$ and $ρ$, the more general $(σ,ρ)$-MinParDomSet has the same complexity as the nonpartial special case (e.g., for Dominating Set), while for other choices, the partial version is significantly harder (e.g., for Perfect Code).

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arXiv2026

Steiner Forest for $H$-Subgraph-Free Graphs

Our main result is a full classification, for every connected graph $H$, of the computational complexity of Steiner Forest on $H$-subgraph-free graphs. To obtain this dichotomy, we establish the following new algorithmic, hardness, and combinatorial results: Algorithms: We identify two new classes of graph-theoretical structures that make it possible to solve Steiner Forest in polynomial time. Roughly speaking, our algorithms handle the following cases: (1) a set $X$ of vertices of bounded size that are pairwise connected by subgraphs of treewidth $2$ or bounded size, possibly together with an independent set of arbitrary size that is connected to $X$ in an arbitrary way; (2) a set $X$ of vertices of arbitrary size that are pairwise connected in a cyclic manner by subgraphs of treewidth $2$ or bounded size. Hardness results: We show that Steiner Forest remains NP-complete for graphs with 2-deletion set number $3$. (The $c$-deletion set number is the size of a smallest cutset $S$ such that every component of $G-S$ has at most $c$ vertices.) Combinatorial results: To establish the dichotomy, we perform a delicate graph-theoretic analysis showing that if $H$ is a path or a subdivided claw, then excluding $H$ as a subgraph either yields one of the two algorithmically favourable structures described above, or yields a graph class for which NP-completeness of Steiner Forest follows from either our new hardness result or a previously known one. Along the way to classifying the hardness for excluded subgraphs, we establish a dichotomy for graphs with $c$-deletion set number at most $k$. Specifically, our results together with pre-existing ones show that Steiner Forest is polynomial-time solvable if (1) $c=1$ and $k\geq 0$, or (2) $c=2$ and $k\leq 2$, or (3) $c\geq 3$ and $k=1$, and is NP-complete otherwise.

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arXiv2026

Multicut Problems in Almost-Planar Graphs: The Dependency of Complexity on the Demand Pattern

Given a graph $G$, a set $T$ of terminal vertices, and a demand graph $H$ on $T$, the \textsc{Multicut} problem asks for a set of edges of minimum weight that separates the pairs of terminals specified by the edges of $H$. The \textsc{Multicut} problem can be solved in polynomial time if the number of terminals and the genus of the graph is bounded (Colin de Verdière [Algorithmica, 2017]). Focke et al.~[SoCG 2024] characterized which special cases of Multicut are fixed-parameter tractable parameterized by the number of terminals on planar graphs. Moreover, they precisely determined how the parameter genus influences the complexity and presented partial results of this form for graphs that can be made planar by the deletion of $π$ edges. We complete the picture on how this parameter $π$ influences the complexity of different special cases and precisely determine the influence of the crossing number. Formally, let $\mathcal{H}$ be any class of graphs (satisfying a mild closure property) and let Multicut$(\mathcal{H})$ be the special case when the demand graph $H$ is in $\mathcal{H}$. Our first main result is showing that if $\mathcal{H}$ has the combinatorial property of having bounded distance to extended bicliques, then Multicut$(\mathcal{H})$ on unweighted graphs is FPT parameterized by the number $t$ of terminals and $π$. For the case when $\mathcal{H}$ does not have this combinatorial property, Focke et al.~[SoCG 2024] showed that $O(\sqrt{t})$ is essentially the best possible exponent of the running time; together with our result, this gives a complete understanding of how the parameter $π$ influences complexity on unweighted graphs. Our second main result is giving an algorithm whose existence shows that the parameter crossing number behaves analogously if we consider weighted graphs.

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arXiv2026

Faster Exponential-Time Approximation Algorithms Using Approximate Monotone Local Search

