The Fine-Grained Complexity of Graph Homomorphism Problems: Towards the Okrasa and Rzążewski Conjecture
arxiv(2024)
摘要
In this paper we are interested in the fine-grained complexity of deciding
whether there is a homomorphism from an input graph G to a fixed graph H
(the H-Coloring problem). The starting point is that these problems can be
viewed as constraint satisfaction problems (CSPs), and that (partial)
polymorphisms of binary relations are of paramount importance in the study of
complexity classes of such CSPs.
Thus, we first investigate the expressivity of binary symmetric relations
E_H and their corresponding (partial) polymorphisms pPol(E_H). For
irreflexive graphs we observe that there is no pair of graphs H and H' such
that pPol(E_H) ⊆ pPol(E_H'), unless E_H'= ∅ or H
=H'. More generally we show the existence of an n-ary relation R whose
partial polymorphisms strictly subsume those of H and such that CSP(R) is
NP-complete if and only if H contains an odd cycle of length at most n.
Motivated by this we also describe the sets of total polymorphisms of
nontrivial cliques, odd cycles, as well as certain cores, and we give an
algebraic characterization of projective cores. As a by-product, we settle the
Okrasa and Rzążewski conjecture for all graphs of at most 7 vertices.
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