arXiv subjects

Michael Lampis

Publications and source records attributed to Michael Lampis.

At least 37 records · Page 2Linked to original sources

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 $(3-ε)^{pw}n^{O(1)}$? (where $pw$ is the pathwidth) (ii) can Coloring be solved in time $pw^{(1-ε)pw}n^{O(1)}$? (iii) can a short reconfiguration between two size-$k$ independent sets be found in time $n^{(1-ε)k}$? 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 $(2-ε)^{pw}n^{O(1)}$, for any $ε>0$, 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

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

Structural Parameterizations for Two Bounded Degree Problems Revisited

We revisit two well-studied problems, Bounded Degree Vertex Deletion and Defective Coloring, where the input is a graph $G$ and a target degree $Δ$ and we are asked either to edit or partition the graph so that the maximum degree becomes bounded by $Δ$. Both are known to be parameterized intractable for treewidth. We revisit the parameterization by treewidth, as well as several related parameters and present a more fine-grained picture of the complexity of both problems. Both admit straightforward DP algorithms with table sizes $(Δ+2)^\mathrm{tw}$ and $(χ_\mathrm{d}(Δ+1))^{\mathrm{tw}}$ respectively, where tw is the input graph's treewidth and $χ_\mathrm{d}$ the number of available colors. We show that both algorithms are optimal under SETH, even if we replace treewidth by pathwidth. Along the way, we also obtain an algorithm for Defective Coloring with complexity quasi-linear in the table size, thus settling the complexity of both problems for these parameters. We then consider the more restricted parameter tree-depth, and bridge the gap left by known lower bounds, by showing that neither problem can be solved in time $n^{o(\mathrm{td})}$ under ETH. In order to do so, we employ a recursive low tree-depth construction that may be of independent interest. Finally, we show that for both problems, an $\mathrm{vc}^{o(\mathrm{vc})}$ algorithm would violate ETH, thus already known algorithms are optimal. Our proof relies on a new application of the technique of $d$-detecting families introduced by Bonamy et al. Our results, although mostly negative in nature, paint a clear picture regarding the complexity of both problems in the landscape of parameterized complexity, since in all cases we provide essentially matching upper and lower bounds.

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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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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 ($Σ_2^p$-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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arXiv2023

New Results on Directed Edge Dominating Set

We study a family of generalizations of Edge Dominating Set on directed graphs called Directed $(p,q)$-Edge Dominating Set. In this problem an arc $(u,v)$ is said to dominate itself, as well as all arcs which are at distance at most $q$ from $v$, or at distance at most $p$ to $u$. First, we give significantly improved FPT algorithms for the two most important cases of the problem, $(0,1)$-dEDS and $(1,1)$-dEDS (that correspond to versions of Dominating Set on line graphs), as well as polynomial kernels. We also improve the best-known approximation for these cases from logarithmic to constant. In addition, we show that $(p,q)$-dEDS is FPT parameterized by $p+q+tw$, but W-hard parameterized by $tw$ (even if the size of the optimal is added as a second parameter), where $tw$ is the treewidth of the underlying graph of the input. We then go on to focus on the complexity of the problem on tournaments. Here, we provide a complete classification for every possible fixed value of $p,q$, which shows that the problem exhibits a surprising behavior, including cases which are in P; cases which are solvable in quasi-polynomial time but not in P; and a single case $(p=q=1)$ which is NP-hard (under randomized reductions) and cannot be solved in sub-exponential time, under standard assumptions.