We generalize the monotone local search approach of Fomin, Gaspers, Lokshtanov and Saurabh [J. ACM 2019], by establishing a connection between parameterized approximation and exponential-time approximation algorithms for monotone subset minimization problems. In a monotone subset minimization problem the input implicitly describes a non-empty set family over a universe of size $n$ which is closed under taking supersets. The task is to find a minimum cardinality set in this family. Broadly speaking, we use approximate monotone local search to show that a parameterized $α$-approximation algorithm that runs in $c^k \cdot n^{O(1)}$ time, where $k$ is the solution size, can be used to derive an $α$-approximation randomized algorithm that runs in $d^n \cdot n^{O(1)}$ time, where $d$ is the unique value in $d \in (1,1+\frac{c-1}α)$ such that $\mathcal{D}(\frac{1}α\|\frac{d-1}{c-1})=\frac{\ln c}α$ and $\mathcal{D}(a \|b)$ is the Kullback-Leibler divergence. This running time matches that of Fomin et al. for $α=1$, and is strictly better when $α>1$, for any $c > 1$. Furthermore, we also show that this result can be derandomized at the expense of a sub-exponential multiplicative factor in the running time. We demonstrate the potential of approximate monotone local search by deriving new and faster exponential approximation algorithms for Vertex Cover, $3$-Hitting Set, Directed Feedback Vertex Set, Directed Subset Feedback Vertex Set, Directed Odd Cycle Transversal and Undirected Multicut. For instance, we get a $1.1$-approximation algorithm for Vertex Cover with running time $1.114^n \cdot n^{O(1)}$, improving upon the previously best known $1.1$-approximation running in time $1.127^n \cdot n^{O(1)}$ by Bourgeois et al. [DAM 2011].

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arXiv2025

Generalized Graph Packing Problems Parameterized by Treewidth

$H$-Packing is the problem of finding a maximum number of vertex-disjoint copies of $H$ in a given graph $G$. $H$-Partition is the special case of finding a set of vertex-disjoint copies that cover each vertex of $G$ exactly once. Our goal is to study these problems and some generalizations on bounded-treewidth graphs. The case of $H$ being a triangle is well understood: given a tree decomposition of $G$ having treewidth $tw$, the $K_3$-Packing problem can be solved in time $2^{tw} \cdot n^{O(1)}$, while Lokshtanov et al.~[{\it ACM Transactions on Algorithms} 2018] showed, under the Strong Exponential-Time Hypothesis (SETH), that there is no $(2-ε)^{tw}\cdot n^{O(1)}$ algorithm for any $ε>0$ even for $K_3$-Partition. Similar results can be obtained for any other clique $K_d$ for $d\ge 3$. We provide generalizations in two directions: - We consider a generalization of the problem where every vertex can be used at most $c$ times for some $c\ge 1$. When $H$ is any clique $K_d$ with $d\ge 3$, then we give upper and lower bounds showing that the optimal running time increases to $(c+1)^{tw}\cdot n^{O(1)}$. We consider two variants depending on whether a copy of $H$ can be used multiple times in the packing. - If $H$ is not a clique, then the dependence of the running time on treewidth may not be even single exponential. Specifically, we show that if $H$ is any fixed graph where not every 2-connected component is a clique, then there is no $2^{o({tw}\log {tw})}\cdot n^{O(1)}$ algorithm for \textsc{$H$-Partition}, assuming the Exponential-Time Hypothesis (ETH).

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arXiv2025

Tight Complexity Bounds for Counting Generalized Dominating Sets in Bounded-Treewidth Graphs Part I: Algorithmic Results

We investigate how efficiently a well-studied family of domination-type problems can be solved on bounded-treewidth graphs. For sets $σ,ρ$ of non-negative integers, a $(σ,ρ)$-set of a graph $G$ is a set $S$ of vertices such that $|N(u)\cap S|\in σ$ for every $u\in S$, and $|N(v)\cap S|\in ρ$ for every $v\not\in S$. The problem of finding a $(σ,ρ)$-set (of a certain size) unifies standard problems such as Independent Set, Dominating Set, Independent Dominating Set, and many others. For all pairs of finite or cofinite sets $(σ,ρ)$, we determine (under standard complexity assumptions) the best possible value $c_{σ,ρ}$ such that there is an algorithm that counts $(σ,ρ)$-sets in time $c_{σ,ρ}^{\sf tw}\cdot n^{O(1)}$ (if a tree decomposition of width ${\sf tw}$ is given in the input). For example, for the Exact Independent Dominating Set problem (also known as Perfect Code) corresponding to $σ=\{0\}$ and $ρ=\{1\}$, we improve the $3^{\sf tw}\cdot n^{O(1)}$ algorithm of [van Rooij, 2020] to $2^{\sf tw}\cdot n^{O(1)}$. Despite the unusually delicate definition of $c_{σ,ρ}$, an accompanying paper shows that our algorithms are most likely optimal, that is, for any pair $(σ, ρ)$ of finite or cofinite sets where the problem is non-trivial, and any $\varepsilon>0$, a $(c_{σ,ρ}-\varepsilon)^{\sf tw}\cdot n^{O(1)}$-algorithm counting the number of $(σ,ρ)$-sets would violate the Counting Strong Exponential-Time Hypothesis (#SETH). For finite sets $σ$ and $ρ$, these lower bounds also extend to the decision version, and hence, our algorithms are optimal in this setting as well. In contrast, for many cofinite sets, we show that further significant improvements for the decision and optimization versions are possible using the technique of representative sets.

