In the Min $k$-Cut problem, input is an edge weighted graph $G$ and an integer $k$, and the task is to partition the vertex set into $k$ non-empty sets, such that the total weight of the edges with endpoints in different parts is minimized. When $k$ is part of the input, the problem is NP-complete and hard to approximate within any factor less than $2$. Recently, the problem has received significant attention from the perspective of parameterized approximation. Gupta et al.~[SODA 2018] initiated the study of FPT-approximation for the Min $k$-Cut problem and gave an $1.9997$-approximation algorithm running in time $2^{\mathcal{O}(k^6)}n^{\mathcal{O}(1)}$. Later, the same set of authors~[FOCS 2018] designed an $(1 +ε)$-approximation algorithm that runs in time $(k/ε)^{\mathcal{O}(k)}n^{k+\mathcal{O}(1)}$, and a $1.81$-approximation algorithm running in time $2^{\mathcal{O}(k^2)}n^{\mathcal{O}(1)}$. More, recently, Kawarabayashi and Lin~[SODA 2020] gave a $(5/3 + ε)$-approximation for Min $k$-Cut running in time $2^{\mathcal{O}(k^2 \log k)}n^{\mathcal{O}(1)}$.
In this paper we give a parameterized approximation algorithm with best possible approximation guarantee, and best possible running time dependence on said guarantee (up to Exponential Time Hypothesis (ETH) and constants in the exponent). In particular, for every $ε> 0$, the algorithm obtains a $(1 +ε)$-approximate solution in time $(k/ε)^{\mathcal{O}(k)}n^{\mathcal{O}(1)}$. The main ingredients of our algorithm are: a simple sparsification procedure, a new polynomial time algorithm for decomposing a graph into highly connected parts, and a new exact algorithm with running time $s^{\mathcal{O}(k)}n^{\mathcal{O}(1)}$ on unweighted (multi-) graphs. Here, $s$ denotes the number of edges in a minimum $k$-cut. The latter two are of independent interest.
We show that the Minimal Length-Bounded L-But problem can be computed in linear time with respect to L and the tree-width of the input graph as parameters. In this problem the task is to find a set of edges of a graph such that after removal of this set, the shortest path between two prescribed vertices is at least L long. We derive an FPT algorithm for a more general multi-commodity length bounded cut problem when parameterized by the number of terminals also. For the former problem we show a W[1]-hardness result when the parameterization is done by the path-width only (instead of the tree-wi…
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 consider two above lower bound parameterizations of the Node Multiway Cut problem - above the maximum separating cut and above a natural LP-relaxation - and prove them to be fixed-parameter tractable. Our results imply O*(4^k) algorithms for Vertex Cover above Maximum Matching and Almost 2-SAT as well as an O*(2^k) algorithm for Node Multiway Cut with a standard parameterization by the solution size, improving previous bounds for these problems.
