Recently, Bojikian and Kratsch [2023] have presented a novel approach to tackle connectivity problems parameterized by clique-width ($\operatorname{cw}$), based on counting small representations of partial solutions (modulo two). Using this technique, they were able to get a tight bound for the Steiner Tree problem, answering an open question posed by Hegerfeld and Kratsch [ESA, 2023]. We use the same technique to solve the Connected Odd Cycle Transversal problem in time $\mathcal{O}^*(12^{\operatorname{cw}})$. We define a new representation of partial solutions by separating the connectivity requirement from the 2-colorability requirement of this problem. Moreover, we prove that our result is tight by providing SETH-based lower bound excluding algorithms with running time $\mathcal{O}^*((12-ε)^{\operatorname{lcw}})$ even when parameterized by linear clique-width. This answers the second question posed by Hegerfeld and Kratsch in the same paper.
University of Primorska（波莫瑞大学）；Kyushu University（九州大学）；Brandenburg University of Technology（勃兰登堡工业大学）；IAM, University of Primorska（波莫瑞大学应用数学研究所）；Osaka Metropolitan University（大阪公立大学）
We present a new combinatorial model for identifying regulatory modules in gene co-expression data using a decomposition into weighted cliques. To capture complex interaction effects, we generalize the previously-studied weighted edge clique partition problem. As a first step, we restrict ourselves to the noise-free setting, and show that the problem is fixed parameter tractable when parameterized by the number of modules (cliques). We present two new algorithms for finding these decompositions, using linear programming and integer partitioning to determine the clique weights. Further, we impl…
The complexity of problems involving global constraints is usually much more difficult to understand than the complexity of problems only involving local constraints. A natural form of global constraints are connectivity constraints. We study connectivity problems from a fine-grained parameterized perspective. In a breakthrough, Cygan et al. (TALG 2022) first obtained algorithms with single-exponential running time c^{tw} n^O(1) for connectivity problems parameterized by treewidth by introducing the cut-and-count-technique. Furthermore, the obtained bases c were shown to be optimal under the S…
In this paper we give upper bounds on the number of minimal separators and potential maximal cliques of graphs w.r.t. two graph parameters, namely vertex cover ($\operatorname{vc}$) and modular width ($\operatorname{mw}$). We prove that for any graph, the number of minimal separators is $\mathcal{O}^*(3^{\operatorname{vc}})$ and $\mathcal{O}^*(1.6181^{\operatorname{mw}})$, and the number of potential maximal cliques is $\mathcal{O}^*(4^{\operatorname{vc}})$ and $\mathcal{O}^*(1.7347^{\operatorname{mw}})$, and these objects can be listed within the same running times. (The $\mathcal{O}^*$ notat…
The Odd Cycle Transversal problem (OCT) asks whether a given graph can be made bipartite (i.e., 2-colorable) by deleting at most l vertices. We study structural parameterizations of OCT with respect to their polynomial kernelizability, i.e., whether instances can be efficiently reduced to a size polynomial in the chosen parameter. It is a major open problem in parameterized complexity whether Odd Cycle Transversal admits a polynomial kernel when parameterized by l. On the positive side, we show a polynomial kernel for OCT when parameterized by the vertex deletion distance to the class of bipar…
We study the parameterized complexity of the classical Edge Hamiltonian Path problem and give several fixed-parameter tractability results. First, we settle an open question of Demaine et al. by showing that Edge Hamiltonian Path is FPT parameterized by vertex cover, and that it also admits a cubic kernel. We then show fixed-parameter tractability even for a generalization of the problem to arbitrary hypergraphs, parameterized by the size of a (supplied) hitting set. We also consider the problem parameterized by treewidth or clique-width. Surprisingly, we show that the problem is FPT for both …
The k-Clique problem is a fundamental combinatorial problem that plays a prominent role in classical as well as in parameterized complexity theory. It is among the most well-known NP-complete and W[1]-complete problems. Moreover, its average-case complexity analysis has created a long thread of research already since the 1970s. Here, we continue this line of research by studying the dependence of the average-case complexity of the k-Clique problem on the parameter k. To this end, we define two natural parameterized analogs of efficient average-case algorithms. We then show that k-Clique admits…
We provide a polynomial-time algorithm for b-Coloring on graphs of constant clique-width. This unifies and extends nearly all previously known polynomial time results on graph classes, and answers open questions posed by Campos and Silva [Algorithmica, 2018] and Bonomo et al. [Graphs Combin., 2009]. This constitutes the first result concerning structural parameterizations of this problem. We show that the problem is FPT when parameterized by the vertex cover number on general graphs, and on chordal graphs when parameterized by the number of colors. Additionally, we observe that our algorithm f…
We show that a graph with $n$ vertices and vertex cover of size $k$ has at most $4^k + n$ potential maximal cliques. We also show that for each positive integer $k$, there exists a graph with vertex cover of size $k$, $O(k^2)$ vertices, and $Ω(4^k)$ potential maximal cliques. Our results extend the results of Fomin, Liedloff, Montealegre, and Todinca [Algorithmica, 80(4):1146--1169, 2018], who proved an upper bound of $poly(n) 4^k$, but left the lower bound as an open problem.
