Testing List H-homomorphisms

• Authors: Yuichi Yoshida
• Venue: CCC '12 Proceedings of the 2012 IEEE Conference on Computational Complexity (CCC)
• Year: 2012

Let $H$ be an undirected graph. In the List $H$-Homomorphism Problem, given an undirected graph $G$ with a list constraint $L(v) \subseteq V(H)$ for each variable $v \in V(G)$, the objective is to find a list $H$-homomorphism $f:V(G) \to V(H)$, that is, $f(v) \in L(v)$ for every $v \in V(G)$ and $(f(u), f(v)) \in E(H)$ whenever $(u, v) \in E(G)$. We consider testing list $H$-homomorphism: given a map $f:V(G) \to V(H)$ as an oracle, the objective is to decide with high probability whether $f$ is a list $H$-homomorphism or \textit{far} from any list $H$-homomorphisms. The efficiency of an algorithm is measured by the number of accesses to $f$. In this paper, we classify graphs $H$ with respect to the query complexity for testing list $H$-homomorphisms. Specifically, we show that (i) list $H$-homomorphisms are testable with a constant number of queries if and only if $H$ is a reflexive complete graph or an irreflexive complete bipartite graph, and (ii) list $H$-homomorphisms are testable with a sub linear number of queries if and only if $H$ is a bi-arc graph. Thus, we give equivalent conditions of graphs $H$ such that list $H$-homomorphisms are testable in constant / sub linear but not constant / linear number of queries.