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…
Given an undirected graph $G$, a collection $\{(s_1,t_1),..., (s_k,t_k)\}$ of pairs of vertices, and an integer $p$, the Edge Multicut problem ask if there is a set $S$ of at most $p$ edges such that the removal of $S$ disconnects every $s_i$ from the corresponding $t_i$. Vertex Multicut is the analogous problem where $S$ is a set of at most $p$ vertices. Our main result is that both problems can be solved in time $2^{O(p^3)}... n^{O(1)}$, i.e., fixed-parameter tractable parameterized by the size $p$ of the cutset in the solution. By contrast, it is unlikely that an algorithm with running time…
The recently introduced graph parameter tree-cut width plays a similar role with respect to immersions as the graph parameter treewidth plays with respect to minors. In this paper, we provide the first algorithmic applications of tree-cut width to hard combinatorial problems. Tree-cut width is known to be lower-bounded by a function of treewidth, but it can be much larger and hence has the potential to facilitate the efficient solution of problems that are not known to be fixed-parameter tractable (FPT) when parameterized by treewidth. We introduce the notion of nice tree-cut decompositions an…
We consider the graph $k$-partitioning problem under the min-max objective, termed as Minmax $k$-cut. The input here is a graph $G=(V,E)$ with non-negative edge weights $w:E\rightarrow \mathbb{R}_+$ and an integer $k\geq 2$ and the goal is to partition the vertices into $k$ non-empty parts $V_1, \ldots, V_k$ so as to minimize $\max_{i=1}^k w(δ(V_i))$. Although minimizing the sum objective $\sum_{i=1}^k w(δ(V_i))$, termed as Minsum $k$-cut, has been studied extensively in the literature, very little is known about minimizing the max objective. We initiate the study of Minmax $k$-cut by showing …
We introduce a novel approach of using important cuts which allowed us to design significantly faster fixed-parameter tractable (FPT) algorithms for the following routing problems: the Mixed Chinese Postman Problem parameterized by the number of directed edges (Gutin et al., JCSS 2017), the Minimum Shared Edges problem (MSE) parameterized by the number p of paths between two specified vertices (Fluschnik et al., JCSS 2019), and the Weighted Min Cut Prevention problem (Gruttemeier et al., WG 2021). The Minimum Vulnerability problem (MV) is a generalization of MSE (Assadi et al., Algorithmica 20…
Given a directed graph $G$, a set of $k$ terminals and an integer $p$, the \textsc{Directed Vertex Multiway Cut} problem asks if there is a set $S$ of at most $p$ (nonterminal) vertices whose removal disconnects each terminal from all other terminals. \textsc{Directed Edge Multiway Cut} is the analogous problem where $S$ is a set of at most $p$ edges. These two problems indeed are known to be equivalent. A natural generalization of the multiway cut is the \emph{multicut} problem, in which we want to disconnect only a set of $k$ given pairs instead of all pairs. Marx (Theor. Comp. Sci. 2006) sh…
We study the \emph{multiterminal cut} problem, which, given an $n$-vertex graph whose edges are integer-weighted and a set of terminals, asks for a partition of the vertex set such that each terminal is in a distinct part, and the total weight of crossing edges is at most $k$. Our weapons shall be two classical results known for decades: \emph{maximum volume minimum ($s,t$)-cuts} by [Ford and Fulkerson, \emph{Flows in Networks}, 1962] and \emph{isolating cuts} by [Dahlhaus et al., \emph{SIAM J. Comp.} 23(4):864-894, 1994]. We sharpen these old weapons with the help of submodular functions, and…
We study the boundary of tractability for the Max-Cut problem in graphs. Our main result shows that Max-Cut above the Edwards-Erdős bound is fixed-parameter tractable: we give an algorithm that for any connected graph with n vertices and m edges finds a cut of size m/2 + (n-1)/4 + k in time 2^O(k)n^4, or decides that no such cut exists. This answers a long-standing open question from parameterized complexity that has been posed several times over the past 15 years. Our algorithm is asymptotically optimal, under the Exponential Time Hypothesis, and is strengthened by a polynomial-time computabl…
We study the problem Symmetric Directed Multicut from a parameterized complexity perspective. In this problem, the input is a digraph $D$, a set of cut requests $C=\{(s_1,t_1),\ldots,(s_\ell,t_\ell)\}$ and an integer $k$, and the task is to find a set $X \subseteq V(D)$ of size at most $k$ such that for every $1 \leq i \leq \ell$, $X$ intersects either all $(s_i,t_i)$-paths or all $(t_i,s_i)$-paths. Equivalently, every strongly connected component of $D-X$ contains at most one vertex out of $s_i$ and $t_i$ for every $i$. This problem is previously known from research in approximation algorithm…
