arXiv subjects

Michał Pilipczuk

Publications and source records attributed to Michał Pilipczuk.

At least 73 records · Page 4Linked to original sources

arXiv2023

Dynamic treewidth

We present a data structure that for a dynamic graph $G$ that is updated by edge insertions and deletions, maintains a tree decomposition of $G$ of width at most $6k+5$ under the promise that the treewidth of $G$ never grows above $k$. The amortized update time is ${\cal O}_k(2^{\sqrt{\log n}\log\log n})$, where $n$ is the vertex count of $G$ and the ${\cal O}_k(\cdot)$ notation hides factors depending on $k$. In addition, we also obtain the dynamic variant of Courcelle's Theorem: for any fixed property $φ$ expressible in the $\mathsf{CMSO}_2$ logic, the data structure can maintain whether $G$ satisfies $φ$ within the same time complexity bounds. To a large extent, this answers a question posed by Bodlaender [WG 1993].

cs.DS↗

arXiv2023

Partitioning edges of a planar graph into linear forests and a matching

We show that the edges of any planar graph of maximum degree at most $9$ can be partitioned into $4$ linear forests and a matching. Combined with known results, this implies that the edges of any planar graph $G$ of odd maximum degree $Δ\ge 9$ can be partitioned into $\tfrac{Δ-1}{2}$ linear forests and one matching. This strengthens well-known results stating that graphs in this class have chromatic index $Δ$ [Vizing, 1965] and linear arboricity at most $\lceil(Δ+1)/2\rceil$ [Wu, 1999].

math.CO↗

arXiv2023

Simpler and faster algorithms for detours in planar digraphs

In the directed detour problem one is given a digraph $G$ and a pair of vertices $s$ and~$t$, and the task is to decide whether there is a directed simple path from $s$ to $t$ in $G$ whose length is larger than $\mathsf{dist}_{G}(s,t)$. The more general parameterized variant, directed long detour, asks for a simple $s$-to-$t$ path of length at least $\mathsf{dist}_{G}(s,t)+k$, for a given parameter $k$. Surprisingly, it is still unknown whether directed detour is polynomial-time solvable on general digraphs. However, for planar digraphs, Wu and Wang~[Networks, '15] proposed an $\mathcal{O}(n^3)$-time algorithm for directed detour, while Fomin et al.~[STACS 2022] gave a $2^{\mathcal{O}(k)}\cdot n^{\mathcal{O}(1)}$-time fpt algorithm for directed long detour. The algorithm of Wu and Wang relies on a nontrivial analysis of how short detours may look like in a plane embedding, while the algorithm of Fomin et al.~is based on a reduction to the $§$-disjoint paths problem on planar digraphs. This latter problem is solvable in polynomial time using the algebraic machinery of Schrijver~[SIAM~J.~Comp.,~'94], but the degree of the obtained polynomial factor is huge. In this paper we propose two simple algorithms: we show how to solve, in planar digraphs, directed detour in time $\mathcal{O}(n^2)$ and directed long detour in time $2^{\mathcal{O}(k)}\cdot n^4 \log n$. In both cases, the idea is to reduce to the $2$-disjoint paths problem in a planar digraph, and to observe that the obtained instances of this problem have a certain topological structure that makes them amenable to a direct greedy strategy.

cs.DM↗

arXiv2022

Parameterized Approximation for Maximum Weight Independent Set of Rectangles and Segments

In the Maximum Weight Independent Set of Rectangles problem (MWISR) we are given a weighted set of $n$ axis-parallel rectangles in the plane. The task is to find a subset of pairwise non-overlapping rectangles with the maximum possible total weight. This problem is NP-hard and the best-known polynomial-time approximation algorithm, due to by Chalermsook and Walczak (SODA 2021), achieves approximation factor $O(\log\log n )$. While in the unweighted setting, constant factor approximation algorithms are known, due to Mitchell (FOCS 2021) and to Gálvez et al. (SODA 2022), it remains open to extend these techniques to the weighted setting. In this paper, we consider MWISR through the lens of parameterized approximation. Grandoni et al. (ESA 2019) gave a $(1-ε)$-approximation algorithm with running time $k^{O(k/ε^8)} n^{O(1/ε^8)}$ time, where $k$ is the number of rectangles in an optimum solution. Unfortunately, their algorithm works only in the unweighted setting and they left it as an open problem to give a parameterized approximation scheme in the weighted setting. Our contribution is a partial answer to the open question of Grandoni et al. (ESA 2019). We give a parameterized approximation algorithm for MWISR that given a parameter $k$, finds a set of non-overlapping rectangles of weight at least $(1-ε) \text{opt}_k$ in $2^{O(k \log(k/ε))} n^{O(1/ε)}$ time, where $\text{opt}_k$ is the maximum weight of a solution of cardinality at most $k$. Note that thus, our algorithm may return a solution consisting of more than $k$ rectangles. To complement this apparent weakness, we also propose a parameterized approximation scheme with running time $2^{O(k^2 \log(k/ε))} n^{O(1)}$ that finds a solution with cardinality at most $k$ and total weight at least $(1-ε)\text{opt}_k$ for the special case of axis-parallel segments.

