arXiv subjects

Marcin Pilipczuk

Publications and source records attributed to Marcin Pilipczuk.

At least 55 records · Page 3Linked to original sources

arXiv2022

Proving a directed analogue of the Gyárfás-Sumner conjecture for orientations of $P_4$

An oriented graph is a digraph that does not contain a directed cycle of length two. An (oriented) graph $D$ is $H$-free if $D$ does not contain $H$ as an induced sub(di)graph. The Gyárfás-Sumner conjecture is a widely-open conjecture on simple graphs, which states that for any forest $F$, there is some function $f$ such that every $F$-free graph $G$ with clique number $ω(G)$ has chromatic number at most $f(ω(G))$. Aboulker, Charbit, and Naserasr [Extension of Gyárfás-Sumner Conjecture to Digraphs; E-JC 2021] proposed an analogue of this conjecture to the dichromatic number of oriented graphs. The dichromatic number of a digraph $D$ is the minimum number of colors required to color the vertex set of $D$ so that no directed cycle in $D$ is monochromatic. Aboulker, Charbit, and Naserasr's $\overrightarrowχ$-boundedness conjecture states that for every oriented forest $F$, there is some function $f$ such that every $F$-free oriented graph $D$ has dichromatic number at most $f(ω(D))$, where $ω(D)$ is the size of a maximum clique in the graph underlying $D$. In this paper, we perform the first step towards proving Aboulker, Charbit, and Naserasr's $\overrightarrowχ$-boundedness conjecture by showing that it holds when $F$ is any orientation of a path on four vertices.

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arXiv2022

On weighted graph separation problems and flow-augmentation

One of the first application of the recently introduced technique of \emph{flow-augmentation} [Kim et al., STOC 2022] is a fixed-parameter algorithm for the weighted version of \textsc{Directed Feedback Vertex Set}, a landmark problem in parameterized complexity. In this note we explore applicability of flow-augmentation to other weighted graph separation problems parameterized by the size of the cutset. We show the following. -- In weighted undirected graphs \textsc{Multicut} is FPT, both in the edge- and vertex-deletion version. -- The weighted version of \textsc{Group Feedback Vertex Set} is FPT, even with an oracle access to group operations. -- The weighted version of \textsc{Directed Subset Feedback Vertex Set} is FPT. Our study reveals \textsc{Directed Symmetric Multicut} as the next important graph separation problem whose parameterized complexity remains unknown, even in the unweighted setting.

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arXiv2022

Fixed-parameter tractability of Directed Multicut with three terminal pairs parameterized by the size of the cutset: twin-width meets flow-augmentation

We show fixed-parameter tractability of the Directed Multicut problem with three terminal pairs (with a randomized algorithm). This problem, given a directed graph $G$, pairs of vertices (called terminals) $(s_1,t_1)$, $(s_2,t_2)$, and $(s_3,t_3)$, and an integer $k$, asks to find a set of at most $k$ non-terminal vertices in $G$ that intersect all $s_1t_1$-paths, all $s_2t_2$-paths, and all $s_3t_3$-paths. The parameterized complexity of this case has been open since Chitnis, Cygan, Hajiaghayi, and Marx proved fixed-parameter tractability of the 2-terminal-pairs case at SODA 2012, and Pilipczuk and Wahlström proved the W[1]-hardness of the 4-terminal-pairs case at SODA 2016. On the technical side, we use two recent developments in parameterized algorithms. Using the technique of directed flow-augmentation [Kim, Kratsch, Pilipczuk, Wahlström, STOC 2022] we cast the problem as a CSP problem with few variables and constraints over a large ordered domain.We observe that this problem can be in turn encoded as an FO model-checking task over a structure consisting of a few 0-1 matrices. We look at this problem through the lenses of twin-width, a recently introduced structural parameter [Bonnet, Kim, Thomassé, Watrigant, FOCS 2020]: By a recent characterization [Bonnet, Giocanti, Ossona de Mendes, Simon, Thomassé, Toruńczyk, STOC 2022] the said FO model-checking task can be done in FPT time if the said matrices have bounded grid rank. To complete the proof, we show an irrelevant vertex rule: If any of the matrices in the said encoding has a large grid minor, a vertex corresponding to the ``middle'' box in the grid minor can be proclaimed irrelevant -- not contained in the sought solution -- and thus reduced.