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arXiv2022

Grundy Distinguishes Treewidth from Pathwidth

Structural graph parameters, such as treewidth, pathwidth, and clique-width, are a central topic of study in parameterized complexity. A main aim of research in this area is to understand the "price of generality" of these widths: as we transition from more restrictive to more general notions, which are the problems that see their complexity status deteriorate from fixed-parameter tractable to intractable? This type of question is by now very well-studied, but, somewhat strikingly, the algorithmic frontier between the two (arguably) most central width notions, treewidth and pathwidth, is still not understood: currently, no natural graph problem is known to be W-hard for one but FPT for the other. Indeed, a surprising development of the last few years has been the observation that for many of the most paradigmatic problems, their complexities for the two parameters actually coincide exactly, despite the fact that treewidth is a much more general parameter. It would thus appear that the extra generality of treewidth over pathwidth often comes "for free". Our main contribution in this paper is to uncover the first natural example where this generality comes with a high price. We consider Grundy Coloring, a variation of coloring where one seeks to calculate the worst possible coloring that could be assigned to a graph by a greedy First-Fit algorithm. We show that this well-studied problem is FPT parameterized by pathwidth; however, it becomes significantly harder (W[1]-hard) when parameterized by treewidth. Furthermore, we show that Grundy Coloring makes a second complexity jump for more general widths, as it becomes para-NP-hard for clique-width. Hence, Grundy Coloring nicely captures the complexity trade-offs between the three most well-studied parameters. Completing the picture, we show that Grundy Coloring is FPT parameterized by modular-width.

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arXiv2022

Hedonic Games and Treewidth Revisited

We revisit the complexity of the well-studied notion of Additively Separable Hedonic Games (ASHGs). Such games model a basic clustering or coalition formation scenario in which selfish agents are represented by the vertices of an edge-weighted digraph $G=(V,E)$, and the weight of an arc $uv$ denotes the utility $u$ gains by being in the same coalition as $v$. We focus on (arguably) the most basic stability question about such a game: given a graph, does a Nash stable solution exist and can we find it efficiently? We study the (parameterized) complexity of ASHG stability when the underlying graph has treewidth $t$ and maximum degree $Δ$. The current best FPT algorithm for this case was claimed by Peters [AAAI 2016], with time complexity roughly $2^{O(Δ^5t)}$. We present an algorithm with parameter dependence $(Δt)^{O(Δt)}$, significantly improving upon the parameter dependence on $Δ$ given by Peters, albeit with a slightly worse dependence on $t$. Our main result is that this slight performance deterioration with respect to $t$ is actually completely justified: we observe that the previously claimed algorithm is incorrect, and that in fact no algorithm can achieve dependence $t^{o(t)}$ for bounded-degree graphs, unless the ETH fails. This, together with corresponding bounds we provide on the dependence on $Δ$ and the joint parameter establishes that our algorithm is essentially optimal for both parameters, under the ETH. We then revisit the parameterization by treewidth alone and resolve a question also posed by Peters by showing that Nash Stability remains strongly NP-hard on stars under additive preferences. Nevertheless, we also discover an island of mild tractability: we show that Connected Nash Stability is solvable in pseudo-polynomial time for constant $t$, though with an XP dependence on $t$ which, as we establish, cannot be avoided.

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arXiv2022

Defective Coloring on Classes of Perfect Graphs

In Defective Coloring we are given a graph $G$ and two integers $χ_d$, $Δ^*$ and are asked if we can $χ_d$-color $G$ so that the maximum degree induced by any color class is at most $Δ^*$. We show that this natural generalization of Coloring is much harder on several basic graph classes. In particular, we show that it is NP-hard on split graphs, even when one of the two parameters $χ_d$, $Δ^*$ is set to the smallest possible fixed value that does not trivialize the problem ($χ_d = 2$ or $Δ^* = 1$). Together with a simple treewidth-based DP algorithm this completely determines the complexity of the problem also on chordal graphs. We then consider the case of cographs and show that, somewhat surprisingly, Defective Coloring turns out to be one of the few natural problems which are NP-hard on this class. We complement this negative result by showing that Defective Coloring is in P for cographs if either $χ_d$ or $Δ^*$ is fixed; that it is in P for trivially perfect graphs; and that it admits a sub-exponential time algorithm for cographs when both $χ_d$ and $Δ^*$ are unbounded.

