Counting Homomorphisms to $K_4$-minor-free Graphs, modulo 2

TL;DR

This paper proves a conjecture about the computational complexity of counting homomorphisms modulo 2 for graphs excluding $K_4$ as a minor, establishing polynomial-time solvability or $ ext{⊕P}$-completeness, and introduces new hardness proof techniques.

Contribution

It verifies Faben and Jerrum's conjecture for $K_4$-minor-free graphs and develops a novel method for deriving hardness from global substructures of the fixed graph.

Findings

Confirmed the conjecture for all $K_4$-minor-free graphs.

Established no subexponential algorithms for $ ext{⊕P}$-complete cases under ETH.

Unified previous partial results and extended classification to list homomorphisms modulo 2.

Abstract

We study the problem of computing the parity of the number of homomorphisms from an input graph $G$ to a fixed graph $H$. Faben and Jerrum [ToC'15] introduced an explicit criterion on the graph $H$ and conjectured that, if satisfied, the problem is solvable in polynomial time and, otherwise, the problem is complete for the complexity class $\oplus\mathrm{P}$ of parity problems. We verify their conjecture for all graphs $H$ that exclude the complete graph on $4$ vertices as a minor. Further, we rule out the existence of a subexponential-time algorithm for the $\oplus\mathrm{P}$-complete cases, assuming the randomised Exponential Time Hypothesis. Our proofs introduce a novel method of deriving hardness from globally defined substructures of the fixed graph $H$. Using this, we subsume all prior progress towards resolving the conjecture (Faben and Jerrum [ToC'15]; G\"obel, Goldberg and Richerby [ToCT'14,'16]). As special cases, our machinery also yields a proof of the conjecture for graphs with maximum degree at most $3$, as well as a full classification for the problem of counting list homomorphisms, modulo $2$.