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arXiv2022

A tight quasi-polynomial bound for Global Label Min-Cut

We study a generalization of the classic Global Min-Cut problem, called Global Label Min-Cut (or sometimes Global Hedge Min-Cut): the edges of the input (multi)graph are labeled (or partitioned into color classes or hedges), and removing all edges of the same label (color or from the same hedge) costs one. The problem asks to disconnect the graph at minimum cost. While the $st$-cut version of the problem is known to be NP-hard, the above global cut version is known to admit a quasi-polynomial randomized $n^{O(\log \mathrm{OPT})}$-time algorithm due to Ghaffari, Karger, and Panigrahi [SODA 2017]. They consider this as ``strong evidence that this problem is in P''. We show that this is actually not the case. We complete the study of the complexity of the Global Label Min-Cut problem by showing that the quasi-polynomial running time is probably optimal: We show that the existence of an algorithm with running time $(np)^{o(\log n/ (\log \log n)^2)}$ would contradict the Exponential Time Hypothesis, where $n$ is the number of vertices, and $p$ is the number of labels in the input. The key step for the lower bound is a proof that Global Label Min-Cut is W[1]-hard when parameterized by the number of uncut labels. In other words, the problem is difficult in the regime where almost all labels need to be cut to disconnect the graph. To turn this lower bound into a quasi-polynomial-time lower bound, we also needed to revisit the framework due to Marx [Theory Comput. 2010] of proving lower bounds assuming Exponential Time Hypothesis through the Subgraph Isomorphism problem parameterized by the number of edges of the pattern. Here, we provide an alternative simplified proof of the hardness of this problem that is more versatile with respect to the choice of the regimes of the parameters.

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arXiv2022

The Influence of Dimensions on the Complexity of Computing Decision Trees

A decision tree recursively splits a feature space $\mathbb{R}^{d}$ and then assigns class labels based on the resulting partition. Decision trees have been part of the basic machine-learning toolkit for decades. A large body of work treats heuristic algorithms to compute a decision tree from training data, usually aiming to minimize in particular the size of the resulting tree. In contrast, little is known about the complexity of the underlying computational problem of computing a minimum-size tree for the given training data. We study this problem with respect to the number $d$ of dimensions of the feature space. We show that it can be solved in $O(n^{2d + 1}d)$ time, but under reasonable complexity-theoretic assumptions it is not possible to achieve $f(d) \cdot n^{o(d / \log d)}$ running time, where $n$ is the number of training examples. The problem is solvable in $(dR)^{O(dR)} \cdot n^{1+o(1)}$ time, if there are exactly two classes and $R$ is an upper bound on the number of tree leaves labeled with the first~class.

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arXiv2022

Taming graphs with no large creatures and skinny ladders

We confirm a conjecture of Gartland and Lokshtanov [arXiv:2007.08761]: if for a hereditary graph class $\mathcal{G}$ there exists a constant $k$ such that no member of $\mathcal{G}$ contains a $k$-creature as an induced subgraph or a $k$-skinny-ladder as an induced minor, then there exists a polynomial $p$ such that every $G \in \mathcal{G}$ contains at most $p(|V(G)|)$ minimal separators. By a result of Fomin, Todinca, and Villanger [SIAM J. Comput. 2015] the latter entails the existence of polynomial-time algorithms for Maximum Weight Independent Set, Feedback Vertex Set and many other problems, when restricted to an input graph from $\mathcal{G}$. Furthermore, as shown by Gartland and Lokshtanov, our result implies a full dichotomy of hereditary graph classes defined by a finite set of forbidden induced subgraphs into tame (admitting a polynomial bound of the number of minimal separators) and feral (containing infinitely many graphs with exponential number of minimal separators).

