arXiv subjects

Michael Lampis

Publications and source records attributed to Michael Lampis.

At least 19 recordsLinked to original sources

arXiv2026

Pure Nash Equilibria in Graphical Games of Bounded Width Revisited

We revisit the complexity of deciding whether a graphical game admits a pure Nash equilibrium (PNE) parameterized by standard measures of the input graph, such as treewidth. The natural dynamic programming algorithm for this problem has parameter dependence $\alpha^{(\Delta+1)\text{tw}}$ where $\alpha$ is the maximum number of strategies available to each player, each player's utility depends on at most $\Delta$ other players, and the input graph has width $\text{tw}$. Our first contribution is to point out that an algorithm by Thomas and van Leeuwen [Algorithmica 2015] claiming to improve this dependence to $\alpha^{O(\text{tw})}$ is flawed and, more strongly, such an algorithm would imply that FPT=W[1]. We then set out to pinpoint the fine-grained complexity of this problem with respect to standard parameters and show that the natural DP algorithm is not optimal, as the problem can be solved with dependence $\alpha^{\lfloor \frac{2\Delta}{3} + 1 \rfloor \text{tw}}$, $\alpha^{\lfloor \frac{\Delta}{2} + 1 \rfloor \text{pw}}$, and $\alpha^{\text{ctw}}$, where $\text{pw}, \text{ctw}$ are the pathwidth and cutwidth of the input respectively. Our main algorithmic tool is a tightening of the relationship between the width of a graph $G$, its maximum degree, and the width of $G^2$, which may be of independent interest. Complementing these results, we show that our algorithms for pathwidth and cutwidth are likely to be optimal, as improving them is equivalent to falsifying the pw-SETH.

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arXiv2026

Parameterized Capacitated Vertex Cover Revisited

Capacitated Vertex Cover is the hard-capacitated variant of Vertex Cover: given a graph, a capacity for every vertex, and an integer $k$, the task is to select at most $k$ vertices that cover all edges and assign each edge to one of its chosen endpoints so that no chosen vertex receives more incident edges than its capacity. This problem is a classical benchmark in parameterized complexity, as it was among the first natural problems shown to be W[1]-hard when parameterized by treewidth. We revisit its exact complexity from a fine-grained parameterized perspective and obtain a much sharper picture for several standard parameters. For the natural parameter $k$, we prove under the Exponential Time Hypothesis (ETH) that no algorithm with running time $k^{o(k)} n^{\mathcal{O}(1)}$ exists. In particular, this shows that the known algorithms with running time $k^{\mathcal{O}(\mathrm{tw})} n^{\mathcal{O}(1)}$ are essentially optimal. We then turn to more general structural parameters. For vertex cover number $\mathrm{vc}$, we give evidence against a $2^{\mathcal{O}(\mathrm{vc}^{2-\varepsilon})} n^{\mathcal{O}(1)}$ algorithm, as such an improvement would imply corresponding progress for a broader class of integer-programming-type problems. We complement this barrier with a nearly matching upper bound for vertex integrity $\mathrm{vi}$, improving the previously known double-exponential dependence to an algorithm with running time $\mathrm{vi}^{\mathcal{O}(\mathrm{vi}^{2})} n^{\mathcal{O}(1)}$ using $N$-fold integer programming. For treewidth, we show that the standard dynamic programming algorithm with running time $n^{\mathcal{O}(\mathrm{tw})}$ is essentially optimal under the ETH, even if one parameterizes by tree-depth. Turning to clique-width, we prove that Capacitated Vertex Cover remains NP-hard already on graphs of linear clique-width $6$...

