License: 
Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.MFCS.2019.26
URN: urn:nbn:de:0030-drops-109703
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2019/10970/
Dörfler, Julian ;
Roth, Marc ;
Schmitt, Johannes ;
Wellnitz, Philip
Counting Induced Subgraphs: An Algebraic Approach to #W[1]-hardness
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}
}
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Keywords: |
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counting complexity, edge-transitive graphs, graph homomorphisms, induced subgraphs, parameterized complexity |
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Collection: |
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44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019) |
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Issue Date: |
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2019 |
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Date of publication: |
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20.08.2019 |