An acyclic r-coloring of a directed graph G=(V,E) is a partition of the vertex set V into r acyclic sets. The dichromatic number of a directed graph G is the smallest r such that G allows an acyclic r-coloring. For symmetric digraphs the dichromatic number equals the well-known chromatic number of the underlying undirected graph. This allows us to carry over the W[1]-hardness and lower bounds for running times of the chromatic number problem parameterized by clique-width to the dichromatic number problem parameterized by directed clique-width. We introduce the first polynomial-time algorithm f…
The field of kernelization studies polynomial-time preprocessing routines for hard problems in the framework of parameterized complexity. Although a framework for proving kernelization lower bounds has been discovered in 2008 and successfully applied multiple times over the last three years, establishing kernelization complexity of many important problems remains open. In this paper we show that, unless NP is a subset of coNP/poly and the polynomial hierarchy collapses up to its third level, the following parameterized problems do not admit a polynomial-time preprocessing algorithm that reduce…
A directed odd cycle transversal of a directed graph (digraph) $D$ is a vertex set $S$ that intersects every odd directed cycle of $D$. In the Directed Odd Cycle Transversal (DOCT) problem, the input consists of a digraph $D$ and an integer $k$. The objective is to determine whether there exists a directed odd cycle transversal of $D$ of size at most $k$. In this paper, we settle the parameterized complexity of DOCT when parameterized by the solution size $k$ by showing that DOCT does not admit an algorithm with running time $f(k)n^{O(1)}$ unless FPT = W[1]. On the positive side, we give a fac…
We continue the study initiated by Bonomo-Braberman and Gonzalez in 2020 on $r$-locally checkable problems. We propose a dynamic programming algorithm that takes as input a graph with an associated clique-width expression and solves a $1$-locally checkable problem under certain restrictions. We show that it runs in polynomial time in graphs of bounded clique-width, when the number of colors of the locally checkable problem is fixed. Furthermore, we present a first extension of our framework to global properties by taking into account the sizes of the color classes, and consequently enlarge the…
Graph Burning asks, given a graph $G = (V,E)$ and an integer $k$, whether there exists $(b_{0},\dots,b_{k-1}) \in V^{k}$ such that every vertex in $G$ has distance at most $i$ from some $b_{i}$. This problem is known to be NP-complete even on connected caterpillars of maximum degree $3$. We study the parameterized complexity of this problem and answer all questions arose by Kare and Reddy [IWOCA 2019] about parameterized complexity of the problem. We show that the problem is W[2]-complete parameterized by $k$ and that it does no admit a polynomial kernel parameterized by vertex cover number un…
Many hard graph problems can be solved efficiently when restricted to graphs of bounded treewidth, and more generally to graphs of bounded clique-width. But there is a price to be paid for this generality, exemplified by the four problems MaxCut, Graph Coloring, Hamiltonian Cycle and Edge Dominating Set that are all FPT parameterized by treewidth but none of which can be FPT parameterized by clique-width unless FPT = W[1], as shown by Fomin et al [7, 8]. We therefore seek a structural graph parameter that shares some of the generality of clique-width without paying this price. Based on splits,…
We study the problem of Imbalance parameterized by the twin cover of a graph. We show that Imbalance is XP parameterized by twin cover, and FPT when parameterized by the twin cover and the size of the largest clique outside the twin cover. In contrast, we introduce a notion of succinct representations of graphs in terms of their twin cover and demonstrate that Imbalance is NP-hard in the setting of succinct representations, even for graphs that have a twin cover of size one.
