The Fine-Grained Complexity of Graph Homomorphism Problems: Towards the Okrasa and Rzążewski Conjecture

TL;DR

The paper advances the fine-grained understanding of $H$-Coloring by framing it as CSP$(E_H)$ and studying the associated (partial) polymorphisms. It proves a largely trivial inclusion structure for ${\mathrm{pPol}}(H)$ among graphs on the same vertex set and identifies a sharp higher-arity criterion tied to the odd-girth that governs nontrivial inclusions relative to arbitrary relations. It also develops an algebraic lens on projective cores, showing that many natural graph families (cliques, odd cycles, and several cores) are projective and that NEQ$_k$ becomes pp-definable exactly in these cases. Finally, the conjecture of Okrasa and Rzażewski is verified for all graphs with at most seven vertices, illustrating the efficacy of the algebraic approach and suggesting a path toward a full characterization of projective cores and their polymorphisms.

Abstract

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($E_H$) $\subseteq$ pPol($E_{H'}$), unless $E_{H'}= \emptyset$ 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.

Figures (11)

• Figure 1: The Grötzsch graph (left) and the Petersen graph (right)
• Figure 2: Every core graph with at most 6 vertices
• Figure 3: The "trivial" 7-cores. We prove in Theorem \ref{['thm:Trivial7coresProjective']} that they are projective.
• Figure 4: The "sporadic" 7-cores. We prove in Theorem \ref{['thm:conjecture_true_7_vertices']} that they are projective.
• Figure 5: Up to isomorphism, any "sporadic" $7$-core (not $K_7,C_7,\overline{C_7}$ or $C_5+2$) $(\{a, b, c, d, e, u, v\}, E$) must satisfy this motif: $\{a,b,c,d,e\}$ induces a $C_5$, $(b,v) \notin E$, all dashed edges are allowed (as long as $(u,v) \notin E$, or $(e,u) \notin E$), that $(c,u) \in E$ or $(c,v) \in E$, and that $(d,u) \in E$ or $(d,v) \in E$.

Theorems & Definitions (78)

• Theorem 1: hell1990complexity
• Definition 2
• Definition 3
• Definition 4
• Theorem 5: poschel2013galois
• Theorem 6: jeavons1998
• Theorem 7: jonsson2017
• Theorem 8
• proof
• Lemma 9