An enumeration kernel as defined by Creignou et al. [Theory Comput. Syst. 2017] for a parameterized enumeration problem consists of an algorithm that transforms each instance into one whose size is bounded by the parameter plus a solution-lifting algorithm that efficiently enumerates all solutions from the set of the solutions of the kernel. We propose to consider two new versions of enumeration kernels by asking that the solutions of the original instance can be enumerated in polynomial time or with polynomial delay from the kernel solutions. Using the NP-hard Matching Cut problem parameteriz…
Given an edge-weighted graph $G$ on $n$ nodes, the NP-hard Max-Cut problem asks for a node bipartition such that the sum of edge weights joining the different partitions is maximized. We propose a fixed-parameter tractable algorithm parameterized by the number $k$ of crossings in a given drawing of $G$. Our algorithm achieves a running time of $O(2^k \cdot p(n + k))$, where $p$ is the polynomial running time for planar Max-Cut. The only previously known similar algorithm [8] is restricted to 1-planar graphs (i.e., at most one crossing per edge) and its dependency on $k$ is of order $3^k$ . A d…
The classical Menger's theorem states that in any undirected (or directed) graph $G$, given a pair of vertices $s$ and $t$, the maximum number of vertex (edge) disjoint paths is equal to the minimum number of vertices (edges) needed to disconnect from $s$ and $t$. This min-max result can be turned into a polynomial time algorithm to find the maximum number of vertex (edge) disjoint paths as well as the minimum number of vertices (edges) needed to disconnect $s$ from $t$. In this paper we study a mixed version of this problem, called Mixed-Cut, where we are given an undirected graph $G$, vertic…
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…
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…
We consider the following natural graph cut problem called Critical Node Cut (CNC): Given a graph $G$ on $n$ vertices, and two positive integers $k$ and $x$, determine whether $G$ has a set of $k$ vertices whose removal leaves $G$ with at most $x$ connected pairs of vertices. We analyze this problem in the framework of parameterized complexity. That is, we are interested in whether or not this problem is solvable in $f(κ) \cdot n^{O(1)}$ time (i.e., whether or not it is fixed-parameter tractable), for various natural parameters $κ$. We consider four such parameters: - The size $k$ of the requi…
In the recent years we have witnessed a rapid development of new algorithmic techniques for parameterized algorithms for graph separation problems. We present experimental evaluation of two cornerstone theoretical results in this area: linear-time branching algorithms guided by half-integral relaxations and kernelization (preprocessing) routines based on representative sets in matroids. A side contribution is a new set of benchmark instances of (unweighted, vertex-deletion) Multiway Cut.
The Max-Cut problem is known to be NP-hard on general graphs, while it can be solved in polynomial time on planar graphs. In this paper, we present a fixed-parameter tractable algorithm for the problem on `almost' planar graphs: Given an $n$-vertex graph and its drawing with $k$ crossings, our algorithm runs in time $O(2^k(n+k)^{3/2} \log (n + k))$. Previously, Dahn, Kriege and Mutzel (IWOCA 2018) obtained an algorithm that, given an $n$-vertex graph and its $1$-planar drawing with $k$ crossings, runs in time $O(3^k n^{3/2} \log n)$. Our result simultaneously improves the running time and remo…
Given a graph $G=(V,E)$, capacities $w(e)$ on edges, and a subset of terminals $\mathcal{T} \subseteq V: |\mathcal{T}| = k$, a mimicking network for $(G,\mathcal{T})$ is a graph $(H,w')$ that contains copies of $\mathcal{T}$ and preserves the value of minimum cuts separating any subset $A, B \subseteq \mathcal{T}$ of terminals. Mimicking networks of size $2^{2^k}$ are known to exist and can be constructed algorithmically, while the best known lower bound is $2^{Ω(k)}$; therefore, an exponential size is required if one aims at preserving cuts exactly. In this paper, we study mimicking networks …
