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

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 parameteriz...

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Bibliographic Details
Published in:Algorithmica Vol. 84; no. 2; pp. 379 - 404
Main Authors: Dörfler, Julian, Roth, Marc, Schmitt, Johannes, Wellnitz, Philip
Format: Journal Article
Language:English
Published: New York Springer US 01.02.2022
Springer Nature B.V
Subjects:
Algorithm Analysis and Problem Complexity
Algorithms
Computer Science
Computer Systems Organization and Communication Networks
Data Structures and Information Theory
Graph theory
Hardness
Mathematics of Computing
Theory of Computation
Counting complexity
Edge-transitive graphs
Graph homomorphisms
Induced subgraphs
Parameterized complexity
ISSN:0178-4617, 1432-0541
Online Access:Get full text
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Summary: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.
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ISSN:0178-4617
1432-0541
DOI:10.1007/s00453-021-00894-9