文章基本信息

标题：Counting Induced Subgraphs: An Algebraic Approach to #W[1]-hardness
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作者：Julian Dörfler ; Marc Roth ; Johannes Schmitt 等
期刊名称：LIPIcs : Leibniz International Proceedings in Informatics
电子版ISSN：1868-8969
出版年度：2019
卷号：138
页码：1-14
DOI：10.4230/LIPIcs.MFCS.2019.26
出版社：Schloss Dagstuhl -- Leibniz-Zentrum fuer Informatik
摘要：We study the problem #IndSub(Phi) of counting all induced subgraphs of size k in a graph G that satisfy the property Phi. This problem was introduced by Jerrum and Meeks and shown to be #W[1]-hard when parameterized by k for some families of properties Phi including, among others, connectivity [JCSS 15] and even- or oddness of the number of edges [Combinatorica 17]. Very recently [IPEC 18], two of the authors introduced a novel technique for the complexity analysis of #IndSub(Phi), inspired by the "topological approach to evasiveness" of Kahn, Saks and Sturtevant [FOCS 83] and the framework of graph motif parameters due to Curticapean, Dell and Marx [STOC 17], allowing them to prove hardness of a wide range of properties Phi. In this work, we refine this technique for graph properties that are non-trivial on edge-transitive graphs with a prime power number of edges. In particular, we fully classify the case of monotone bipartite graph properties: It is shown that, given any graph property Phi that is closed under the removal of vertices and edges, and that is non-trivial for bipartite graphs, the problem #IndSub(Phi) is #W[1]-hard and cannot be solved in time f(k)* n^{o(k)} for any computable function f, unless the Exponential Time Hypothesis fails. This holds true even if the input graph is restricted to be bipartite and counting is done modulo a fixed prime. A similar result is shown for properties that are closed under the removal of edges only.
关键词：counting complexity; edge-transitive graphs; graph homomorphisms; induced subgraphs; parameterized complexity