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arXiv2026

d-QBF with Few Existential Variables Revisited

Quantified Boolean Formula (QBF) is a notoriously hard generalization of \textsc{SAT}, especially from the point of view of parameterized complexity, where the problem remains intractable for most standard parameters. A recent work by Eriksson et al.~[IJCAI 24] addressed this by considering the case where the propositional part of the formula is in CNF and we parameterize by the number $k$ of existentially quantified variables. One of their main results was that this natural (but so far overlooked) parameter does lead to fixed-parameter tractability, if we also bound the maximum arity $d$ of the clauses of the given CNF. Unfortunately, their algorithm has a \emph{double-exponential} dependence on $k$ ($2^{2^k}$), even when $d$ is an absolute constant. Since the work of Eriksson et al.\ only complemented this with a SETH-based lower bound implying that a $2^{O(k)}$ dependence is impossible, this left a large gap as an open question. Our main result in this paper is to close this gap by showing that the double-exponential dependence is optimal, assuming the ETH: even for CNFs of arity $4$, QBF with $k$ existential variables cannot be solved in time $2^{2^{o(k)}}|\phi|^{O(1)}$. Complementing this, we also consider the further restricted case of QBF with only two quantifier blocks ($\forall\exists$-QBF). We show that in this case the situation improves dramatically: for each $d\ge 3$ we show an algorithm with running time $k^{O_d(k ^{d-1})}|\phi|^{O(1)}$ and a lower bound under the ETH showing our algorithm is almost optimal.

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arXiv2025

k-SUM Hardness Implies Treewidth-SETH

We show that if k-SUM is hard, in the sense that the standard algorithm is essentially optimal, then a variant of the SETH called the Primal Treewidth SETH is true. Formally: if there is an $\varepsilon>0$ and an algorithm which solves SAT in time $(2-\varepsilon)^{tw}|\phi|^{O(1)}$, where $tw$ is the width of a given tree decomposition of the primal graph of the input, then there exists a randomized algorithm which solves k-SUM in time $n^{(1-\delta)\frac{k}{2}}$ for some $\delta>0$ and all sufficiently large $k$. We also establish an analogous result for the k-XOR problem, where integer addition is replaced by component-wise addition modulo $2$. As an application of our reduction we are able to revisit tight lower bounds on the complexity of several fundamental problems parameterized by treewidth (Independent Set, Max Cut, $k$-Coloring). Our results imply that these bounds, which were initially shown under the SETH, also hold if one assumes the k-SUM or k-XOR Hypotheses, arguably increasing our confidence in their validity.

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arXiv2025

Forcing a unique minimum spanning tree and a unique shortest path

A forcing set $S$ in a combinatorial problem is a set of elements such that there is a unique solution that contains all the elements in $S$. An anti-forcing set is the symmetric concept: a set $S$ of elements is called an anti-forcing set if there is a unique solution disjoint from $S$. There are extensive studies on the computational complexity of finding a minimum forcing set in various combinatorial problems, and the known results indicate that many problems are harder than their classical counterparts: the decision version of finding a minimum forcing set for perfect matchings is NP-complete [Adams et al., Discrete Mathematics, 2004], and that of finding a minimum forcing set for satisfying assignments for 3CNF formulas is $\Sigma_2^P$-complete [Hatami-Maserrat, Discrete Applied Mathematics, 2005]. In this paper, we investigate the complexity of finding minimum forcing and anti-forcing sets for the shortest $s$-$t$ path problem and the minimum-weight spanning tree problem. We show that, unlike the aforementioned results, these problems are tractable, with the exception of the decision version of finding a minimum anti-forcing set for shortest $s$-$t$ paths, which is NP-complete. To complement this intractability result, we design fixed-parameter tractable algorithms for finding a minimum anti-forcing set for shortest $s$-$t$ paths.

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arXiv2025

Structural Parameters for Steiner Orientation

We consider the \textsc{Steiner Orientation} problem, where we are given as input a mixed graph $G=(V,E,A)$ and a set of $k$ demand pairs $(s_i,t_i)$, $i\in[k]$. The goal is to orient the undirected edges of $G$ in a way that the resulting directed graph has a directed path from $s_i$ to $t_i$ for all $i\in[k]$. We adopt the point of view of structural parameterized complexity and investigate the complexity of \textsc{Steiner Orientation} for standard measures, such as treewidth. Our results indicate that \textsc{Steiner Orientation} is a surprisingly hard problem from this point of view. In particular, our main contributions are the following: (1) We show that \textsc{Steiner Orientation} is NP-complete on instances where the underlying graph has feedback vertex number 2, treewidth 2, pathwidth 3, and vertex integrity 6; (2) We present an XP algorithm parameterized by vertex cover number $\mathrm{vc}$ of complexity $n^{\mathcal{O}(\mathrm{vc}^2)}$. Furthermore, we show that this running time is essentially optimal by proving that a running time of $n^{o(\mathrm{vc}^2)}$ would refute the ETH; (3) We consider parameterizations by the number of undirected or directed edges ($|E|$ or $|A|$) and we observe that the trivial $2^{|E|}n^{\mathcal{O}(1)}$-time algorithm for the former parameter is optimal under the SETH. Complementing this, we show that the problem admits a $2^{\mathcal{O}(|A|)}n^{\mathcal{O}(1)}$-time algorithm. In addition to the above, we consider the complexity of \textsc{Steiner Orientation} parameterized by $\mathrm{tw}+k$ (FPT), distance to clique (FPT), and $\mathrm{vc}+k$ (FPT with a polynomial kernel).