A breakthrough result of Cygan et al. (FOCS 2011) showed that connectivity problems parameterized by treewidth can be solved much faster than the previously best known time $\mathcal{O}^*(2^{\mathcal{O}(tw \log(tw))})$. Using their inspired Cut\&Count technique, they obtained $\mathcal{O}^*(α^{tw})$ time algorithms for many such problems. Moreover, they proved these running times to be optimal assuming the Strong Exponential-Time Hypothesis. Unfortunately, like other dynamic programming algorithms on tree decompositions, these algorithms also require exponential space, and this is widely belie…
We study the {\sc multicut on trees} and the {\sc generalized multiway Cut on trees} problems. For the {\sc multicut on trees} problem, we present a parameterized algorithm that runs in time $O^{*}(ρ^k)$, where $ρ= \sqrt{\sqrt{2} + 1} \approx 1.555$ is the positive root of the polynomial $x^4-2x^2-1$. This improves the current-best algorithm of Chen et al. that runs in time $O^{*}(1.619^k)$. For the {\sc generalized multiway cut on trees} problem, we show that this problem is solvable in polynomial time if the number of terminal sets is fixed; this answers an open question posed in a recent pa…
Decompositional parameters such as treewidth are commonly used to obtain fixed-parameter algorithms for NP-hard graph problems. For problems that are W[1]-hard parameterized by treewidth, a natural alternative would be to use a suitable analogue of treewidth that is based on edge cuts instead of vertex separators. While tree-cut width has been coined as such an analogue of treewidth for edge cuts, its algorithmic applications have often led to disappointing results: out of twelve problems where one would hope for fixed-parameter tractability parameterized by an edge-cut based analogue to treew…
In this paper we design {\sf FPT}-algorithms for two parameterized problems. The first is \textsc{List Digraph Homomorphism}: given two digraphs $G$ and $H$ and a list of allowed vertices of $H$ for every vertex of $G$, the question is whether there exists a homomorphism from $G$ to $H$ respecting the list constraints. The second problem is a variant of \textsc{Multiway Cut}, namely \textsc{Min-Max Multiway Cut}: given a graph $G$, a non-negative integer $\ell$, and a set $T$ of $r$ terminals, the question is whether we can partition the vertices of $G$ into $r$ parts such that (a) each part c…
Cutwidth of a digraph is a width measure introduced by Chudnovsky, Fradkin, and Seymour [4] in connection with development of a structural theory for tournaments, or more generally, for semi-complete digraphs. In this paper we provide an algorithm with running time 2^{O(\sqrt{k log k})} * n^{O(1)} that tests whether the cutwidth of a given n-vertex semi-complete digraph is at most k, improving upon the currently fastest algorithm of the second author [18] that works in 2^{O(k)} * n^2 time. As a byproduct, we obtain a new algorithm for Feedback Arc Set in tournaments (FAST) with running time 2^…
We study a broad class of graph partitioning problems, where each problem is specified by a graph $G=(V,E)$, and parameters $k$ and $p$. We seek a subset $U\subseteq V$ of size $k$, such that $α_1m_1 + α_2m_2$ is at most (or at least) $p$, where $α_1,α_2\in\mathbb{R}$ are constants defining the problem, and $m_1, m_2$ are the cardinalities of the edge sets having both endpoints, and exactly one endpoint, in $U$, respectively. This class of fixed cardinality graph partitioning problems (FGPP) encompasses Max $(k,n-k)$-Cut, Min $k$-Vertex Cover, $k$-Densest Subgraph, and $k$-Sparsest Subgraph. O…
Poljak and Turzík (Discrete Math. 1986) introduced the notion of λ-extendible properties of graphs as a generalization of the property of being bipartite. They showed that for any 0<λ<1 and λ-extendible property Π, any connected graph G on n vertices and m edges contains a subgraph H \in Π with at least λm+ (1-λ)/2 (n-1) edges. The property of being bipartite is 1/2-extendible, and thus this bound generalizes the Edwards-Erdős bound for Max-Cut. We define a variant, namely strong λ-extendibility, to which the bound applies. For a strongly λ-extendible graph property Π, we define the parameteri…
We study multi-budgeted variants of the classic minimum cut problem and graph separation problems that turned out to be important in parameterized complexity: Skew Multicut and Directed Feedback Arc Set. In our generalization, we assign colors $1,2,...,\ell$ to some edges and give separate budgets $k_{1},k_{2},...,k_{\ell}$. Let $E_{i}$ be the set of edges of color $i$. The solution $C$ for the multi-budgeted variant of a graph separation problem not only needs to satisfy the usual separation requirements, but also needs to satisfy that $|C\cap E_{i}|\leq k_{i}$ for every $i\in \{1,...,\ell\}$…
