Counting Induced Subgraphs: An Algebraic Approach to #W[1]-Hardness - Dörfler, Julian; Roth, Marc; Schmitt, Johannes; Wellnitz, Philip (Algorithmica, 2022)
We study the problem # I N D S U B ( 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, # I N D S U B ( Φ ) is hard for the class # W [ 1 ] , i.e., the parameterized counting equivalent of N P . 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 # I N D S U B ( Φ ) cannot be solved in time f ( k ) · n o ( 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.