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arXiv2022

Constant Congestion Brambles

A bramble in an undirected graph $G$ is a family of connected subgraphs of $G$ such that for every two subgraphs $H_1$ and $H_2$ in the bramble either $V(H_1) \cap V(H_2) \neq \emptyset$ or there is an edge of $G$ with one endpoint in $V(H_1)$ and the second endpoint in $V(H_2)$. The order of the bramble is the minimum size of a vertex set that intersects all elements of a bramble. Brambles are objects dual to treewidth: As shown by Seymour and Thomas, the maximum order of a bramble in an undirected graph $G$ equals one plus the treewidth of $G$. However, as shown by Grohe and Marx, brambles of high order may necessarily be of exponential size: In a constant-degree $n$-vertex expander a bramble of order $Ω(n^{1/2+δ})$ requires size exponential in $Ω(n^{2δ})$ for any fixed $δ\in (0,\frac{1}{2}]$. On the other hand, the combination of results of Grohe and Marx and Chekuri and Chuzhoy shows that a graph of treewidth $k$ admits a bramble of order $\widetildeΩ(k^{1/2})$ and size $\widetilde{\mathcal{O}}(k^{3/2})$. ($\widetildeΩ$ and $\widetilde{\mathcal{O}}$ hide polylogarithmic factors and divisors, respectively.) In this note, we first sharpen the second bound by proving that every graph $G$ of treewidth at least $k$ contains a bramble of order $\widetildeΩ(k^{1/2})$ and congestion $2$, i.e., every vertex of $G$ is contained in at most two elements of the bramble (thus the bramble is of size linear in its order). Second, we provide a tight upper bound for the lower bound of Grohe and Marx: For every $δ\in (0,\frac{1}{2}]$, every graph $G$ of treewidth at least $k$ contains a bramble of order $\widetildeΩ(k^{1/2+δ})$ and size $2^{\widetilde{\mathcal{O}}(k^{2δ})}$.

math.CO↗

arXiv2022

Max Weight Independent Set in graphs with no long claws: An analog of the Gyárfás' path argument

We revisit recent developments for the Maximum Weight Independent Set problem in graphs excluding a subdivided claw $S_{t,t,t}$ as an induced subgraph [Chudnovsky, Pilipczuk, Pilipczuk, Thomassé, SODA 2020] and provide a subexponential-time algorithm with improved running time $2^{\mathcal{O}(\sqrt{n}\log n)}$ and a quasipolynomial-time approximation scheme with improved running time $2^{\mathcal{O}(\varepsilon^{-1} \log^{5} n)}$. The Gyárfás' path argument, a powerful tool that is the main building block for many algorithms in $P_t$-free graphs, ensures that given an $n$-vertex $P_t$-free graph, in polynomial time we can find a set $P$ of at most $t-1$ vertices, such that every connected component of $G-N[P]$ has at most $n/2$ vertices. Our main technical contribution is an analog of this result for $S_{t,t,t}$-free graphs: given an $n$-vertex $S_{t,t,t}$-free graph, in polynomial time we can find a set $P$ of $\mathcal{O}(t \log n)$ vertices and an extended strip decomposition (an appropriate analog of the decomposition into connected components) of $G-N[P]$ such that every particle (an appropriate analog of a connected component to recurse on) of the said extended strip decomposition has at most $n/2$ vertices.

