Testing List H-Homomorphisms

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

Social Collaborative Retrieval

Socially-based recommendation systems have recently attracted significant interest, and a number of studies have shown that social information can dramatically improve a system's predictions of user interests. Meanwhile, there are now many potential applications that involve aspects of both recommendation and information retrieval, and the task of collaborative retrieval---a combination of these two traditional problems---has recently been introduced. Successful collaborative retrieval requires overcoming severe data sparsity, making additional sources of information, such as social graphs, particularly valuable. In this paper we propose a new model for collaborative retrieval, and show that our algorithm outperforms current state-of-the-art approaches by incorporating information from social networks. We also provide empirical analyses of the ways in which cultural interests propagate along a social graph using a real-world music dataset.Comment: 10 page

On The Power of Tree Projections: Structural Tractability of Enumerating CSP Solutions

The problem of deciding whether CSP instances admit solutions has been deeply studied in the literature, and several structural tractability results have been derived so far. However, constraint satisfaction comes in practice as a computation problem where the focus is either on finding one solution, or on enumerating all solutions, possibly projected to some given set of output variables. The paper investigates the structural tractability of the problem of enumerating (possibly projected) solutions, where tractability means here computable with polynomial delay (WPD), since in general exponentially many solutions may be computed. A general framework based on the notion of tree projection of hypergraphs is considered, which generalizes all known decomposition methods. Tractability results have been obtained both for classes of structures where output variables are part of their specification, and for classes of structures where computability WPD must be ensured for any possible set of output variables. These results are shown to be tight, by exhibiting dichotomies for classes of structures having bounded arity and where the tree decomposition method is considered