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arXiv2022

Digraph Coloring and Distance to Acyclicity

In $k$-Digraph Coloring we are given a digraph and are asked to partition its vertices into at most $k$ sets, so that each set induces a DAG. This well-known problem is NP-hard, as it generalizes (undirected) $k$-Coloring, but becomes trivial if the input digraph is acyclic. This poses the natural parameterized complexity question what happens when the input is "almost" acyclic. In this paper we study this question using parameters that measure the input's distance to acyclicity in either the directed or the undirected sense. It is already known that, for all $k\ge 2$, $k$-Digraph Coloring is NP-hard on digraphs of DFVS at most $k+4$. We strengthen this result to show that, for all $k\ge 2$, $k$-Digraph Coloring is NP-hard for DFVS $k$. Refining our reduction we obtain two further consequences: (i) for all $k\ge 2$, $k$-Digraph Coloring is NP-hard for graphs of feedback arc set (FAS) at most $k^2$; interestingly, this leads to a dichotomy, as we show that the problem is FPT by $k$ if FAS is at most $k^2-1$; (ii) $k$-Digraph Coloring is NP-hard for graphs of DFVS $k$, even if the maximum degree $Δ$ is at most $4k-1$; we show that this is also almost tight, as the problem becomes FPT for DFVS $k$ and $Δ\le 4k-3$. We then consider parameters that measure the distance from acyclicity of the underlying graph. We show that $k$-Digraph Coloring admits an FPT algorithm parameterized by treewidth, whose parameter dependence is $(tw!)k^{tw}$. Then, we pose the question of whether the $tw!$ factor can be eliminated. Our main contribution in this part is to settle this question in the negative and show that our algorithm is essentially optimal, even for the much more restricted parameter treedepth and for $k=2$. Specifically, we show that an FPT algorithm solving $2$-Digraph Coloring with dependence $td^{o(td)}$ would contradict the ETH.

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arXiv2021

Determining a Slater Winner is Complete for Parallel Access to NP

We consider the complexity of deciding the winner of an election under the Slater rule. In this setting we are given a tournament $T = (V, A)$, where the vertices of V represent candidates and the direction of each arc indicates which of the two endpoints is preferable for the majority of voters. The Slater score of a vertex $v\in V$ is defined as the minimum number of arcs that need to be reversed so that $T$ becomes acyclic and $v$ becomes the winner. We say that $v$ is a Slater winner in $T$ if $v$ has minimum Slater score in $T$. Deciding if a vertex is a Slater winner in a tournament has long been known to be NP-hard. However, the best known complexity upper bound for this problem is the class $Θ_2^p$, which corresponds to polynomial-time Turing machines with parallel access to an NP oracle. In this paper we close this gap by showing that the problem is $Θ_2^p$-complete, and that this hardness applies to instances constructible by aggregating the preferences of 7 voters.

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arXiv2021

Filling Crosswords is Very Hard

We revisit a classical crossword filling puzzle which already appeared in Garey\&Jonhson's book. We are given a grid with $n$ vertical and horizontal slots and a dictionary with $m$ words and are asked to place words from the dictionary in the slots so that shared cells are consistent. We attempt to pinpoint the source of intractability of this problem by taking into account the structure of the grid graph, which contains a vertex for each slot and an edge if two slots intersect. Our main approach is to consider the case where this graph has a tree-like structure. Unfortunately, if we impose the common rule that words cannot be reused, we show that the problem remains NP-hard even under very severe structural restrictions. The problem becomes slightly more tractable if word reuse is allowed, as we obtain an $m^{tw}$ algorithm in this case, where $tw$ is the treewidth of the grid graph. However, even in this case, we show that our algorithm cannot be improved. More strongly, we show that under the ETH the problem cannot be solved in time $m^{o(k)}$, where $k$ is the number of horizontal slots of the instance. Motivated by these mostly negative results, we consider the much more restricted case where the problem is parameterized by the number of slots $n$. Here, we show that the problem becomes FPT, but the parameter dependence is exponential in $n^2$. We show that this dependence is also justified: the existence of an algorithm with running time $2^{o(n^2)}$ would contradict the randomized ETH. Finally, we consider an optimization version of the problem, where we seek to place as many words on the grid as possible. Here it is easy to obtain a $\frac{1}{2}$-approximation, even on weighted instances. We show that this algorithm is also likely to be optimal, as obtaining a better approximation ratio in polynomial time would contradict the Unique Games Conjecture.

