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2014 IEEE 55th Annual Symposium on Foundations of Computer Science (FOCS) (2014)
Philadelphia, PA, USA
Oct. 18, 2014 to Oct. 21, 2014
ISSN: 0272-5428
ISBN: 978-1-4799-6517-5
pp: 130-139
For a class C of graphs, #Sub(C) is the counting problem that, given a graph H from C and an arbitrary graph G, asks for the number of subgraphs of G isomorphic to H. It is known that if C has bounded vertex-cover number (equivalently, the size of the maximum matching in C is bounded), then #Sub(C) is polynomial-time solvable. We complement this result with a corresponding lower bound: if C is any recursively enumerable class of graphs with unbounded vertex-cover number, then #Sub(C) is #W[1]-hard parameterized by the size of H and hence not polynomial-time solvable and not even fixed-parameter tractable, unless FPT is equal to #W[1]. As a first step of the proof, we show that counting k-matchings in bipartite graphs is #W[1]-hard. Recently, Curticapean [ICALP 2013] proved the #W[1]-hardness of counting k-matchings in general graphs, our result strengthens this statement to bipartite graphs with a considerably simpler proof and even shows that, assuming the Exponential Time Hypothesis (ETH), there is no f(k)*no(k/log(k)) time algorithm for counting k-matchings in bipartite graphs for any computable function f. As a consequence, we obtain an independent and somewhat simpler proof of the classical result of Flum and Grohe [SICOMP 2004] stating that counting paths of length k is #W[1]-hard, as well as a similar almost-tight ETH-based lower bound on the exponent.
Color, Bipartite graph, Complexity theory, Polynomials, Computer science, Context, Standards

R. Curticapean and D. Marx, "Complexity of Counting Subgraphs: Only the Boundedness of the Vertex-Cover Number Counts," 2014 IEEE 55th Annual Symposium on Foundations of Computer Science (FOCS), Philadelphia, PA, USA, 2014, pp. 130-139.
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