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arXiv2025

Faster algorithms for k-Orthogonal Vectors in low dimension

In the Orthogonal Vectors problem (OV), we are given two families $A, B$ of subsets of $\{1,\ldots,d\}$, each of size $n$, and the task is to decide whether there exists a pair $a \in A$ and $b \in B$ such that $a \cap b = \emptyset$. Straightforward algorithms for this problem run in $\mathcal{O}(n^2 \cdot d)$ or $\mathcal{O}(2^d \cdot n)$ time, and assuming SETH, there is no $2^{o(d)}\cdot n^{2-\varepsilon}$ time algorithm that solves this problem for any constant $\varepsilon > 0$. Williams (FOCS 2024) presented a $\tilde{\mathcal{O}}(1.35^d \cdot n)$-time algorithm for the problem, based on the succinct equality-rank decomposition of the disjointness matrix. In this paper, we present a combinatorial algorithm that runs in randomized time $\tilde{\mathcal{O}}(1.25^d n)$. This can be improved to $\mathcal{O}(1.16^d \cdot n)$ using computer-aided evaluations. We generalize our result to the $k$-Orthogonal Vectors problem, where given $k$ families $A_1,\ldots,A_k$ of subsets of $\{1,\ldots,d\}$, each of size $n$, the task is to find elements $a_i \in A_i$ for every $i \in \{1,\ldots,k\}$ such that $a_1 \cap a_2 \cap \ldots \cap a_k = \emptyset$. We show that for every fixed $k \ge 2$, there exists $\varepsilon_k > 0$ such that the $k$-OV problem can be solved in time $\mathcal{O}(2^{(1 - \varepsilon_k)\cdot d}\cdot n)$. We also show that, asymptotically, this is the best we can hope for: for any $\varepsilon > 0$ there exists a $k \ge 2$ such that $2^{(1 - \varepsilon)\cdot d} \cdot n^{\mathcal{O}(1)}$ time algorithm for $k$-Orthogonal Vectors would contradict the Set Cover Conjecture.

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arXiv2025

Broadcasting under Structural Restrictions

In the Telephone Broadcast problem we are given a graph $G=(V,E)$ with a designated source vertex $s\in V$. Our goal is to transmit a message, which is initially known only to $s$, to all vertices of the graph by using a process where in each round an informed vertex may transmit the message to one of its uninformed neighbors. The optimization objective is to minimize the number of rounds. Following up on several recent works, we investigate the structurally parameterized complexity of Telephone Broadcast. In particular, we first strengthen existing NP-hardness results by showing that the problem remains NP-complete on graphs of bounded tree-depth and also on cactus graphs which are one vertex deletion away from being path forests. Motivated by this (severe) hardness, we study several other parameterizations of the problem and obtain FPT algorithms parameterized by vertex integrity (generalizing a recent FPT algorithm parameterized by vertex cover by Fomin, Fraigniaud, and Golovach [TCS 2024]) and by distance to clique, as well as FPT approximation algorithms parameterized by clique-cover and cluster vertex deletion. Furthermore, we obtain structural results that relate the length of the optimal broadcast protocol of a graph $G$ with its pathwidth and tree-depth. By presenting a substantial improvement over the best previously known bound for pathwidth (Aminian, Kamali, Seyed-Javadi, and Sumedha [arXiv 2025]) we exponentially improve the approximation ratio achievable in polynomial time on graphs of bounded pathwidth from $\mathcal{O}(4^\mathrm{pw})$ to $\mathcal{O}(\mathrm{pw})$.