cs.DS↗

arXiv2022

Highly unbreakable graph with a fixed excluded minor are almost rigid

A set $X \subseteq V(G)$ in a graph $G$ is $(q,k)$-unbreakable if every separation $(A,B)$ of order at most $k$ in $G$ satisfies $|A \cap X| \leq q$ or $|B \cap X| \leq q$. In this paper, we prove the following result: If a graph $G$ excludes a fixed complete graph $K_h$ as a minor and satisfies certain unbreakability guarantees, then $G$ is almost rigid in the following sense: the vertices of $G$ can be partitioned in an isomorphism-invariant way into a part inducing a graph of bounded treewidth and a part that admits a small isomorphism-invariant family of labelings. This result is the key ingredient in the fixed-parameter algorithm for Graph Isomorphism parameterized by the Hadwiger number of the graph, which is presented in a companion paper.

math.CO↗

arXiv2022

Fixed-parameter tractability of Graph Isomorphism in graphs with an excluded minor

We prove that Graph Isomorphism and Canonization in graphs excluding a fixed graph $H$ as a minor can be solved by an algorithm working in time $f(H)\cdot n^{O(1)}$, where $f$ is some function. In other words, we show that these problems are fixed-parameter tractable when parameterized by the size of the excluded minor, with the caveat that the bound on the running time is not necessarily computable. The underlying approach is based on decomposing the graph in a canonical way into unbreakable (intuitively, well-connected) parts, which essentially provides a reduction to the case where the given $H$-minor-free graph is unbreakable itself. This is complemented by an analysis of unbreakable $H$-minor-free graphs, performed in a second subordinate manuscript, which reveals that every such graph can be canonically decomposed into a part that admits few automorphisms and a part that has bounded treewidth.

cs.DS↗

arXiv2022

On Rational Recursive Sequences

We study the class of rational recursive sequences (ratrec) over the rational numbers. A ratrec sequence is defined via a system of sequences using mutually recursive equations of depth 1, where the next values are computed as rational functions of the previous values. An alternative class is that of simple ratrec sequences, where one uses a single recursive equation, however of depth k: the next value is defined as a rational function of k previous values. We conjecture that the classes ratrec and simple ratrec coincide. The main contribution of this paper is a proof of a variant of this conjecture where the initial conditions are treated symbolically, using a formal variable per sequence, while the sequences themselves consist of rational functions over those variables. While the initial conjecture does not follow from this variant, we hope that the introduced algebraic techniques may eventually be helpful in resolving the problem. The class ratrec strictly generalises a well-known class of polynomial recursive sequences (polyrec). These are defined like ratrec, but using polynomial functions instead of rational ones. One can observe that if our conjecture is true and effective, then we can improve the complexities of the zeroness and the equivalence problems for polyrec sequences. Currently, the only known upper bound is Ackermanian, which follows from results on polynomial automata. We complement this observation by proving a PSPACE lower bound for both problems for polyrec. Our lower bound construction also implies that the Skolem problem is PSPACE-hard for the polyrec class.

cs.FL↗

arXiv2022

A subexponential parameterized algorithm for Directed Subset Traveling Salesman Problem on planar graphs

There are numerous examples of the so-called ``square root phenomenon'' in the field of parameterized algorithms: many of the most fundamental graph problems, parameterized by some natural parameter $k$, become significantly simpler when restricted to planar graphs and in particular the best possible running time is exponential in $O(\sqrt{k})$ instead of $O(k)$ (modulo standard complexity assumptions). We consider a classic optimization problem Subset Traveling Salesman, where we are asked to visit all the terminals $T$ by a minimum-weight closed walk. We investigate the parameterized complexity of this problem in planar graphs, where the number $k=|T|$ of terminals is regarded as the parameter. We show that Subset TSP can be solved in time $2^{O(\sqrt{k}\log k)}\cdot n^{O(1)}$ even on edge-weighted directed planar graphs. This improves upon the algorithm of Klein and Marx [SODA 2014] with the same running time that worked only on undirected planar graphs with polynomially large integer weights.