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arXiv2022

Bounding twin-width for bounded-treewidth graphs, planar graphs, and bipartite graphs

Twin-width is a newly introduced graph width parameter that aims at generalizing a wide range of "nicely structured" graph classes. In this work, we focus on obtaining good bounds on twin-width $\text{tww}(G)$ for graphs $G$ from a number of classic graph classes. We prove the following: - $\text{tww}(G) \leq 3\cdot 2^{\text{tw}(G)-1}$, where $\text{tw}(G)$ is the treewidth of $G$, - $\text{tww}(G) \leq \max(4\text{bw}(G),\frac{9}{2}\text{bw}(G)-3)$ for a planar graph $G$ with $\text{bw}(G) \geq 2$, where $\text{bw}(G)$ is the branchwidth of $G$, - $\text{tww}(G) \leq 183$ for a planar graph $G$, - the twin-width of a universal bipartite graph $(X,2^X,E)$ with $|X|=n$ is $n - \log_2(n) + \mathcal{O}(1)$ . An important idea behind the bounds for planar graphs is to use an embedding of the graph and sphere-cut decompositions to obtain good bounds on neighbourhood complexity.

cs.DM↗

arXiv2021

Close relatives (of Feedback Vertex Set), revisited

At IPEC 2020, Bergougnoux, Bonnet, Brettell, and Kwon showed that a number of problems related to the classic Feedback Vertex Set (FVS) problem do not admit a $2^{o(k \log k)} \cdot n^{\mathcal{O}(1)}$-time algorithm on graphs of treewidth at most $k$, assuming the Exponential Time Hypothesis. This contrasts with the $3^{k} \cdot k^{\mathcal{O}(1)} \cdot n$-time algorithm for FVS using the Cut&Count technique. During their live talk at IPEC 2020, Bergougnoux et al.~posed a number of open questions, which we answer in this work. - Subset Even Cycle Transversal, Subset Odd Cycle Transversal, Subset Feedback Vertex Set can be solved in time $2^{\mathcal{O}(k \log k)} \cdot n$ in graphs of treewidth at most $k$. This matches a lower bound for Even Cycle Transversal of Bergougnoux et al.~and improves the polynomial factor in some of their upper bounds. - Subset Feedback Vertex Set and Node Multiway Cut can be solved in time $2^{\mathcal{O}(k \log k)} \cdot n$, if the input graph is given as a clique-width expression of size $n$ and width $k$. - Odd Cycle Transversal can be solved in time $4^k \cdot k^{\mathcal{O}(1)} \cdot n$ if the input graph is given as a clique-width expression of size $n$ and width $k$. Furthermore, the existence of a constant $\varepsilon > 0$ and an algorithm performing this task in time $(4-\varepsilon)^k \cdot n^{\mathcal{O}(1)}$ would contradict the Strong Exponential Time Hypothesis.

cs.DM↗

arXiv2021

Constant congestion brambles in directed graphs

The Directed Grid Theorem, stating that there is a function $f$ such that a directed graphs of directed treewidth at least $f(k)$ contains a directed grid of size at least $k$ as a butterfly minor, after being a conjecture for nearly 20 years, has been proven in 2015 by Kawarabayashi and Kreutzer. However, the function $f$ obtained in the proof is very fast growing. In this work, we show that if one relaxes directed grid to bramble of constant congestion, one can obtain a polynomial bound. More precisely, we show that for every $k \geq 1$ there exists $t = \mathcal{O}(k^{48} \log^{13} k)$ such that every directed graph of directed treewidth at least $t$ contains a bramble of congestion at most $8$ and size at least $k$.

math.CO↗

arXiv2021

Hardness of Metric Dimension in Graphs of Constant Treewidth

The Metric Dimension problem asks for a minimum-sized resolving set in a given (unweighted, undirected) graph $G$. Here, a set $S \subseteq V(G)$ is resolving if no two distinct vertices of $G$ have the same distance vector to $S$. The complexity of Metric Dimension in graphs of bounded treewidth remained elusive in the past years. Recently, Bonnet and Purohit [IPEC 2019] showed that the problem is W[1]-hard under treewidth parameterization. In this work, we strengthen their lower bound to show that Metric Dimension is NP-hard in graphs of treewidth 24.

