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 nontrivial on edgetransitive 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 nontrivial 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 = {26:126:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959771177},
ISSN = {18688969},
year = {2019},
volume = {138},
editor = {Peter Rossmanith and Pinar Heggernes and JoostPieter Katoen},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2019/10970},
URN = {urn:nbn:de:0030drops109703},
doi = {10.4230/LIPIcs.MFCS.2019.26},
annote = {Keywords: counting complexity, edgetransitive graphs, graph homomorphisms, induced subgraphs, parameterized complexity}
}
Keywords: 

counting complexity, edgetransitive graphs, graph homomorphisms, induced subgraphs, parameterized complexity 
Collection: 

44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019) 
Issue Date: 

2019 
Date of publication: 

20.08.2019 