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arXiv2021

New Algorithms for Mixed Dominating Set

A mixed dominating set is a collection of vertices and edges that dominates all vertices and edges of a graph. We study the complexity of exact and parameterized algorithms for \textsc{Mixed Dominating Set}, resolving some open questions. In particular, we settle the problem's complexity parameterized by treewidth and pathwidth by giving an algorithm running in time $O^*(5^{tw})$ (improving the current best $O^*(6^{tw})$), as well as a lower bound showing that our algorithm cannot be improved under the Strong Exponential Time Hypothesis (SETH), even if parameterized by pathwidth (improving a lower bound of $O^*((2 - \varepsilon)^{pw})$). Furthermore, by using a simple but so far overlooked observation on the structure of minimal solutions, we obtain branching algorithms which improve both the best known FPT algorithm for this problem, from $O^*(4.172^k)$ to $O^*(3.510^k)$, and the best known exponential-time exact algorithm, from $O^*(2^n)$ and exponential space, to $O^*(1.912^n)$ and polynomial space.

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arXiv2021

(In)approximability of Maximum Minimal FVS

We study the approximability of the NP-complete \textsc{Maximum Minimal Feedback Vertex Set} problem. Informally, this natural problem seems to lie in an intermediate space between two more well-studied problems of this type: \textsc{Maximum Minimal Vertex Cover}, for which the best achievable approximation ratio is $\sqrt{n}$, and \textsc{Upper Dominating Set}, which does not admit any $n^{1-ε}$ approximation. We confirm and quantify this intuition by showing the first non-trivial polynomial time approximation for \textsc{Max Min FVS} with a ratio of $O(n^{2/3})$, as well as a matching hardness of approximation bound of $n^{2/3-ε}$, improving the previous known hardness of $n^{1/2-ε}$. The approximation algorithm also gives a cubic kernel when parameterized by the solution size. Along the way, we also obtain an $O(Δ)$-approximation and show that this is asymptotically best possible, and we improve the bound for which the problem is NP-hard from $Δ\ge 9$ to $Δ\ge 6$. Having settled the problem's approximability in polynomial time, we move to the context of super-polynomial time. We devise a generalization of our approximation algorithm which, for any desired approximation ratio $r$, produces an $r$-approximate solution in time $n^{O(n/r^{3/2})}$. This time-approximation trade-off is essentially tight: we show that under the ETH, for any ratio $r$ and $ε>0$, no algorithm can $r$-approximate this problem in time $n^{O((n/r^{3/2})^{1-ε})}$, hence we precisely characterize the approximability of the problem for the whole spectrum between polynomial and sub-exponential time, up to an arbitrarily small constant in the second exponent.

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arXiv2021

Upper Dominating Set: Tight Algorithms for Pathwidth and Sub-Exponential Approximation

An upper dominating set is a minimal dominating set in a graph. In the \textsc{Upper Dominating Set} problem, the goal is to find an upper dominating set of maximum size. We study the complexity of parameterized algorithms for \textsc{Upper Dominating Set}, as well as its sub-exponential approximation. First, we prove that, under ETH, \textsc{$k$-Upper Dominating Set} cannot be solved in time $O(n^{o(k)})$ (improving on $O(n^{o(\sqrt{k})})$), and in the same time we show under the same complexity assumption that for any constant ratio $r$ and any $\varepsilon > 0$, there is no $r$-approximation algorithm running in time $O(n^{k^{1-\varepsilon}})$. Then, we settle the problem's complexity parameterized by pathwidth by giving an algorithm running in time $O^*(6^{pw})$ (improving the current best $O^*(7^{pw})$), and a lower bound showing that our algorithm is the best we can get under the SETH. Furthermore, we obtain a simple sub-exponential approximation algorithm for this problem: an algorithm that produces an $r$-approximation in time $n^{O(n/r)}$, for any desired approximation ratio $r < n$. We finally show that this time-approximation trade-off is tight, up to an arbitrarily small constant in the second exponent: under the randomized ETH, and for any ratio $r > 1$ and $\varepsilon > 0$, no algorithm can output an $r$-approximation in time $n^{(n/r)^{1-\varepsilon}}$. Hence, we completely characterize the approximability of the problem in sub-exponential time.