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arXiv2021

Finding large induced sparse subgraphs in $C_{>t}$-free graphs in quasipolynomial time

For an integer $t$, a graph $G$ is called {\em{$C_{>t}$-free}} if $G$ does not contain any induced cycle on more than~$t$ vertices. We prove the following statement: for every pair of integers $d$ and $t$ and a CMSO$_2$ statement~$ϕ$, there exists an algorithm that, given an $n$-vertex $C_{>t}$-free graph $G$ with weights on vertices, finds in time $n^{O(\log^4 n)}$ a maximum-weight vertex subset $S$ such that $G[S]$ has degeneracy at most $d$ and satisfies $ϕ$. The running time can be improved to $n^{O(\log^2 n)}$ assuming $G$ is $P_t$-free, that is, $G$ does not contain an induced path on $t$ vertices. This expands the recent results of the authors [to appear at FOCS 2020 and SOSA 2021] on the {\sc{Maximum Weight Independent Set}} problem on $P_t$-free graphs in two directions: by encompassing the more general setting of $C_{>t}$-free graphs, and by being applicable to a much wider variety of problems, such as {\sc{Maximum Weight Induced Forest}} or {\sc{Maximum Weight Induced Planar Graph}}.

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arXiv2020

Quasi-polynomial-time algorithm for Independent Set in $P_t$-free graphs via shrinking the space of induced paths

In a recent breakthrough work, Gartland and Lokshtanov [FOCS 2020] showed a quasi-polynomial-time algorithm for Maximum Weight Independent Set in $P_t$-free graphs, that is, graphs excluding a fixed path as an induced subgraph. Their algorithm runs in time $n^{\mathcal{O}(\log^3 n)}$, where $t$ is assumed to be a constant. Inspired by their ideas, we present an arguably simpler algorithm with an improved running time bound of $n^{\mathcal{O}(\log^2 n)}$. Our main insight is that a connected $P_t$-free graph always contains a vertex $w$ whose neighborhood intersects, for a constant fraction of pairs $\{u,v\} \in \binom{V(G)}{2}$, a constant fraction of induced $u-v$ paths. Since a $P_t$-free graph contains $\mathcal{O}(n^{t-1})$ induced paths in total, branching on such a vertex and recursing independently on the connected components leads to a quasi-polynomial running time bound. We also show that the same approach can be used to obtain quasi-polynomial-time algorithms for related problems, including Maximum Weight Induced Matching and 3-Coloring.

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arXiv2020

Two lower bounds for $p$-centered colorings

Given a graph $G$ and an integer $p$, a coloring $f : V(G) \to \mathbb{N}$ is \emph{$p$-centered} if for every connected subgraph $H$ of $G$, either $f$ uses more than $p$ colors on $H$ or there is a color that appears exactly once in $H$. The notion of $p$-centered colorings plays a central role in the theory of sparse graphs. In this note we show two lower bounds on the number of colors required in a $p$-centered coloring. First, we consider monotone classes of graphs whose shallow minors have average degree bounded polynomially in the radius, or equivalently (by a result of Dvořák and Norin), admitting strongly sublinear separators. We construct such a class such that $p$-centered colorings require a number of colors super-polynomial in $p$. This is in contrast with a recent result of Pilipczuk and Siebertz, who established a polynomial upper bound in the special case of graphs excluding a fixed minor. Second, we consider graphs of maximum degree $Δ$. Dębski, Felsner, Micek, and Schröder recently proved that these graphs have $p$-centered colorings with $O(Δ^{2-1/p} p)$ colors. We show that there are graphs of maximum degree $Δ$ that require $Ω(Δ^{2-1/p} p \ln^{-1/p}Δ)$ colors in any $p$-centered coloring, thus matching their upper bound up to a logarithmic factor.