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arXiv2025

Multicut Problems in Embedded Graphs: The Dependency of Complexity on the Demand Pattern

The Multicut problem asks for a minimum cut separating certain pairs of vertices: formally, given a graph $G$ and demand graph $H$ on a set $T\subseteq V(G)$ of terminals, the task is to find a minimum-weight set $C$ of edges of $G$ such that whenever two vertices of $T$ are adjacent in $H$, they are in different components of $G\setminus C$. Colin de Verdière [Algorithmica, 2017] showed that Multicut with $t$ terminals on a graph $G$ of genus $g$ can be solved in time $f(t,g)n^{O(\sqrt{g^2+gt+t})}$. Cohen-Addad et al. [JACM, 2021] proved a matching lower bound showing that the exponent of $n$ is essentially best possible (for every fixed value of $t$ and $g$), even in the special case of Multiway Cut, where the demand graph $H$ is a complete graph. However, this lower bound tells us nothing about other special cases of Multicut such as Group 3-Terminal Cut (where three groups of terminals need to be separated from each other). We show that if the demand pattern is, in some sense, close to being a complete bipartite graph, then Multicut can be solved faster than $f(t,g)n^{O(\sqrt{g^2+gt+t})}$, and furthermore this is the only property that allows such an improvement. Formally, for a class $\mathcal{H}$ of graphs, Multicut$(\mathcal{H})$ is the special case where the demand graph $H$ is in $\mathcal{H}$. For every fixed class $\mathcal{H}$ (satisfying some mild closure property), fixed $g$, and fixed $t$, our main result gives tight upper and lower bounds on the exponent of $n$ in algorithms solving Multicut$(\mathcal{H})$.

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arXiv2025

From Chinese Postman to Salesman and Beyond II: Inapproximability and Parameterized Complexity

A well-studied continuous model of graphs considers each edge as a continuous unit-length interval of points. In the problem $δ$-Tour defined within this model, the objective to find a shortest tour that comes within a distance of $δ$ of every point on every edge. This parameterized problem was introduced in the predecessor to this article and shown to be essentially equivalent to the Chinese Postman problem for $δ= 0$, to the graphic Travel Salesman Problem (TSP) for $δ= 1/2$, and close to first Vertex Cover and then Dominating Set for even larger $δ$. Moreover, approximation algorithms for multiple parameter ranges were provided. In this article, we provide complementing inapproximability bounds and examine the fixed-parameter tractability of the problem. On the one hand, we show the following: (1) For every fixed $0 < δ< 3/2$, the problem $δ$-Tour is APX-hard, while for every fixed $δ\geq 3/2$, the problem has no polynomial-time $o(\log{n})$-approximation unless P = NP. Our techniques also yield the new result that TSP remains APX-hard on cubic (and even cubic bipartite) graphs. (2) For every fixed $0 < δ< 3/2$, the problem $δ$-Tour is fixed-parameter tractable (FPT) when parameterized by the length of a shortest tour, while it is W[2]-hard for every fixed $δ\geq 3/2$ and para-NP-hard for $δ$ being part of the input. On the other hand, if $δ$ is considered to be part of the input, then an interesting nontrivial phenomenon occurs when $δ$ is a constant fraction of the number of vertices: (3) If $δ$ is part of the input, then the problem can be solved in time $f(k)n^{O(k)}$, where $k = \lceil n/δ\rceil$; however, assuming the Exponential-Time Hypothesis (ETH), there is no algorithm that solves the problem and runs in time $f(k)n^{o(k/\log k)}$.