In an edge-colored graph, the cost incurred at a vertex on a path when two incident edges with different colors are traversed is called reload or changeover cost. The "Minimum Changeover Cost Arborescence" (MINCCA) problem consists in finding an arborescence with a given root vertex such that the total changeover cost of the internal vertices is minimized. It has been recently proved by Gözüpek et al. [TCS 2016] that the problem is FPT when parameterized by the treewidth and the maximum degree of the input graph. In this article we present the following results for the MINCCA problem: - the pr…
Every square matrix $A=(a_{uv})\in \mathcal{C}^{n\times n}$ can be represented as a digraph having $n$ vertices. In the digraph, a block (or 2-connected component) is a maximally connected subdigraph that has no cut-vertex. The determinant and the permanent of a matrix can be calculated in terms of the determinant and the permanent of some specific induced subdigraphs of the blocks in the digraph. Interestingly, these induced subdigraphs are vertex-disjoint and they partition the digraph. Such partitions of the digraph are called the $\mathcal{B}$-partitions. In this paper, first, we develop a…
In the Colored Clustering problem, one is asked to cluster edge-colored (hyper-)graphs whose colors represent interaction types. More specifically, the goal is to select as many edges as possible without choosing two edges that share an endpoint and are colored differently. Equivalently, the goal can also be described as assigning colors to the vertices in a way that fits the edge-coloring as well as possible. As this problem is NP-hard, we build on previous work by studying its parameterized complexity. We give a $2^{\mathcal O(k)} \cdot n^{\mathcal O(1)}$-time algorithm where $k$ is the numb…
The parameterized complexity of problems is often studied with respect to the size of their optimal solutions. However, for a maximization problem, the size of the optimal solution can be very large, rendering algorithms parameterized by it inefficient. Therefore, we suggest to study the parameterized complexity of maximization problems with respect to the size of the optimal solutions to their minimization versions. We examine this suggestion by considering the Maximal Minimal Vertex Cover (MMVC) problem, whose minimization version, Vertex Cover, is one of the most studied problems in the fie…
The parameterized complexity of counting minimum cuts stands as a natural question because Ball and Provan showed its #P-completeness. For any undirected graph $G=(V,E)$ and two disjoint sets of its vertices $S,T$, we design a fixed-parameter tractable algorithm which counts minimum edge $(S,T)$-cuts parameterized by their size $p$. Our algorithm operates on a transformed graph instance. This transformation, called drainage, reveals a collection of at most $n=\left| V \right|$ successive minimum $(S,T)$-cuts $Z_i$. We prove that any minimum $(S,T)$-cut $X$ contains edges of at least one cut $Z…
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…
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…
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 …
The k-CO-PATH SET problem asks, given a graph G and a positive integer k, whether one can delete k edges from G so that the remainder is a collection of disjoint paths. We give a linear-time fpt algorithm with complexity O^*(1.588^k) for deciding k-CO-PATH SET, significantly improving the previously best known O^*(2.17^k) of Feng, Zhou, and Wang (2015). Our main tool is a new O^*(4^{tw(G)}) algorithm for CO-PATH SET using the Cut&Count framework, where tw(G) denotes treewidth. In general graphs, we combine this with a branching algorithm which refines a 6k-kernel into reduced instances, which …
The Minimum Fill-in problem is to decide if a graph can be triangulated by adding at most k edges. Kaplan, Shamir, and Tarjan [FOCS 1994] have shown that the problem is solvable in time O(2^(O(k)) + k2 * nm) on graphs with n vertices and m edges and thus is fixed parameter tractable. Here, we give the first subexponential parameterized algorithm solving Minimum Fill-in in time O(2^(O(\sqrt{k} log k)) + k2 * nm). This substantially lower the complexity of the problem. Techniques developed for Minimum Fill-in can be used to obtain subexponential parameterized algorithms for several related probl…
The NP-hard Minimum Common String Partition problem asks whether two strings $x$ and $y$ can each be partitioned into at most $k$ substrings, called blocks, such that both partitions use exactly the same blocks in a different order. We present the first fixed-parameter algorithm for Minimum Common String Partition using only parameter $k$.
