The Complexity of Homomorphism Indistinguishability
For every graph class {F}, let HomInd({F}) be the problem of deciding whether two given graphs are homomorphism-indistinguishable over {F}, i.e., for every graph F in {F}, the number hom(F, G) of homomorphisms from F to G equals the corresponding number hom(F, H) for H. For several natural graph classes (such as paths, trees, bounded treewidth graphs), homomorphism-indistinguishability over the class has an efficient structural characterization, resulting in polynomial time solvability [H. Dell et al., 2018].
In particular, it is known that two non-isomorphic graphs are homomorphism-indistinguishable over the class {T}_k of graphs of treewidth k if and only if they are not distinguished by k-dimensional Weisfeiler-Leman algorithm, a central heuristic for isomorphism testing: this characterization implies a polynomial time algorithm for HomInd({T}_k), for every fixed k in N. In this paper, we show that there is a polynomial-time-decidable class {F} of undirected graphs of bounded treewidth such that HomInd({F}) is undecidable.
Our second hardness result concerns the class {K} of complete graphs. We show that HomInd({K}) is co-NP-hard, and in fact, we show completeness for the class C_=P (under P-time Turing reductions). On the algorithmic side, we show that HomInd({P}) can be solved in polynomial time for the class {P} of directed paths. We end with a brief study of two variants of the HomInd({F}) problem: (a) the problem of lexographic-comparison of homomorphism numbers of two graphs, and (b) the problem of computing certain distance-measures (defined via homomorphism numbers) between two graphs.
graph homomorphism numbers
counting complexity
treewidth
Theory of computation~Graph algorithms analysis
Mathematics of computing~Graph theory
54:1-54:13
Regular Paper
Jan
Böker
Jan Böker
RWTH Aachen University, Aachen, Germany
https://orcid.org/0000-0003-4584-121X
Yijia
Chen
Yijia Chen
Fudan University, Shanghai, China
https://orcid.org/0000-0001-7033-9593
Martin
Grohe
Martin Grohe
RWTH Aachen University, Aachen, Germany
https://orcid.org/0000-0002-0292-9142
Gaurav
Rattan
Gaurav Rattan
RWTH Aachen University, Aachen, Germany
https://orcid.org/0000-0002-5095-860X
Supported by DFG grant RA 3242/1-1.
10.4230/LIPIcs.MFCS.2019.54
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Jan Böker, Yijia Chen, Martin Grohe, and Gaurav Rattan
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