We consider the problem of counting, in a given graph, the number of induced k-vertex subgraphs which have an even number of edges, and also the complementary problem of counting the k-vertex induced subgraphs having an odd number of edges. We demonstrate that both problems are #W-hard when parameterised by k, in fact proving a somewhat stronger result about counting subgraphs with a property that only holds for some subset of k-vertex subgraphs which have an even (respectively odd) number of edges. On the other hand, we show that each of the problems admits an FPTRAS. These approximation schemes are based on a surprising structural result, which exploits ideas from Ramsey theory.
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Research supported by EPSRC grant EP/I011935/1 (“Computational Counting”)
This work was undertaken when Kitty Meeks was affiliated with the School of Mathematical Sciences, Queen Mary University of London
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Jerrum, M., Meeks, K. The parameterised complexity of counting even and odd induced subgraphs. Combinatorica 37, 965–990 (2017). https://doi.org/10.1007/s00493-016-3338-5
Mathematics Subject Classification (2000)