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arXiv2025

Structural Parameterizations for Induced and Acyclic Matching

We revisit the (structurally) parameterized complexity of Induced Matching and Acyclic Matching, two problems where we seek to find a maximum independent set of edges whose endpoints induce, respectively, a matching and a forest. Chaudhary and Zehavi [WG '23] recently studied these problems parameterized by treewidth, denoted by $\mathrm{tw}$. We resolve several of the problems left open in their work and extend their results as follows: (i) for Acyclic Matching, Chaudhary and Zehavi gave an algorithm of running time $6^{\mathrm{tw}}n^{\mathcal{O}(1)}$ and a lower bound of $(3-\varepsilon)^{\mathrm{tw}}n^{\mathcal{O}(1)}$ (under the SETH); we close this gap by, on the one hand giving a more careful analysis of their algorithm showing that its complexity is actually $5^{\mathrm{tw}} n^{\mathcal{O}(1)}$, and on the other giving a pw-SETH-based lower bound showing that this running time cannot be improved (even for pathwidth), (ii) for Induced Matching we show that their $3^{\mathrm{tw}} n^{\mathcal{O}(1)}$ algorithm is optimal under the pw-SETH (in fact improving over this for pathwidth or even for cutwidth is equivalent to falsifying the pw-SETH) by adapting a recent reduction for Bounded Degree Vertex Deletion, (iii) for both problems we give FPT algorithms with single-exponential dependence when parameterized by clique-width and in particular for Induced Matching our algorithm has running time $3^{\mathrm{cw}} n^{\mathcal{O}(1)}$, which is optimal under the pw-SETH from our previous result.

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arXiv2025

Satisfactory Budget Division

A divisible budget must be allocated to several projects, and agents are asked for their opinion on how much they would give to each project. We consider that an agent is satisfied by a division of the budget if, for at least a certain predefined number $\tau$ of projects, the part of the budget actually allocated to each project is at least as large as the amount the agent requested. The objective is to find a budget division that ``best satisfies'' the agents. In this context, different problems can be stated and we address the following ones. We study $(i)$ the largest proportion of agents that can be satisfied for any instance, $(ii)$ classes of instances admitting a budget division that satisfies all agents, $(iii)$ the complexity of deciding if, for a given instance, every agent can be satisfied, and finally $(iv)$ the question of finding, for a given instance, the smallest total budget to satisfy all agents. We provide answers to these complementary questions for several natural values of the parameter $\tau$, capturing scenarios where we seek to satisfy for each agent all; almost all; half; or at least one of her requests.

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arXiv2024

On the Tractability Landscape of the Conditional Minisum Approval Voting Rule

This work examines the Conditional Approval Framework for elections involving multiple interdependent issues, specifically focusing on the Conditional Minisum Approval Voting Rule. We first conduct a detailed analysis of the computational complexity of this rule, demonstrating that no approach can significantly outperform the brute-force algorithm under common computational complexity assumptions and various natural input restrictions. In response, we propose two practical restrictions (the first in the literature) that make the problem computationally tractable and show that these restrictions are essentially tight. Overall, this work provides a clear picture of the tractability landscape of the problem, contributing to a comprehensive understanding of the complications introduced by conditional ballots and indicating that conditional approval voting can be applied in practice, albeit under specific conditions.

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arXiv2024

Parameterized Spanning Tree Congestion

In this paper we study the Spanning Tree Congestion problem, where we are given a graph $G=(V,E)$ and are asked to find a spanning tree $T$ of minimum maximum congestion. Here, the congestion of an edge $e\in T$ is the number of edges $uv\in E$ such that the (unique) path from $u$ to $v$ in $T$ traverses $e$. We consider this well-studied NP-hard problem from the point of view of (structural) parameterized complexity and obtain the following results. We resolve a natural open problem by showing that Spanning Tree Congestion is not FPT parameterized by treewidth (under standard assumptions). More strongly, we present a generic reduction which applies to (almost) any parameter of the form ``vertex-deletion distance to class $\mathcal{C}$'', thus obtaining W[1]-hardness for parameters more restricted than treewidth, including tree-depth plus feedback vertex set, or incomparable to treewidth, such as twin cover. Via a slight tweak of the same reduction we also show that the problem is NP-complete on interval graphs of modular-width $4$. Even though it is known that Spanning Tree Congestion remains NP-hard on instances with only one vertex of unbounded degree, it is currently open whether the problem remains hard on bounded-degree graphs. We resolve this question by showing NP-hardness on graphs of maximum degree 8. Complementing the problem's W[1]-hardness for treewidth...

