The Complexity of Counting Homomorphisms to Cactus Graphs Modulo 2

TL;DR

This paper characterizes the computational complexity of counting graph homomorphisms modulo 2 for cactus graphs, identifying which cases are polynomial-time solvable and which are parity-P complete, thus advancing understanding of modular counting problems.

Contribution

It provides a complete complexity classification for counting homomorphisms modulo 2 to cactus graphs, extending previous results from trees to this broader class.

Findings

Some cactus graphs allow polynomial-time counting

Most cactus graphs lead to parity-P complete problems

Polynomial-time determination of tractability for given H

Abstract

A homomorphism from a graph G to a graph H is a function from V(G) to V(H) that preserves edges. Many combinatorial structures that arise in mathematics and computer science can be represented naturally as graph homomorphisms and as weighted sums of graph homomorphisms. In this paper, we study the complexity of counting homomorphisms modulo 2. The complexity of modular counting was introduced by Papadimitriou and Zachos and it has been pioneered by Valiant who famously introduced a problem for which counting modulo 7 is easy but counting modulo 2 is intractable. Modular counting provides a rich setting in which to study the structure of homomorphism problems. In this case, the structure of the graph H has a big influence on the complexity of the problem. Thus, our approach is graph-theoretic. We give a complete solution for the class of cactus graphs, which are connected graphs in which every edge belongs to at most one cycle. Cactus graphs arise in many applications such as the modelling of wireless sensor networks and the comparison of genomes. We show that, for some cactus graphs H, counting homomorphisms to H modulo 2 can be done in polynomial time. For every other fixed cactus graph H, the problem is complete for the complexity class parity-P which is a wide complexity class to which every problem in the polynomial hierarchy can be reduced (using randomised reductions). Determining which H lead to tractable problems can be done in polynomial time. Our result builds upon the work of Faben and Jerrum, who gave a dichotomy for the case in which H is a tree.