Checking whether a system of linear equations is consistent is a basic computational problem with ubiquitous applications. When dealing with inconsistent systems, one may seek an assignment that minimizes the number of unsatisfied equations. This problem is NP-hard and UGC-hard to approximate within any constant even for two-variable equations over the two-element field. We study this problem from the point of view of parameterized complexity, with the parameter being the number of unsatisfied equations. We consider equations defined over Euclidean domains - a family of commutative rings that …
Given a graph $G=(V,E)$ and an integer $k$, the Minimum Membership Dominating Set (MMDS) problem seeks to find a dominating set $S \subseteq V$ of $G$ such that for each $v \in V$, $|N[v] \cap S|$ is at most $k$. We investigate the parameterized complexity of the problem and obtain the following results about MMDS: W[1]-hardness of the problem parameterized by the pathwidth (and thus, treewidth) of the input graph. W[1]-hardness parameterized by $k$ on split graphs. An algorithm running in time $2^{\mathcal{O}(\textbf{vc})} |V|^{\mathcal{O}(1)}$, where $\textbf{vc}$ is the size of a minimum-si…
(see paper for full abstract) Cut problems and connectivity problems on digraphs are two well-studied classes of problems from the viewpoint of parameterized complexity. After a series of papers over the last decade, we now have (almost) tight bounds for the running time of several standard variants of these problems parameterized by two parameters: the number $k$ of terminals and the size $p$ of the solution. When there is evidence of FPT intractability, then the next natural alternative is to consider FPT approximations. In this paper, we show two types of results for several directed cut an…
We study the puzzle game Buttons and Scissors in which the goal is to remove all buttons from an $n\times m$ grid by a series of horizontal and vertical cuts. We show that the corresponding parameterized problem has an algorithm with time complexity $2^{O(k^2 \log k)} (n+m)^{O(1)}$, where $k$ is an upper bound on the number of cuts.
We design the first subexponential-time (parameterized) algorithms for several cut and cycle-hitting problems on $H$-minor free graphs. In particular, we obtain the following results (where $k$ is the solution-size parameter). 1. $2^{O(\sqrt{k}\log k)} \cdot n^{O(1)}$ time algorithms for Edge Bipartization and Odd Cycle Transversal; 2. a $2^{O(\sqrt{k}\log^4 k)} \cdot n^{O(1)}$ time algorithm for Edge Multiway Cut and a $2^{O(r \sqrt{k} \log k)} \cdot n^{O(1)}$ time algorithm for Vertex Multiway Cut, where $r$ is the number of terminals to be separated; 3. a $2^{O((r+\sqrt{k})\log^4 (rk))} \cd…
Computing the smallest number $q$ such that the vertices of a given graph can be properly $q$-colored is one of the oldest and most fundamental problems in combinatorial optimization. The $q$-Coloring problem has been studied intensively using the framework of parameterized algorithmics, resulting in a very good understanding of the best-possible algorithms for several parameterizations based on the structure of the graph. While there is an abundance of work for parameterizations based on decompositions of the graph by vertex separators, almost nothing is known about parameterizations based on…
We show the existence of an exact mimicking network of $k^{O(\log k)}$ edges for minimum multicuts over a set of terminals in an undirected graph, where $k$ is the total capacity of the terminals, as well as a method for computing a mimicking network of quasipolynomial size in polynomial time. As a consequence of the latter, several problems are shown to have quasipolynomial kernels, including Edge Multiway Cut, Group Feedback Edge Set for an arbitrary group, and Edge Multicut parameterized by the solution and the number of cut requests. The result combines the matroid-based irrelevant edge ap…
