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

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Abstract

We study the problem #INDSUB(Φ) of counting all induced subgraphs of size k in a graph G that satisfy the property Φ. It is shown that, given any graph property Φ that distinguishes independent sets from bicliques, #INDSUB(Φ) is hard for the class # W[1] , i.e., the parameterized counting equivalent of NP. Under additional suitable density conditions on Φ, satisfied e.g. by non-trivial monotone properties on bipartite graphs, we strengthen # W[1] -hardness by establishing that #INDSUB(Φ) cannot be solved in time f(k) · no(k) for any computable function f, unless the Exponential Time Hypothesis fails. Finally, we observe that our results remain true even if the input graph G is restricted to be bipartite and counting is done modulo a fixed prime.

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Dörfler, J., Roth, M., Schmitt, J., & Wellnitz, P. (2022). Counting Induced Subgraphs: An Algebraic Approach to #W[1]-Hardness. Algorithmica, 84(2), 379–404. https://doi.org/10.1007/s00453-021-00894-9

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