cs.DS↗

arXiv2022

Stable graphs of bounded twin-width

We prove that every class of graphs $\mathscr C$ that is monadically stable and has bounded twin-width can be transduced from some class with bounded sparse twin-width. This generalizes analogous results for classes of bounded linear cliquewidth and of bounded cliquewidth. It also implies that monadically stable classes of bounded twin-widthare linearly $χ$-bounded.

cs.LO↗

arXiv2022

Polynomial kernel for immersion hitting in tournaments

For a fixed simple digraph $H$ without isolated vertices, we consider the problem of deleting arcs from a given tournament to get a digraph which does not contain $H$ as an immersion. We prove that for every $H$, this problem admits a polynomial kernel when parameterized by the number of deleted arcs. The degree of the bound on the kernel size depends on $H$.

cs.DS↗

arXiv2022

On objects dual to tree-cut decompositions

Tree-cut width is a graph parameter introduced by Wollan that is an analogue of treewidth for the immersion order on graphs in the following sense: the tree-cut width of a graph is functionally equivalent to the largest size of a wall that can be found in it as an immersion. In this work we propose a variant of the definition of tree-cut width that is functionally equivalent to the original one, but for which we can state and prove a tight duality theorem relating it to naturally defined dual objects: appropriately defined brambles and tangles. Using this result we also propose a game characterization of tree-cut width.

math.CO↗

arXiv2022

Twin-width and types

We study problems connected to first-order logic in graphs of bounded twin-width. Inspired by the approach of Bonnet et al. [FOCS 2020], we introduce a robust methodology of local types and describe their behavior in contraction sequences -- the decomposition notion underlying twin-width. We showcase the applicability of the methodology by proving the following two algorithmic results. In both statements, we fix a first-order formula $φ(x_1,\ldots,x_k)$ and a constant $d$, and we assume that on input we are given a graph $G$ together with a contraction sequence of width at most $d$. (A) One can in time $O(n)$ construct a data structure that can answer the following queries in time $O(\log \log n)$: given $w_1,\ldots,w_k$, decide whether $ϕ(w_1,\ldots,w_k)$ holds in $G$. (B) After $O(n)$-time preprocessing, one can enumerate all tuples $w_1,\ldots,w_k$ that satisfy $ϕ(x_1,\ldots,x_k)$ in $G$ with $O(1)$ delay. In the case of (A), the query time can be reduced to $O(1/\varepsilon)$ at the expense of increasing the construction time to $O(n^{1+\varepsilon})$, for any fixed $\varepsilon>0$. Finally, we also apply our tools to prove the following statement, which shows optimal bounds on the VC density of set systems that are first-order definable in graphs of bounded twin-width. (C) Let $G$ be a graph of twin-width $d$, $A$ be a subset of vertices of $G$, and $φ(x_1,\ldots,x_k,y_1,\ldots,y_l)$ be a first-order formula. Then the number of different subsets of $A^k$ definable by $ϕ$ using $l$-tuples of vertices from $G$ as parameters, is bounded by $O(|A|^l)$.

cs.LO↗

arXiv2022

Independence number of intersection graphs of axis-parallel segments

We prove that for any triangle-free intersection graph of $n$ axis-parallel segments in the plane, the independence number $α$ of this graph is at least $α\ge n/4 + Ω(\sqrt{n})$. We complement this with a construction of a graph in this class satisfying $α\le n/4 + c \sqrt{n}$ for an absolute constant $c$, which demonstrates the optimality of our result.

math.CO↗

arXiv2022

Computing treedepth in polynomial space and linear fpt time

The treedepth of a graph $G$ is the least possible depth of an elimination forest of $G$: a rooted forest on the same vertex set where every pair of vertices adjacent in $G$ is bound by the ancestor/descendant relation. We propose an algorithm that given a graph $G$ and an integer $d$, either finds an elimination forest of $G$ of depth at most $d$ or concludes that no such forest exists; thus the algorithm decides whether the treedepth of $G$ is at most $d$. The running time is $2^{O(d^2)}\cdot n^{O(1)}$ and the space usage is polynomial in $n$. Further, by allowing randomization, the time and space complexities can be improved to $2^{O(d^2)}\cdot n$ and $d^{O(1)}\cdot n$, respectively. This improves upon the algorithm of Reidl et al. [ICALP 2014], which also has time complexity $2^{O(d^2)}\cdot n$, but uses exponential space.