math.CO↗

arXiv2020

(Theta, triangle)-free and (even hole, $K_4$)-free graphs. Part 2 : bounds on treewidth

A {\em theta} is a graph made of three internally vertex-disjoint chordless paths $P_1 = a \dots b$, $P_2 = a \dots b$, $P_3 = a \dots b$ of length at least~2 and such that no edges exist between the paths except the three edges incident to $a$ and the three edges incident to $b$. A {\em pyramid} is a graph made of three chordless paths $P_1 = a \dots b_1$, $P_2 = a \dots b_2$, $P_3 = a \dots b_3$ of length at least~1, two of which have length at least 2, vertex-disjoint except at $a$, and such that $b_1b_2b_3$ is a triangle and no edges exist between the paths except those of the triangle and the three edges incident to~$a$. An \emph{even hole} is a chordless cycle of even length. For three non-negative integers $i\leq j\leq k$, let $S_{i,j,k}$ be the tree with a vertex $v$, from which start three paths with $i$, $j$, and $k$ edges respectively. We denote by $K_t$ the complete graph on $t$ vertices. We prove that for all non-negative integers $i, j, k$, the class of graphs that contain no theta, no $K_3$, and no $S_{i, j, k}$ as induced subgraphs have bounded treewidth. We prove that for all non-negative integers $i, j, k, t$, the class of graphs that contain no even hole, no pyramid, no $K_t$, and no $S_{i, j, k}$ as induced subgraphs have bounded treewidth. To bound the treewidth, we prove that every graph of large treewidth must contain a large clique or a minimal separator of large cardinality.

cs.DM↗

arXiv2020

The Complexity of Connectivity Problems in Forbidden-Transition Graphs and Edge-Colored Graphs

The notion of forbidden-transition graphs allows for a robust generalization of walks in graphs. In a forbidden-transition graph, every pair of edges incident to a common vertex is permitted or forbidden; a walk is compatible if all pairs of consecutive edges on the walk are permitted. Forbidden-transition graphs and related models have found applications in a variety of fields, such as routing in optical telecommunication networks, road networks, and bio-informatics. We initiate the study of fundamental connectivity problems from the point of view of parameterized complexity, including an in-depth study of tractability with regards to various graph-width parameters. Among several results, we prove that finding a simple compatible path between given endpoints in a forbidden-transition graph is W[1]-hard when parameterized by the vertex-deletion distance to a linear forest (so it is also hard when parameterized by pathwidth or treewidth). On the other hand, we show an algebraic trick that yields tractability when parameterized by treewidth of finding a properly colored Hamiltonian cycle in an edge-colored graph; properly colored walks in edge-colored graphs is one of the most studied special cases of compatible walks in forbidden-transition graphs.

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arXiv2020

Randomized contractions meet lean decompositions

We show an algorithm that, given an $n$-vertex graph $G$ and a parameter $k$, in time $2^{O(k \log k)} n^{O(1)}$ finds a tree decomposition of $G$ with the following properties: * every adhesion of the tree decomposition is of size at most $k$, and * every bag of the tree decomposition is $(i,i)$-unbreakable in $G$ for every $1 \leq i \leq k$. Here, a set $X \subseteq V(G)$ is $(a,b)$-unbreakable in $G$ if for every separation $(A,B)$ of order at most $b$ in $G$, we have $|A \cap X| \leq a$ or $|B \cap X| \leq a$. The resulting tree decomposition has arguably best possible adhesion size boundsand unbreakability guarantees. Furthermore, the parametric factor in the running time bound is significantly smaller than in previous similar constructions. These improvements allow us to present parameterized algorithms for Minimum Bisection, Steiner Cut, and Steiner Multicut with improved parameteric factor in the running time bound. The main technical insight is to adapt the notion of lean decompositions of Thomas and the subsequent construction algorithm of Bellenbaum and Diestel to the parameterized setting.

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