In this work we start the investigation of tight complexity bounds for connectivity problems parameterized by cutwidth assuming the Strong Exponential-Time Hypothesis (SETH). Van Geffen et al. posed this question for odd cycle transversal and feedback vertex set. We answer it for these two and four further problems, namely connected vertex cover, connected domintaing set, steiner tree, and connected odd cycle transversal. For the latter two problems it sufficed to prove lower bounds that match the running time inherited from parameterization by treewidth; for the others we provide faster algor…
We consider problems that can be formulated as a task of finding an optimal triangulation of a graph w.r.t. some notion of optimality. We present algorithms parameterized by the size of a minimum edge clique cover ($cc$) to such problems. This parameterization occurs naturally in many problems in this setting, e.g., in the perfect phylogeny problem $cc$ is at most the number of taxa, in fractional hypertreewidth $cc$ is at most the number of hyperedges, and in treewidth of Bayesian networks $cc$ is at most the number of non-root nodes. We show that the number of minimal separators of graphs is…
In the Cluster Vertex Deletion problem the input is a graph $G$ and an integer $k$. The goal is to decide whether there is a set of vertices $S$ of size at most $k$ such that the deletion of the vertices of $S$ from $G$ results a graph in which every connected component is a clique. We give an algorithm for Cluster Vertex Deletion whose running time is $O^*(1.811^k)$.
Many NP-hard problems, such as Dominating Set, are FPT parameterized by clique-width. For graphs of clique-width $k$ given with a $k$-expression, Dominating Set can be solved in $4^k n^{O(1)}$ time. However, no FPT algorithm is known for computing an optimal $k$-expression. For a graph of clique-width $k$, if we rely on known algorithms to compute a $(2^{3k}-1)$-expression via rank-width and then solving Dominating Set using the $(2^{3k}-1)$-expression, the above algorithm will only give a runtime of $4^{2^{3k}} n^{O(1)}$. There have been results which overcome this exponential jump; the best …
We revisit the complexity of the classical $k$-Coloring problem parameterized by clique-width. This is a very well-studied problem that becomes highly intractable when the number of colors $k$ is large. However, much less is known on its complexity for small, concrete values of $k$. In this paper, we completely determine the complexity of $k$-Coloring parameterized by clique-width for any fixed $k$, under the SETH. Specifically, we show that for all $k\ge 3,ε>0$, $k$-Coloring cannot be solved in time $O^*((2^k-2-ε)^{cw})$, and give an algorithm running in time $O^*((2^k-2)^{cw})$. Thus, if the…
Clustering is a well-known and important problem with numerous applications. The graph-based model is one of the typical cluster models. In the graph model, clusters are generally defined as cliques. However, such an approach might be too restrictive as in some applications, not all objects from the same cluster must be connected. That is why different types of cliques relaxations often considered as clusters. In our work, we consider a problem of partitioning graph into clusters and a problem of isolating cluster of a special type whereby cluster we mean highly connected subgraph. Initially, …
Clique-width is one of the most important parameters that describes structural complexity of a graph. Probably, only treewidth is more studied graph width parameter. In this paper we study how clique-width influences the complexity of the Maximum Happy Vertices (MHV) and Maximum Happy Edges (MHE) problems. We answer a question of Choudhari and Reddy '18 about parameterization by the distance to threshold graphs by showing that MHE is NP-complete on threshold graphs. Hence, it is not even in XP when parameterized by clique-width, since threshold graphs have clique-width at most two. As a comple…
It is known that a number of natural graph problems which are FPT parameterized by treewidth become W-hard when parameterized by clique-width. It is therefore desirable to find a different structural graph parameter which is as general as possible, covers dense graphs but does not incur such a heavy algorithmic penalty. The main contribution of this paper is to consider a parameter called modular-width, defined using the well-known notion of modular decompositions. Using a combination of ILPs and dynamic programming we manage to design FPT algorithms for Coloring and Partitioning into paths (a…
An $(r, \ell)$-partition of a graph $G$ is a partition of its vertex set into $r$ independent sets and $\ell$ cliques. A graph is $(r, \ell)$ if it admits an $(r, \ell)$-partition. A graph is well-covered if every maximal independent set is also maximum. A graph is $(r,\ell)$-well-covered if it is both $(r,\ell)$ and well-covered. In this paper we consider two different decision problems. In the $(r,\ell)$-Well-Covered Graph problem ($(r,\ell)$WCG for short), we are given a graph $G$, and the question is whether $G$ is an $(r,\ell)$-well-covered graph. In the Well-Covered $(r,\ell)$-Graph prob…
