Testing List H-Homomorphisms

📝 Abstract

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 the following problem: given a map $f:V(G) \to V(H)$ as an oracle access, 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 and show the following trichotomy holds: (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. (ii) List $H $-homomorphisms are testable with a sublinear number of queries if and only if $H$ is a bi-arc graph. (iii) Testing list $H $-homomorphisms requires a linear number of queries if $H$ is not a bi-arc graph.

💡 Analysis

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 the following problem: given a map $f:V(G) \to V(H)$ as an oracle access, 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 and show the following trichotomy holds: (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. (ii) List $H $-homomorphisms are testable with a sublinear number of queries if and only if $H$ is a bi-arc graph. (iii) Testing list $H $-homomorphisms requires a linear number of queries if $H$ is not a bi-arc graph.

📄 Content

arXiv:1106.3126v1 [cs.DS] 16 Jun 2011 Testing List H-Homomorphisms Yuichi Yoshida∗ School of Informatics, Kyoto University, and Preferred Infrastructure, Inc. yyoshida@kuis.kyoto-u.ac.jp November 13, 2018 Abstract Let H be an undirected graph. In the List H-Homomorphism Problem, given an undirected graph G with a list constraint L(v) ⊆V (H) for each variable v ∈V (G), the objective is to find a list H-homomorphism f : V (G) →V (H), that is, f(v) ∈L(v) for every v ∈V (G) and (f(u), f(v)) ∈E(H) whenever (u, v) ∈E(G). We consider the following problem: given a map f : V (G) →V (H) as an oracle access, the objective is to decide with high probability whether f is a list H-homomorphism or 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 and show the following trichotomy holds: (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. (ii) List H-homomorphisms are testable with a sublinear number of queries if and only if H is a bi-arc graph. (iii) Testing list H-homomorphisms requires a linear number of queries if H is not a bi-arc graph. ∗Supported by MSRA Fellowship 2010. 0 1 Introduction For two graphs G = (V (G), E(G)) and H = (V (H), E(H)), a map f : V (G) →V (H) is called a ho- momorphism from G to H if (f(u), f(v)) ∈E(H) whenever (u, v) ∈E(G). In the H-Homomorphism Problem (HOM(H) for short), given an undirected graph G, the objective is to decide whether there exists a homomorphism from G to H. It is well known that HOM(H) is in P if H is a bipartite graph and in NP-Complete if H is not a bipartite graph [3, 8, 21]. List H-Homomorphism Problem (LHOM(H) for short) is a variant of H-Homomorphism Problem, in which we are also given a list L(v) ⊆V (H) for each vertex v in G. A map f : V (G) →V (H) is called a list-homomorphism from G to H if f is a homomorphism from G to H and f(v) ∈L(v) for every v ∈V (G). The objective is to decide whether there exists a list-homomorphism f from G to H. There are many results on the relationship between the graph H and the computational complexity of LHOM(H) [11, 12, 13, 14]. In particular, LHOM(H) is in P iffH is a bi-arc graph [14]. In this paper, we consider testing list-homomorphisms. See [17, 27] for surveys on property testing. In our setting, a map f is given as an oracle access, i.e., the oracle returns f(v) if we specify a vertex v ∈V (G). A map f is called ǫ-far from list-homomorphisms if we must modify at least an ǫ-fraction of f to make f a list-homomorphism. An algorithm is called a tester for LHOM(H) if it accepts with probability at least 2/3 if f is a list-homomorphism from G to H and rejects with probability at least 2/3 if f is ǫ-far from list-homomorphisms. The efficiency of an algorithm is measured by the number of accesses to the oracle f. When we say that a query complexity is constant/sublinear/linear, it always means constant/sublinear/linear in |V (G)|, i.e., the domain size of f. We can assume that there exists a list-homomorphism from G to H. If otherwise, we can reject immediately without any query. In this paper, we completely classify graphs H with respect to the query complexity for testing LHOM(H). Our result consists of the following two theorems. Theorem 1.1. LHOM(H) is testable with a constant number of queries iffH is an irreflexive complete bipartite graph or a reflexive complete graph. Theorem 1.2. LHOM(H) is testable with a sublinear number of queries iffH is a bi-arc graph. The central question in the area of property testing is to classify properties into the following three categories: properties testable with a constant/sublinear/linear number of queries. Our result first establishes such a classification for a natural and general combinatorial problem. We note that, from Theorem 1.2 and results given by [14], LHOM(H) is testable with a sublinear number of queries iffLHOM(H) is in P. However, it is not clear whether there is a computational class corresponding to properties testable with a constant number of queries. To obtain our results, we exploit universal algebra, which is now a common tool to study computational complexity of constraint satisfaction problems (see, e.g., [24]). Another contribution of this paper is showing that universal algebraic approach is quite useful in the setting of property testing. Related works: It is rare that we succeed to obtain characterizations of properties testable with a constant/sublinear number of queries. The only such a characterization we are aware of is one for graph properties in the dense model [18]. In this model, it is revealed that Szemer´edi’s regularity lemma [28] plays a crucial role [1]. Roughly speaking, the regularity lemma gives the constant-size sketch of a graph. It turns out that a property is testable in the dense m

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