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arXiv2024

Circuits and Backdoors: Five Shades of the SETH

The Strong Exponential Time Hypothesis (SETH) is a standard assumption in (fine-grained) parameterized complexity and many tight lower bounds are based on it. We consider a number of reasonable weakenings of the SETH, with sources from (i) circuit complexity (ii) backdoors for SAT-solving (iii) graph width parameters and (iv) weighted satisfiability problems. Our goal is to arrive at formulations which are simultaneously more plausible as hypotheses, but also capture interesting and robust notions of complexity. Using several tools from classical complexity theory we are able to consolidate these numerous hypotheses into a hierarchy of five main equivalence classes of increasing solidity. This framework serves as a step towards structurally classifying a variety of SETH-based lower bounds into intermediate equivalence classes. To illustrate the applicability of our framework, for each of our classes we give at least one (non-SAT) problem which is equivalent to the class as a characteristic example application. As our main showcase, we consider a natural parameterization of Independent Set by vertex deletion distance from several standard graph classes. We provide precise characterizations of the difficulty of breaking such bounds, in particular proving that obtaining $(2-\varepsilon)^kn^{O(1)}$ time algorithms for Cograph$+kv$ or Block$+kv$ graphs is equivalent to obtaining a fast satisfiability algorithm for circuits of depth $\varepsilon n$; while solving the weighted version for Interval$+kv$ graphs is equivalent to the (seemingly) harder problem of obtaining a fast satisfiability algorithm for SAT parameterized by a 2-SAT backdoor.

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arXiv2024

Parameterized Maximum Node-Disjoint Paths

We revisit the Maximum Node-Disjoint Paths problem, the natural optimization version of Node-Disjoint Paths, where we are given a graph $G$, $k$ pairs of vertices $(s_i, t_i)$ and an integer $\ell$, and are asked whether there exist at least $\ell$ vertex-disjoint paths in $G$ whose endpoints are given pairs. We present several results, with an emphasis towards FPT approximation. Our main positive contribution is to show that the problem's intractability can be overcome using approximation and that for several of the structural parameters for which the problem is hard, most notably tree-depth, it admits an efficient FPT approximation scheme, returning a $(1-\varepsilon)$-approximate solution in time $f(td,\varepsilon)n^{O(1)}$. We manage to obtain these results by comprehensively mapping out the structural parameters for which the problem is FPT if $\ell$ is also a parameter, hence showing that understanding $\ell$ as a parameter is key to the problem's approximability. This, in turn, is a problem we are able to solve via a surprisingly simple color-coding algorithm, which relies on identifying an insightful problem-specific variant of the natural parameter, namely the number of vertices used in the solution. A natural question is whether the FPT approximation algorithm we devised for tree-depth can be extended to pathwidth. We resolve this negatively, showing that under the Parameterized Inapproximability Hypothesis no FPT approximation scheme for this parameter is possible, even in time $f(pw,\varepsilon)n^{g(\varepsilon)}$, thus precisely determining the parameter border where the problem transitions from ``hard but approximable'' to ``inapproximable''. Lastly, we strengthen existing lower bounds by replacing W[1]-hardness by XNLP-completeness for parameter pathwidth, and improving the $n^{o(\sqrt{td})}$ ETH-based lower bound for tree-depth to $n^{o(td)}$.

