首页/论文/2005.13938

arXiv 2005.139382020-05cs.DScs.DMmath.CO

Computing Subset Transversals in $H$-Free Graphs

Nick Brettell, Matthew Johnson, Giacomo Paesani, Daniël Paulusma

备注：: Minor changes incorporating suggestions of reviewers

英文摘要：

We study the computational complexity of two well-known graph transversal problems, namely Subset Feedback Vertex Set and Subset Odd Cycle Transversal, by restricting the input to $H$-free graphs, that is, to graphs that do not contain some fixed graph~$H$ as an induced subgraph. By combining known and new results, we determine the computational complexity of both problems on $H$-free graphs for every graph $H$ except when $H=sP_1+P_4$ for some $s\geq 1$. As part of our approach, we introduce the Subset Vertex Cover problem and prove that it is polynomial-time solvable for $(sP_1+P_4)$-free graphs for every $s\geq 1$.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

Computing Weighted Subset Transversals in $H$-Free Graphs

For the Odd Cycle Transversal problem, the task is to find a small set $S$ of vertices in a graph that intersects every cycle of odd length. The Subset Odd Cycle Transversal problem requires S to intersect only those odd cycles that include a vertex of a distinguished vertex subset $T$. If we are given weights for the vertices, we ask instead that $S$ has small weight: this is the problem Weighted Subset Odd Cycle Transversal. We prove an almost-complete complexity dichotomy for Weighted Subset Odd Cycle Transversal for graphs that do not contain a graph $H$ as an induced subgraph. Our general…

2021-06-14cs.DMcs.DSmath.CO

Inapproximability of $H$-Transversal/Packing

Given an undirected graph $G = (V_G, E_G)$ and a fixed "pattern" graph $H = (V_H, E_H)$ with $k$ vertices, we consider the $H$-Transversal and $H$-Packing problems. The former asks to find the smallest $S \subseteq V_G$ such that the subgraph induced by $V_G \setminus S$ does not have $H$ as a subgraph, and the latter asks to find the maximum number of pairwise disjoint $k$-subsets $S_1, ..., S_m \subseteq V_G$ such that the subgraph induced by each $S_i$ has $H$ as a subgraph. We prove that if $H$ is 2-connected, $H$-Transversal and $H$-Packing are almost as hard to approximate as general $k$…

2015-06-20cs.CCcs.DS

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, Sub…

2021-06-30cs.CCcs.DMcs.DS

Multi-transversals for Triangles and the Tuza's Conjecture

In this paper, we study a primal and dual relationship about triangles: For any graph $G$, let $ν(G)$ be the maximum number of edge-disjoint triangles in $G$, and $τ(G)$ be the minimum subset $F$ of edges such that $G \setminus F$ is triangle-free. It is easy to see that $ν(G) \leq τ(G) \leq 3 ν(G)$, and in fact, this rather obvious inequality holds for a much more general primal-dual relation between $k$-hyper matching and covering in hypergraphs. Tuza conjectured in $1981$ that $τ(G) \leq 2 ν(G)$, and this question has received attention from various groups of researchers in discrete mathema…

2021-02-03cs.DMcs.DSmath.CO

New Results on Monotone Dualization and Generating Hypergraph Transversals

We consider the problem of dualizing a monotone CNF (equivalently, computing all minimal transversals of a hypergraph), whose associated decision problem is a prominent open problem in NP-completeness. We present a number of new polynomial time resp. output-polynomial time results for significant cases, which largely advance the tractability frontier and improve on previous results. Furthermore, we show that duality of two monotone CNFs can be disproved with limited nondeterminism. More precisely, this is feasible in polynomial time with O(chi(n) * log n) suitably guessed bits, where chi(n) is…

2002-04-26cs.CCcs.DS

On Polynomial Kernels for Structural Parameterizations of Odd Cycle Transversal

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…

2011-07-19cs.DS

更多相关论文收起更多论文

Hitting Weighted Even Cycles in Planar Graphs

A classical branch of graph algorithms is graph transversals, where one seeks a minimum-weight subset of nodes in a node-weighted graph $G$ which intersects all copies of subgraphs~$F$ from a fixed family $\mathcal F$. Many such graph transversal problems have been shown to admit polynomial-time approximation schemes (PTAS) for planar input graphs $G$, using a variety of techniques like the shifting technique (Baker, J. ACM 1994), bidimensionality (Fomin et al., SODA 2011), or connectivity domination (Cohen-Addad et al., STOC 2016). These techniques do not seem to apply to graph transversals w…

