2,486 research outputs found

    On the Parameterized Complexity of Simultaneous Deletion Problems

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    For a family of graphs F, an n-vertex graph G, and a positive integer k, the F-Deletion problem asks whether we can delete at most k vertices from G to obtain a graph in F. F-Deletion generalizes many classical graph problems such as Vertex Cover, Feedback Vertex Set, and Odd Cycle Transversal. A (multi) graph G = (V, cup_{i=1}^{alpha} E_{i}), where the edge set of G is partitioned into alpha color classes, is called an alpha-edge-colored graph. A natural extension of the F-Deletion problem to edge-colored graphs is the Simultaneous (F_1, ldots, F_alpha)-Deletion problem. In the latter problem, we are given an alpha-edge-colored graph G and the goal is to find a set S of at most k vertices such that each graph G_i - S, where G_i = (V, E_i) and 1 leq i leq alpha, is in F_i. Recently, a subset of the authors considered the aforementioned problem with F_1 = ldots = F_alpha being the family of all forests. They showed that the problem is fixed-parameter tractable when parameterized by k and alpha, and can be solved in O(2^{O(alpha k)}n^{O(1)}) time. In this work, we initiate the investigation of the complexity of Simultaneous (F_1, ldots, F_alpha)-Deletion with different families of graphs. In the process, we obtain a complete characterization of the parameterized complexity of this problem when one or more of the F_i\u27s is the class of bipartite graphs and the rest (if any) are forests. We show that if F_1 is the family of all bipartite graphs and each of F_2 = F_3 = ldots = F_alpha is the family of all forests then the problem is fixed-parameter tractable parameterized by k and alpha. However, even when F_1 and F_2 are both the family of all bipartite graphs, then the Simultaneous (F_1, F_2)-Deletion} problem itself is already W[1]-hard

    Simultaneous Feedback Vertex Set: A Parameterized Perspective

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    Given a family of graphs F\mathcal{F}, a graph GG, and a positive integer kk, the F\mathcal{F}-Deletion problem asks whether we can delete at most kk vertices from GG to obtain a graph in F\mathcal{F}. F\mathcal{F}-Deletion generalizes many classical graph problems such as Vertex Cover, Feedback Vertex Set, and Odd Cycle Transversal. A graph G=(V,i=1αEi)G = (V, \cup_{i=1}^{\alpha} E_{i}), where the edge set of GG is partitioned into α\alpha color classes, is called an α\alpha-edge-colored graph. A natural extension of the F\mathcal{F}-Deletion problem to edge-colored graphs is the α\alpha-Simultaneous F\mathcal{F}-Deletion problem. In the latter problem, we are given an α\alpha-edge-colored graph GG and the goal is to find a set SS of at most kk vertices such that each graph GiSG_i \setminus S, where Gi=(V,Ei)G_i = (V, E_i) and 1iα1 \leq i \leq \alpha, is in F\mathcal{F}. In this work, we study α\alpha-Simultaneous F\mathcal{F}-Deletion for F\mathcal{F} being the family of forests. In other words, we focus on the α\alpha-Simultaneous Feedback Vertex Set (α\alpha-SimFVS) problem. Algorithmically, we show that, like its classical counterpart, α\alpha-SimFVS parameterized by kk is fixed-parameter tractable (FPT) and admits a polynomial kernel, for any fixed constant α\alpha. In particular, we give an algorithm running in 2O(αk)nO(1)2^{O(\alpha k)}n^{O(1)} time and a kernel with O(αk3(α+1))O(\alpha k^{3(\alpha + 1)}) vertices. The running time of our algorithm implies that α\alpha-SimFVS is FPT even when αo(logn)\alpha \in o(\log n). We complement this positive result by showing that for αO(logn)\alpha \in O(\log n), where nn is the number of vertices in the input graph, α\alpha-SimFVS becomes W[1]-hard. Our positive results answer one of the open problems posed by Cai and Ye (MFCS 2014)