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arXiv2024

The Primal Pathwidth SETH

Motivated by the importance of dynamic programming (DP) in parameterized complexity, we consider several fine-grained questions, such as the following examples: (i) can Dominating Set be solved in time $(3-\epsilon)^{pw}n^{O(1)}$? (where $pw$ is the pathwidth) (ii) can Coloring be solved in time $pw^{(1-\epsilon)pw}n^{O(1)}$? (iii) can a short reconfiguration between two size-$k$ independent sets be found in time $n^{(1-\epsilon)k}$? Such questions are well-studied: in some cases the answer is No under the SETH, while in others coarse-grained lower bounds are known under the ETH. Even though questions such as the above seem "morally equivalent" as they all ask if a simple DP can be improved, the problems concerned have wildly varying time complexities, ranging from single-exponential FPT to XNLP-complete. This paper's main contribution is to show that, despite their varying complexities, these questions are not just morally equivalent, but in fact they are the same question in disguise. We achieve this by putting forth a natural complexity assumption which we call the Primal Pathwidth-Strong Exponential Time Hypothesis (pw-SETH) and which states that 3-SAT cannot be solved in time $(2-\epsilon)^{pw}n^{O(1)}$, for any $\epsilon>0$, where $pw$ is the pathwidth of the primal graph of the input. We then show that numerous fine-grained questions in parameterized complexity, including the ones above, are equivalent to the pw-SETH, and hence to each other. This allows us to obtain sharp fine-grained lower bounds for problems for which previous lower bounds left a constant in the exponent undetermined, but also to increase our confidence in bounds which were previously known under the SETH, because we show that breaking any one such bound requires breaking all (old and new) bounds; and because we show that the pw-SETH is more plausible than the SETH.

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arXiv2024

Core Stability in Additively Separable Hedonic Games of Low Treewidth

Additively Separable Hedonic Game (ASHG) are coalition-formation games where we are given a graph whose vertices represent $n$ selfish agents and the weight of each edge $uv$ denotes how much agent $u$ gains (or loses) when she is placed in the same coalition as agent $v$. We revisit the computational complexity of the well-known notion of core stability of ASHGs, where the goal is to construct a partition of the agents into coalitions such that no group of agents would prefer to diverge from the given partition and form a new (blocking) coalition. Since both finding a core stable partition and verifying that a given partition is core stable are intractable problems ($\Sigma_2^p$-complete and coNP-complete respectively) we study their complexity from the point of view of structural parameterized complexity, using standard graph-theoretic parameters, such as treewidth.

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arXiv2024

Parameterized Algorithms for Steiner Forest in Bounded Width Graphs

In this paper we reassess the parameterized complexity and approximability of the well-studied Steiner Forest problem in several graph classes of bounded width. The problem takes an edge-weighted graph and pairs of vertices as input, and the aim is to find a minimum cost subgraph in which each given vertex pair lies in the same connected component. It is known that this problem is APX-hard in general, and NP-hard on graphs of treewidth 3, treedepth 4, and feedback vertex set size 2. However, Bateni, Hajiaghayi and Marx [JACM, 2011] gave an approximation scheme with a runtime of $n^{O(\frac{k^2}{\varepsilon})}$ on graphs of treewidth $k$. Our main result is a much faster efficient parameterized approximation scheme (EPAS) with a runtime of $2^{O(\frac{k^2}{\varepsilon} \log \frac{k^2}{\varepsilon})} \cdot n^{O(1)}$. If $k$ instead is the vertex cover number of the input graph, we show how to compute the optimum solution in $2^{O(k \log k)} \cdot n^{O(1)}$ time, and we also prove that this runtime dependence on $k$ is asymptotically best possible, under ETH. Furthermore, if $k$ is the size of a feedback edge set, then we obtain a faster $2^{O(k)} \cdot n^{O(1)}$ time algorithm, which again cannot be improved under ETH.

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arXiv2024

Parameterized Vertex Integrity Revisited

Vertex integrity is a graph parameter that measures the connectivity of a graph. Informally, its meaning is that a graph has small vertex integrity if it has a small separator whose removal disconnects the graph into connected components which are themselves also small. Graphs with low vertex integrity are extremely structured; this renders many hard problems tractable and has recently attracted interest in this notion from the parameterized complexity community. In this paper we revisit the NP-complete problem of computing the vertex integrity of a given graph from the point of view of structural parameterizations. We present a number of new results, which also answer some recently posed open questions from the literature. Specifically: We show that unweighted vertex integrity is W[1]-hard parameterized by treedepth; we show that the problem remains W[1]-hard if we parameterize by feedback edge set size (via a reduction from a Bin Packing variant which may be of independent interest); and complementing this we show that the problem is FPT by max-leaf number. Furthermore, for weighted vertex integrity, we show that the problem admits a single-exponential FPT algorithm parameterized by vertex cover or by modular width, the latter result improving upon a previous algorithm which required weights to be polynomially bounded.

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