2021-08-01cs.DMcs.DS

在 $P_k$-自由图中的奇环横贯问题

Odd Cycle Transversal in $P_k$-Free Graphs

Indiana State University（印第安纳州立大学）

针对$P_k$-自由图，提出基于二分图环分解的常数因子近似算法，奇数$k$时近似比为$k-2$，偶数$k$时为$k-3$。

2026-06-05cs.DMcs.DS

关于最小三角形边横截的分布式计算

On Distributed Computation of the Minimum Triangle Edge Transversal

本文研究分布式网络中最小三角形边横截的计算，证明精确求解在CONGEST模型中需近二次方轮数，并提出在poly log n轮内获得(1+ε)-近似及通过归约到超图顶点覆盖实现更快近似的方法。

2024-02-21cs.DCcs.DS

$P_5$-free图上的多项式时间奇环横截面

Odd Cycle Transversal on $P_5$-free Graphs in Polynomial Time

本文证明了Odd Cycle Transversal问题在$P_5$-free图上存在多项式时间算法，解决了此前提出的开放问题。

2024-02-18cs.DS

Polynomial Delay Algorithm for Listing Minimal Edge Dominating sets in Graphs

The Transversal problem, i.e, the enumeration of all the minimal transversals of a hypergraph in output-polynomial time, i.e, in time polynomial in its size and the cumulated size of all its minimal transversals, is a fifty years old open problem, and up to now there are few examples of hypergraph classes where the problem is solved. A minimal dominating set in a graph is a subset of its vertex set that has a non empty intersection with the closed neighborhood of every vertex. It is proved in [M. M. Kanté, V. Limouzy, A. Mary, L. Nourine, On the Enumeration of Minimal Dominating Sets and Relat…

2014-07-08cs.DS

Computing the Union Join and Subset Graph of Acyclic Hypergraphs in Subquadratic Time

We investigate the two problems of computing the union join graph as well as computing the subset graph for acyclic hypergraphs and their subclasses. In the union join graph $G$ of an acyclic hypergraph $H$, each vertex of $G$ represents a hyperedge of $H$ and two vertices of $G$ are adjacent if there exits a join tree $T$ for $H$ such that the corresponding hyperedges are adjacent in $T$. The subset graph of a hypergraph $H$ is a directed graph where each vertex represents a hyperedge of $H$ and there is a directed edge from a vertex $u$ to a vertex $v$ if the hyperedge corresponding to $u$ i…

2021-04-14cs.DS

Computing the Expected Value and Variance of Geometric Measures

Let P be a set of points in R^d, and let M be a function that maps any subset of P to a positive real number. We examine the problem of computing the exact mean and variance of M when a subset of points in P is selected according to a well-defined random distribution. We consider two distributions; in the first distribution (which we call the Bernoulli distribution), each point p in P is included in the random subset independently, with probability pi(p). In the second distribution (the fixed-size distribution), a subset of exactly s points is selected uniformly at random among all possible su…

2016-10-12cs.DS

Parameterized Algorithms for Deletion to (r,l)-graphs

For fixed integers $r,\ell \geq 0$, a graph $G$ is called an {\em $(r,\ell)$-graph} if the vertex set $V(G)$ can be partitioned into $r$ independent sets and $\ell$ cliques. This brings us to the following natural parameterized questions: {\sc Vertex $(r,\ell)$-Partization} and {\sc Edge $(r,\ell)$-Partization}. An input to these problems consist of a graph $G$ and a positive integer $k$ and the objective is to decide whether there exists a set $S\subseteq V(G)$ ($S\subseteq E(G)$) such that the deletion of $S$ from $G$ results in an $(r,\ell)$-graph. These problems generalize well studied pro…

2015-04-30cs.CCcs.DS