    Finding Even Subgraphs Even Faster

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    Problems of the following kind have been the focus of much recent research in the realm of parameterized complexity: Given an input graph (digraph) on nn vertices and a positive integer parameter kk, find if there exist kk edges (arcs) whose deletion results in a graph that satisfies some specified parity constraints. In particular, when the objective is to obtain a connected graph in which all the vertices have even degrees---where the resulting graph is \emph{Eulerian}---the problem is called Undirected Eulerian Edge Deletion. The corresponding problem in digraphs where the resulting graph should be strongly connected and every vertex should have the same in-degree as its out-degree is called Directed Eulerian Edge Deletion. Cygan et al. [\emph{Algorithmica, 2014}] showed that these problems are fixed parameter tractable (FPT), and gave algorithms with the running time 2O(klogk)nO(1)2^{O(k \log k)}n^{O(1)}. They also asked, as an open problem, whether there exist FPT algorithms which solve these problems in time 2O(k)nO(1)2^{O(k)}n^{O(1)}. In this paper we answer their question in the affirmative: using the technique of computing \emph{representative families of co-graphic matroids} we design algorithms which solve these problems in time 2O(k)nO(1)2^{O(k)}n^{O(1)}. The crucial insight we bring to these problems is to view the solution as an independent set of a co-graphic matroid. We believe that this view-point/approach will be useful in other problems where one of the constraints that need to be satisfied is that of connectivity

    Simultaneous Feedback Edge Set: A Parameterized Perspective

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    In a recent article Agrawal et al. (STACS 2016) studied a simultaneous variant of the classic Feedback Vertex Set problem, called Simultaneous Feedback Vertex Set (Sim-FVS). In this problem the input is an n-vertex graph G, an integer k and a coloring function col : E(G) -> 2^[alpha]and the objective is to check whether there exists a vertex subset S of cardinality at most k in G such that for all i in [alpha], G_i - S is acyclic. Here, G_i = (V (G), {e in E(G) | i in col(e)}) and [alpha] = {1,...,alpha}. In this paper we consider the edge variant of the problem, namely, Simultaneous Feedback Edge Set (Sim-FES). In this problem, the input is same as the input of Sim-FVS and the objective is to check whether there is an edge subset S of cardinality at most k in G such that for all i in [alpha], G_i - S is acyclic. Unlike the vertex variant of the problem, when alpha = 1, the problem is equivalent to finding a maximal spanning forest and hence it is polynomial time solvable. We show that for alpha = 3 Sim-FES is NP-hard by giving a reduction from Vertex Cover on cubic-graphs. The same reduction shows that the problem does not admit an algorithm of running time O(2^o(k) n^O(1)) unless ETH fails. This hardness result is complimented by an FPT algorithm for Sim-FES running in time O(2^((omega k alpha) + (alpha log k)) n^O(1)), where omega is the exponent in the running time of matrix multiplication. The same algorithm gives a polynomial time algorithm for the case when alpha = 2. We also give a kernel for Sim-FES with (k alpha)^O(alpha) vertices. Finally, we consider the problem Maximum Simultaneous Acyclic Subgraph. Here, the input is a graph G, an integer q and, a coloring function col : E(G) -> 2^[alpha] . The question is whether there is a edge subset F of cardinality at least q in G such that for all i in [alpha], G[F_i] is acyclic. Here, F_i = {e in F | i in col(e)}. We give an FPT algorithm for Maximum Simultaneous Acyclic Subgraph running in time O(2^(omega q alpha) n^O(1) ). All our algorithms are based on parameterized version of the Matroid Parity problem

    Parameterized Algorithmics for Computational Social Choice: Nine Research Challenges

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    Computational Social Choice is an interdisciplinary research area involving Economics, Political Science, and Social Science on the one side, and Mathematics and Computer Science (including Artificial Intelligence and Multiagent Systems) on the other side. Typical computational problems studied in this field include the vulnerability of voting procedures against attacks, or preference aggregation in multi-agent systems. Parameterized Algorithmics is a subfield of Theoretical Computer Science seeking to exploit meaningful problem-specific parameters in order to identify tractable special cases of in general computationally hard problems. In this paper, we propose nine of our favorite research challenges concerning the parameterized complexity of problems appearing in this context
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