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arXiv2025

From Chinese Postman to Salesman and Beyond I: Approximating Shortest Tours $δ$-Covering All Points on All Edges

A well-studied continuous model of graphs, introduced by Dearing and Francis [Transportation Science, 1974], considers each edge as a continuous unit-length interval of points. For $δ\geq 0$, we introduce the problem $δ$-Tour, where the objective is to find the shortest tour that comes within a distance of $δ$ of every point on every edge. It can be observed that 0-Tour is essentially equivalent to the Chinese Postman Problem, which is solvable in polynomial time. In contrast, 1/2-Tour is essentially equivalent to the Graphic Traveling Salesman Problem (TSP), which is NP-hard but admits a constant-factor approximation in polynomial time. We investigate $δ$-Tour for other values of $δ$, noting that the problem's behavior and the insights required to understand it differ significantly across various $δ$ regimes. We design polynomial-time approximation algorithms summarized as follows: (1) For every fixed $0 < δ< 3/2$, the problem $δ$-Tour admits a constant-factor approximation. (2) For every fixed $δ\geq 3/2$, the problem admits an $O(\log{n})$-approximation. (3) If $δ$ is considered to be part of the input, then the problem admits an $O(\log^3{n})$-approximation. This is the first of two articles on the $δ$-Tour problem. In the second one we complement the approximation algorithms presented here with inapproximability results and related to parameterized complexity.

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arXiv2024

Robust Contraction Decomposition for Minor-Free Graphs and its Applications

We prove a robust contraction decomposition theorem for $H$-minor-free graphs, which states that given an $H$-minor-free graph $G$ and an integer $p$, one can partition in polynomial time the vertices of $G$ into $p$ sets $Z_1,\dots,Z_p$ such that $\operatorname{tw}(G/(Z_i \setminus Z')) = O(p + |Z'|)$ for all $i \in [p]$ and $Z' \subseteq Z_i$. Here, $\operatorname{tw}(\cdot)$ denotes the treewidth of a graph and $G/(Z_i \setminus Z')$ denotes the graph obtained from $G$ by contracting all edges with both endpoints in $Z_i \setminus Z'$. Our result generalizes earlier results by Klein [SICOMP 2008] and Demaine et al. [STOC 2011] based on partitioning $E(G)$, and some recent theorems for planar graphs by Marx et al. [SODA 2022], for bounded-genus graphs (more generally, almost-embeddable graphs) by Bandyapadhyay et al. [SODA 2022], and for unit-disk graphs by Bandyapadhyay et al. [SoCG 2022]. The robust contraction decomposition theorem directly results in parameterized algorithms with running time $2^{\widetilde{O}(\sqrt{k})} \cdot n^{O(1)}$ or $n^{O(\sqrt{k})}$ for every vertex/edge deletion problems on $H$-minor-free graphs that can be formulated as Permutation CSP Deletion or 2-Conn Permutation CSP Deletion. Consequently, we obtain the first subexponential-time parameterized algorithms for Subset Feedback Vertex Set, Subset Odd Cycle Transversal, Subset Group Feedback Vertex Set, 2-Conn Component Order Connectivity on $H$-minor-free graphs. For other problems which already have subexponential-time parameterized algorithms on $H$-minor-free graphs (e.g., Odd Cycle Transversal, Vertex Multiway Cut, Vertex Multicut, etc.), our theorem gives much simpler algorithms of the same running time.

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arXiv2024

Parameterized Approximation for Robust Clustering in Discrete Geometric Spaces

We consider the well-studied Robust $(k, z)$-Clustering problem, which generalizes the classic $k$-Median, $k$-Means, and $k$-Center problems. Given a constant $z\ge 1$, the input to Robust $(k, z)$-Clustering is a set $P$ of $n$ weighted points in a metric space $(M,δ)$ and a positive integer $k$. Further, each point belongs to one (or more) of the $m$ many different groups $S_1,S_2,\ldots,S_m$. Our goal is to find a set $X$ of $k$ centers such that $\max_{i \in [m]} \sum_{p \in S_i} w(p) δ(p,X)^z$ is minimized. This problem arises in the domains of robust optimization [Anthony, Goyal, Gupta, Nagarajan, Math. Oper. Res. 2010] and in algorithmic fairness. For polynomial time computation, an approximation factor of $O(\log m/\log\log m)$ is known [Makarychev, Vakilian, COLT $2021$], which is tight under a plausible complexity assumption even in the line metrics. For FPT time, there is a $(3^z+ε)$-approximation algorithm, which is tight under GAP-ETH [Goyal, Jaiswal, Inf. Proc. Letters, 2023]. Motivated by the tight lower bounds for general discrete metrics, we focus on \emph{geometric} spaces such as the (discrete) high-dimensional Euclidean setting and metrics of low doubling dimension, which play an important role in data analysis applications. First, for a universal constant $η_0 >0.0006$, we devise a $3^z(1-η_{0})$-factor FPT approximation algorithm for discrete high-dimensional Euclidean spaces thereby bypassing the lower bound for general metrics. We complement this result by showing that even the special case of $k$-Center in dimension $Θ(\log n)$ is $(\sqrt{3/2}- o(1))$-hard to approximate for FPT algorithms. Finally, we complete the FPT approximation landscape by designing an FPT $(1+ε)$-approximation scheme (EPAS) for the metric of sub-logarithmic doubling dimension.