The Cut & Count technique and the rank-based approach have lead to single-exponential FPT algorithms parameterized by treewidth, that is, running in time $2^{O(tw)}n^{O(1)}$, for Feedback Vertex Set and connected versions of the classical graph problems (such as Vertex Cover and Dominating Set). We show that Subset Feedback Vertex Set, Subset Odd Cycle Transversal, Restricted Edge-Subset Feedback Edge Set, Node Multiway Cut, and Multiway Cut are unlikely to have such running times. More precisely, we match algorithms running in time $2^{O(tw \log tw)}n^{O(1)}$ with tight lower bounds under the…
We show that Odd Cycle Transversal and Vertex Multiway Cut admit deterministic polynomial kernels when restricted to planar graphs and parameterized by the solution size. This answers a question of Saurabh. On the way to these results, we provide an efficient sparsification routine in the flavor of the sparsification routine used for the Steiner Tree problem in planar graphs (FOCS 2014). It differs from the previous work because it preserves the existence of low-cost subgraphs that are not necessarily Steiner trees in the original plane graph, but structures that turn into (supergraphs of) Ste…
Golovach, Paulusma and Song (Inf. Comput. 2014) asked to determine the parameterized complexity of the following problems parameterized by $k$: (1) Given a graph $G$, a clique modulator $D$ (a clique modulator is a set of vertices, whose removal results in a clique) of size $k$ for $G$, and a list $L(v)$ of colors for every $v\in V(G)$, decide whether $G$ has a proper list coloring; (2) Given a graph $G$, a clique modulator $D$ of size $k$ for $G$, and a pre-coloring $λ_P: X \rightarrow Q$ for $X \subseteq V(G),$ decide whether $λ_P$ can be extended to a proper coloring of $G$ using only color…
We study two variants of \textsc{Maximum Cut}, which we call \textsc{Connected Maximum Cut} and \textsc{Maximum Minimal Cut}, in this paper. In these problems, given an unweighted graph, the goal is to compute a maximum cut satisfying some connectivity requirements. Both problems are known to be NP-complete even on planar graphs whereas \textsc{Maximum Cut} on planar graphs is solvable in polynomial time. We first show that these problems are NP-complete even on planar bipartite graphs and split graphs. Then we give parameterized algorithms using graph parameters such as clique-width, tree-wid…
In this paper, we study several coloring problems on graphs from the viewpoint of parameterized complexity. We show that Precoloring Extension is fixed-parameter tractable (FPT) parameterized by distance to clique and Equitable Coloring is FPT parameterized by the distance to threshold graphs. We also study the List k-Coloring and show that the problem is NP-complete on split graphs and it is FPT parameterized by solution size on split graphs.
We study the parameterized complexity of a robust generalization of the classical Feedback Vertex Set problem, namely the Group Feedback Vertex Set problem; we are given a graph G with edges labeled with group elements, and the goal is to compute the smallest set of vertices that hits all cycles of G that evaluate to a non-null element of the group. This problem generalizes not only Feedback Vertex Set, but also Subset Feedback Vertex Set, Multiway Cut and Odd Cycle Transversal. Completing the results of Guillemot [Discr. Opt. 2011], we provide a fixed-parameter algorithm for the parameterizat…
Parameterized complexity seeks to use input structure to obtain faster algorithms for NP-hard problems. This has been most successful for graphs of low treewidth: Many problems admit fast algorithms relative to treewidth and many of them are optimal under SETH. Fewer such results are known for more general structure such as low clique-width and more restrictive structure such as low deletion distance to a sparse graph class. Despite these successes, such results remain "islands'' within the realm of possible structure. Rather than adding more islands, we seek to determine the transitions betwe…
Many hard graph problems, such as Hamiltonian Cycle, become FPT when parameterized by treewidth, a parameter that is bounded only on sparse graphs. When parameterized by the more general parameter clique-width, Hamiltonian Cycle becomes W[1]-hard, as shown by Fomin et al. [5]. Sæther and Telle address this problem in their paper [13] by introducing a new parameter, split-matching-width, which lies between treewidth and clique-width in terms of generality. They show that even though graphs of restricted split-matching-width might be dense, solving problems such as Hamiltonian Cycle can be done …
A polynomial Turing kernel for some parameterized problem $P$ is a polynomial-time algorithm that solves $P$ using queries to an oracle of $P$ whose sizes are upper-bounded by some polynomial in the parameter. Here the term "polynomial" refers to the bound on the query sizes, as the running time of any kernel is required to be polynomial. One of the most important open goals in parameterized complexity is to understand the applicability and limitations of polynomial Turing Kernels. As any fixed-parameter tractable problem admits a Turing kernel of some size, the focus has mostly being on deter…
