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DOI: 10.4230/LIPIcs.MFCS.2019.26
URN: urn:nbn:de:0030-drops-109703
URL: http://drops.dagstuhl.de/opus/volltexte/2019/10970/
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Dörfler, Julian ; Roth, Marc ; Schmitt, Johannes ; Wellnitz, Philip

Counting Induced Subgraphs: An Algebraic Approach to #W[1]-hardness

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Abstract

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.

BibTeX - Entry

@InProceedings{drfler_et_al:LIPIcs:2019:10970,
  author =	{Julian D{\"o}rfler and Marc Roth and Johannes Schmitt and Philip Wellnitz},
  title =	{{Counting Induced Subgraphs: An Algebraic Approach to #W[1]-hardness}},
  booktitle =	{44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)},
  pages =	{26:1--26:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-117-7},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{138},
  editor =	{Peter Rossmanith and Pinar Heggernes and Joost-Pieter Katoen},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2019/10970},
  URN =		{urn:nbn:de:0030-drops-109703},
  doi =		{10.4230/LIPIcs.MFCS.2019.26},
  annote =	{Keywords: counting complexity, edge-transitive graphs, graph homomorphisms, induced subgraphs, parameterized complexity}
}

Keywords: counting complexity, edge-transitive graphs, graph homomorphisms, induced subgraphs, parameterized complexity
Seminar: 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)
Issue Date: 2019
Date of publication: 23.08.2019


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