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arXiv2024

From Graph Properties to Graph Parameters: Tight Bounds for Counting on Small Subgraphs

A graph property is a function $Φ$ that maps every graph to {0, 1} and is invariant under isomorphism. In the $\#IndSub(Φ)$ problem, given a graph $G$ and an integer $k$, the task is to count the number of $k$-vertex induced subgraphs $G'$ with $Φ(G')=1$. $\#IndSub(Φ)$ can be naturally generalized to graph parameters, that is, to functions $Φ$ on graphs that do not necessarily map to {0, 1}: now the task is to compute the sum $\sum_{G'} Φ(G')$ taken over all $k$-vertex induced subgraphs $G'$. This problem setting can express a wider range of counting problems (for instance, counting $k$-cycles or $k$-matchings) and can model problems involving expected values (for instance, the expected number of components in a subgraph induced by $k$ random vertices). Our main results are lower bounds on $\#IndSub(Φ)$ in this setting, which simplify, generalize, and tighten the recent lower bounds of Döring, Marx, and Wellnitz [STOC'24] in various ways. (1) We show a lower bound for every nontrivial edge-monotone graph parameter $Φ$ with finite codomain (not only for parameters that take value in {0, 1}). (2) The lower bound is tight: we show that, assuming ETH, there is no $f(k)n^{o(k)}$ time algorithm. (3) The lower bound applies also to the modular counting versions of the problem. (4) The lower bound applies also to the multicolored version of the problem. We can extend the #W[1]-hardness result to the case when the codomain of $Φ$ is not finite, but has size at most $(1 - \varepsilon)\sqrt{k}$ on $k$-vertex graphs. However, if there is no bound on the size of the codomain, the situation changes significantly: for example, there is a nontrivial edge-monotone function $Φ$ where the size of the codomain is $k$ on $k$-vertex graphs and $\#IndSub(Φ)$ is FPT.

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arXiv2024

Conditional lower bounds for sparse parameterized 2-CSP: A streamlined proof

Assuming the Exponential Time Hypothesis (ETH), a result of Marx (ToC'10) implies that there is no $f(k)\cdot n^{o(k/\log k)}$ time algorithm that can solve 2-CSPs with $k$ constraints (over a domain of arbitrary large size $n$) for any computable function $f$. This lower bound is widely used to show that certain parameterized problems cannot be solved in time $f(k)\cdot n^{o(k/\log k)}$ time (assuming the ETH). The purpose of this note is to give a streamlined proof of this result.

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arXiv2024

Hitting Meets Packing: How Hard Can it Be?

We study a general family of problems that form a common generalization of classic hitting (also referred to as covering or transversal) and packing problems. An instance of X-HitPack asks: Can removing k (deletable) vertices of a graph G prevent us from packing $\ell$ vertex-disjoint objects of type X? This problem captures a spectrum of problems with standard hitting and packing on opposite ends. Our main motivating question is whether the combination X-HitPack can be significantly harder than these two base problems. Already for a particular choice of X, this question can be posed for many different complexity notions, leading to a large, so-far unexplored domain in the intersection of the areas of hitting and packing problems. On a high-level, we present two case studies: (1) X being all cycles, and (2) X being all copies of a fixed graph H. In each, we explore the classical complexity, as well as the parameterized complexity with the natural parameters k+l and treewidth. We observe that the combined problem can be drastically harder than the base problems: for cycles or for H being a connected graph with at least 3 vertices, the problem is Σ_2^P-complete and requires double-exponential dependence on the treewidth of the graph (assuming the Exponential-Time Hypothesis). In contrast, the combined problem admits qualitatively similar running times as the base problems in some cases, although significant novel ideas are required. For example, for X being all cycles, we establish a 2^poly(k+l)n^O(1) algorithm using an involved branching method. Also, for X being all edges (i.e., H = K_2; this combines Vertex Cover and Maximum Matching) the problem can be solved in time 2^\poly(tw)n^O(1) on graphs of treewidth tw. The key step enabling this running time relies on a combinatorial bound obtained from an algebraic (linear delta-matroid) representation of possible matchings.

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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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