In graph modification problems, one is given a graph G and the goal is to apply a minimum number of modification operations (such as edge deletions) to G such that the resulting graph fulfills a certain property. For example, the Cluster Deletion problem asks to delete as few edges as possible such that the resulting graph is a disjoint union of cliques. Graph modification problems appear in numerous applications, including the analysis of biological and social networks. Typically, graph modification problems are NP-hard, making them natural candidates for parameterized complexity studies. We …
A graph is $k$-clique-extendible if there is an ordering of the vertices such that whenever two $k$-sized overlapping cliques $A$ and $B$ have $k-1$ common vertices, and these common vertices appear between the two vertices $a,b\in (A\setminus B)\cup (B\setminus A)$ in the ordering, there is an edge between $a$ and $b$, implying that $A\cup B$ is a $(k+1)$-sized clique. Such an ordering is said to be a $k$-C-E ordering. These graphs arise in applications related to modelling preference relations. Recently, it has been shown that a maximum sized clique in such a graph can be found in $n^{O(k)}$…
A clique coloring of a graph is an assignment of colors to its vertices such that no maximal clique is monochromatic. We initiate the study of structural parameterizations of the Clique Coloring problem which asks whether a given graph has a clique coloring with $q$ colors. For fixed $q \ge 2$, we give an $\mathcal{O}^{\star}(q^{tw})$-time algorithm when the input graph is given together with one of its tree decompositions of width $tw$. We complement this result with a matching lower bound under the Strong Exponential Time Hypothesis. We furthermore show that (when the number of colors is unb…
Many common graph data mining tasks take the form of identifying dense subgraphs (e.g. clustering, clique-finding, etc). In biological applications, the natural model for these dense substructures is often a complete bipartite graph (biclique), and the problem requires enumerating all maximal bicliques (instead of just identifying the largest or densest). The best known algorithm in general graphs is due to Dias et al., and runs in time O(M |V|^4 ), where M is the number of maximal induced bicliques (MIBs) in the graph. When the graph being searched is itself bipartite, Zhang et al. give a fas…
We extend the notion of lossy kernelization, introduced by Lokshtanov et al. [STOC 2017], to approximate Turing kernelization. An $α$-approximate Turing kernel for a parameterized optimization problem is a polynomial-time algorithm that, when given access to an oracle that outputs $c$-approximate solutions in $O(1)$ time, obtains an $(α\cdot c)$-approximate solution to the considered problem, using calls to the oracle of size at most $f(k)$ for some function $f$ that only depends on the parameter. Using this definition, we show that Independent Set parameterized by treewidth $\ell$ has a $(1+\…
The Cluster Editing problem seeks a transformation of a given undirected graph into a disjoint union of cliques via a minimum number of edge additions or deletions. A multi-parameterized version of the problem is studied, featuring a number of input parameters that bound the amount of both edge-additions and deletions per single vertex, as well as the size of a clique-cluster. We show that the problem remains NP-hard even when only one edge can be deleted and at most two edges can be added per vertex. However, the new formulation allows us to solve Cluster Editing (exactly) in polynomial time …
Combining the techniques of approximation algorithms and parameterized complexity has long been considered a promising research area, but relatively few results are currently known. In this paper we study the parameterized approximability of a number of problems which are known to be hard to solve exactly when parameterized by treewidth or clique-width. Our main contribution is to present a natural randomized rounding technique that extends well-known ideas and can be used for both of these widths. Applying this very generic technique we obtain approximation schemes for a number of problems, e…
Subgraph detection has recently been one of the most studied problems in the CONGEST model of distributed computing. In this work, we study the distributed complexity of problems closely related to subgraph detection, mainly focusing on induced subgraph detection. The main line of this work presents lower bounds and parameterized algorithms w.r.t structural parameters of the input graph: -- On general graphs, we give unconditional lower bounds for induced detection of cycles and patterns of treewidth 2 in CONGEST. Moreover, by adapting reductions from centralized parameterized complexity, we p…
We present a parallel k-clique listing algorithm with improved work bounds (for the same depth) in sparse graphs with low degeneracy or arboricity. We achieve this by introducing and analyzing a new pruning criterion for a backtracking search. Our algorithm has better asymptotic performance, especially for larger cliques (when k is not constant), where we avoid the straightforwardly exponential runtime growth with respect to the clique size. In particular, for cliques that are a constant factor smaller than the graph's degeneracy, the work improvement is an exponential factor in the clique siz…