cs.DS↗

arXiv2022

Dynamic data structures for parameterized string problems

We revisit classic string problems considered in the area of parameterized complexity, and study them through the lens of dynamic data structures. That is, instead of asking for a static algorithm that solves the given instance efficiently, our goal is to design a data structure that efficiently maintains a solution, or reports a lack thereof, upon updates in the instance. We first consider the Closest String problem, for which we design randomized dynamic data structures with amortized update times $d^{\mathcal{O}(d)}$ and $|Σ|^{\mathcal{O}(d)}$, respectively, where $Σ$ is the alphabet and $d$ is the assumed bound on the maximum distance. These are obtained by combining known static approaches to Closest String with color-coding. Next, we note that from a result of Frandsen et al.~[J. ACM'97] one can easily infer a meta-theorem that provides dynamic data structures for parameterized string problems with worst-case update time of the form $\mathcal{O}(\log \log n)$, where $k$ is the parameter in question and $n$ is the length of the string. We showcase the utility of this meta-theorem by giving such data structures for problems Disjoint Factors and Edit Distance. We also give explicit data structures for these problems, with worst-case update times $\mathcal{O}(k2^{k}\log \log n)$ and $\mathcal{O}(k^2\log \log n)$, respectively. Finally, we discuss how a lower bound methodology introduced by Amarilli et al.~[ICALP'21] can be used to show that obtaining update time $\mathcal{O}(f(k))$ for Disjoint Factors and Edit Distance is unlikely already for a constant value of the parameter $k$.

cs.DS↗

arXiv2022

On the Erdős-Pósa property for immersions and topological minors in tournaments

We consider the Erdős-Pósa property for immersions and topological minors in tournaments. We prove that for every simple digraph $H$, $k\in \mathbb{N}$, and tournament $T$, the following statements hold: (i) If in $T$ one cannot find $k$ arc-disjoint immersion copies of $H$, then there exists a set of $\mathcal{O}_H(k^3)$ arcs that intersects all immersion copies of $H$ in $T$. (ii) If in $T$ one cannot find $k$ vertex-disjoint topological minor copies of $H$, then there exists a set of $\mathcal{O}_H(k\log k)$ vertices that intersects all topological minor copies of $H$ in $T$. This improves the results of Raymond [DMTCS '18], who proved similar statements under the assumption that $H$ is strongly connected.

math.CO↗

arXiv2022

Transducing paths in graph classes with unbounded shrubdepth

Transductions are a general formalism for expressing transformations of graphs (and more generally, of relational structures) in logic. We prove that a graph class $\mathscr{C}$ can be $\mathsf{FO}$-transduced from a class of bounded-height trees (that is, has bounded shrubdepth) if, and only if, from $\mathscr{C}$ one cannot $\mathsf{FO}$-transduce the class of all paths. This establishes one of the three remaining open questions posed by Blumensath and Courcelle about the $\mathsf{MSO}$-transduction quasi-order, even in the stronger form that concerns $\mathsf{FO}$-transductions instead of $\mathsf{MSO}$-transductions. The backbone of our proof is a graph-theoretic statement that says the following: If a graph $G$ excludes a path, the bipartite complement of a path, and a half-graph as semi-induced subgraphs, then the vertex set of $G$ can be partitioned into a bounded number of parts so that every part induces a cograph of bounded height, and every pair of parts semi-induce a bi-cograph of bounded height. This statement may be of independent interest; for instance, it implies that the graphs in question form a class that is linearly $χ$-bounded.

math.CO↗

arXiv2022

Graphs of bounded twin-width are quasi-polynomially $χ$-bounded

We prove that for every $t\in \mathbb{N}$ there is a constant $γ_t$ such that every graph with twin-width at most $t$ and clique number $ω$ has chromatic number bounded by $2^{γ_t \log^{4t+3} ω}$. In other words, we prove that graph classes of bounded twin-width are quasi-polynomially $χ$-bounded. This provides a significant step towards resolving the question of Bonnet et al. [ICALP 2021] about whether they are polynomially $χ$-bounded.

math.CO↗