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arXiv2020

Parameterized Complexity of $(A,\ell)$-Path Packing

Given a graph $G = (V,E)$, $A \subseteq V$, and integers $k$ and $\ell$, the \textsc{$(A,\ell)$-Path Packing} problem asks to find $k$ vertex-disjoint paths of length $\ell$ that have endpoints in $A$ and internal points in $V \setminus A$. We study the parameterized complexity of this problem with parameters $|A|$, $\ell$, $k$, treewidth, pathwidth, and their combinations. We present sharp complexity contrasts with respect to these parameters. Among other results, we show that the problem is polynomial-time solvable when $\ell \le 3$, while it is NP-complete for constant $\ell \ge 4$. We also show that the problem is W[1]-hard parameterized by pathwidth${}+|A|$, while it is fixed-parameter tractable parameterized by treewidth${}+\ell$.

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arXiv2020

Parameterized Orientable Deletion

A graph is $d$-orientable if its edges can be oriented so that the maximum in-degree of the resulting digraph is at most $d$. $d$-orientability is a well-studied concept with close connections to fundamental graph-theoretic notions and applications as a load balancing problem. In this paper we consider the d-ORIENTABLE DELETION problem: given a graph $G=(V,E)$, delete the minimum number of vertices to make $G$ $d$-orientable. We contribute a number of results that improve the state of the art on this problem. Specifically: - We show that the problem is W[2]-hard and $\log n$-inapproximable with respect to $k$, the number of deleted vertices. This closes the gap in the problem's approximability. - We completely characterize the parameterized complexity of the problem on chordal graphs: it is FPT parameterized by $d+k$, but W-hard for each of the parameters $d,k$ separately. - We show that, under the SETH, for all $d,ε$, the problem does not admit a $(d+2-ε)^{tw}$, algorithm where $tw$ is the graph's treewidth, resolving as a special case an open problem on the complexity of PSEUDOFOREST DELETION. - We show that the problem is W-hard parameterized by the input graph's clique-width. Complementing this, we provide an algorithm running in time $d^{O(d\cdot cw)}$, showing that the problem is FPT by $d+cw$, and improving the previously best known algorithm for this case.

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arXiv2019

Improved (In-)Approximability Bounds for d-Scattered Set

In the $d$-Scattered Set problem we are asked to select at least $k$ vertices of a given graph, so that the distance between any pair is at least $d$. We study the problem's (in-)approximability and offer improvements and extensions of known results for Independent Set, of which the problem is a generalization. Specifically, we show: - A lower bound of $Δ^{\lfloor d/2\rfloor-ε}$ on the approximation ratio of any polynomial-time algorithm for graphs of maximum degree $Δ$ and an improved upper bound of $O(Δ^{\lfloor d/2\rfloor})$ on the approximation ratio of any greedy scheme for this problem. - A polynomial-time $2\sqrt{n}$-approximation for bipartite graphs and even values of $d$, that matches the known lower bound by considering the only remaining case. - A lower bound on the complexity of any $ρ$-approximation algorithm of (roughly) $2^{\frac{n^{1-ε}}{ρd}}$ for even $d$ and $2^{\frac{n^{1-ε}}{ρ(d+ρ)}}$ for odd $d$ (under the randomized ETH), complemented by $ρ$-approximation algorithms of running times that (almost) match these bounds.

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