The Parameterized Complexity of s-Club with Triangle and Seed Constraints

IF 0.6 4区计算机科学 Q4 COMPUTER SCIENCE, THEORY & METHODS

Theory of Computing Systems Pub Date : 2023-08-12 DOI:10.1007/s00224-023-10135-x

Jaroslav Garvardt, Christian Komusiewicz, Frank Sommer

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引用次数: 1

Abstract

Abstract The s - Club problem asks whether a given undirected graph G contains a vertex set S of size at least k such that G [ S ], the subgraph of G induced by S , has diameter at most s . We consider variants of s - Club where one additionally demands that each vertex of G [ S ] is contained in at least $$\ell $$ ℓ triangles in G [ S ], that each edge of G [ S ] is contained in at least $$\ell $$ ℓ triangles in G [ S ], or that S contains a given set W of seed vertices. We show that in general these variants are W[1]-hard when parameterized by the solution size k , making them significantly harder than the unconstrained s - Club problem. On the positive side, we obtain some FPT algorithms for the case when $$\ell =1$$ ℓ = 1 and for the case when G [ W ], the graph induced by the set of seed vertices, is a clique.

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具有三角形和种子约束的s-Club的参数化复杂度

s - Club问题是问给定的无向图G是否包含一个大小至少为k的顶点集s，使得由s诱导的G的子图G [s]的直径最大为s。我们考虑s - Club的变体，其中一个额外要求G [s]的每个顶点至少包含在G [s]中的$$\ell $$个三角形中，G [s]的每条边至少包含在G [s]中的$$\ell $$个三角形中，或者s包含给定集合W的种子顶点。我们表明，当解大小k参数化时，这些变量通常是W[1]-困难的，这使得它们比无约束的s - Club问题要困难得多。在积极的方面，我们得到了$$\ell =1$$ = 1和G [W](由种子顶点集合诱导的图)为团的情况下的FPT算法。

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来源期刊

Theory of Computing Systems 工程技术-计算机：理论方法

CiteScore

1.90

自引率

0.00%

发文量

审稿时长

6-12 weeks

期刊介绍： TOCS is devoted to publishing original research from all areas of theoretical computer science, ranging from foundational areas such as computational complexity, to fundamental areas such as algorithms and data structures, to focused areas such as parallel and distributed algorithms and architectures.