Pure Nash Equilibria: Hard and Easy Games

We investigate complexity issues related to pure Nash equilibria of strategic games. We show that, even in very restrictive settings, determining whether a game has a pure Nash Equilibrium is NP-hard, while deciding whether a game has a strong Nash equilibrium is SigmaP2-complete. We then study practically relevant restrictions that lower the complexity. In particular, we are interested in quantitative and qualitative restrictions of the way each players payoff depends on moves of other players. We say that a game has small neighborhood if the utility function for each player depends only on (the actions of) a logarithmically small number of other players. The dependency structure of a game G can be expressed by a graph DG(G) or by a hypergraph H(G). By relating Nash equilibrium problems to constraint satisfaction problems (CSPs), we show that if G has small neighborhood and if H(G) has bounded hypertree width (or if DG(G) has bounded treewidth), then finding pure Nash and Pareto equilibria is feasible in polynomial time. If the game is graphical, then these problems are LOGCFL-complete and thus in the class NC2 of highly parallelizable problems

An Analytical Study of Large SPARQL Query Logs

With the adoption of RDF as the data model for Linked Data and the Semantic Web, query specification from end- users has become more and more common in SPARQL end- points. In this paper, we conduct an in-depth analytical study of the queries formulated by end-users and harvested from large and up-to-date query logs from a wide variety of RDF data sources. As opposed to previous studies, ours is the first assessment on a voluminous query corpus, span- ning over several years and covering many representative SPARQL endpoints. Apart from the syntactical structure of the queries, that exhibits already interesting results on this generalized corpus, we drill deeper in the structural char- acteristics related to the graph- and hypergraph represen- tation of queries. We outline the most common shapes of queries when visually displayed as pseudographs, and char- acterize their (hyper-)tree width. Moreover, we analyze the evolution of queries over time, by introducing the novel con- cept of a streak, i.e., a sequence of queries that appear as subsequent modifications of a seed query. Our study offers several fresh insights on the already rich query features of real SPARQL queries formulated by real users, and brings us to draw a number of conclusions and pinpoint future di- rections for SPARQL query evaluation, query optimization, tuning, and benchmarking

A Backtracking-Based Algorithm for Computing Hypertree-Decompositions

Hypertree decompositions of hypergraphs are a generalization of tree decompositions of graphs. The corresponding hypertree-width is a measure for the cyclicity and therefore tractability of the encoded computation problem. Many NP-hard decision and computation problems are known to be tractable on instances whose structure corresponds to hypergraphs of bounded hypertree-width. Intuitively, the smaller the hypertree-width, the faster the computation problem can be solved. In this paper, we present the new backtracking-based algorithm det-k-decomp for computing hypertree decompositions of small width. Our benchmark evaluations have shown that det-k-decomp significantly outperforms opt-k-decomp, the only exact hypertree decomposition algorithm so far. Even compared to the best heuristic algorithm, we obtained competitive results as long as the hypergraphs are not too large.Comment: 19 pages, 6 figures, 3 table

Tractable Combinations of Global Constraints

We study the complexity of constraint satisfaction problems involving global constraints, i.e., special-purpose constraints provided by a solver and represented implicitly by a parametrised algorithm. Such constraints are widely used; indeed, they are one of the key reasons for the success of constraint programming in solving real-world problems. Previous work has focused on the development of efficient propagators for individual constraints. In this paper, we identify a new tractable class of constraint problems involving global constraints of unbounded arity. To do so, we combine structural restrictions with the observation that some important types of global constraint do not distinguish between large classes of equivalent solutions.Comment: To appear in proceedings of CP'13, LNCS 8124. arXiv admin note: text overlap with arXiv:1307.179

A Graph Based Backtracking Algorithm for Solving General CSPs

Many AI tasks can be formalized as constraint satisfaction problems (CSPs), which involve finding values for variables subject to constraints. While solving a CSP is an NP-complete task in general, tractable classes of CSPs have been identified based on the structure of the underlying constraint graphs. Much effort has been spent on exploiting structural properties of the constraint graph to improve the efficiency of finding a solution. These efforts contributed to development of a class of CSP solving algorithms called decomposition algorithms. The strength of CSP decomposition is that its worst-case complexity depends on the structural properties of the constraint graph and is usually better than the worst-case complexity of search methods. Its practical application is limited, however, since it cannot be applied if the CSP is not decomposable. In this paper, we propose a graph based backtracking algorithm called omega-CDBT, which shares merits and overcomes the weaknesses of both decomposition and search approaches

GYM: A Multiround Distributed Join Algorithm

Multiround algorithms are now commonly used in distributed data processing systems, yet the extent to which algorithms can benefit from running more rounds is not well understood. This paper answers this question for several rounds for the problem of computing the equijoin of n relations. Given any query Q with width w, intersection width iw, input size IN, output size OUT, and a cluster of machines with M=Omega(IN frac{1}{epsilon}) memory available per machine, where epsilon > 1 and w ge 1 are constants, we show that: 1. Q can be computed in O(n) rounds with O(n(INw + OUT)2/M) communication cost with high probability. Q can be computed in O(log(n)) rounds with O(n(INmax(w, 3iw) + OUT)2/M) communication cost with high probability. Intersection width is a new notion we introduce for queries and generalized hypertree decompositions (GHDs) of queries that captures how connected the adjacent components of the GHDs are. We achieve our first result by introducing a distributed and generalized version of Yannakakis\u27s algorithm, called GYM. GYM takes as input any GHD of Q with width w and depth d, and computes Q in O(d + log(n)) rounds and O(n (INw + OUT)2/M) communication cost. We achieve our second result by showing how to construct GHDs of Q with width max(w, 3iw) and depth O(log(n)). We describe another technique to construct GHDs with longer widths and lower depths, demonstrating other tradeoffs one can make between communication and the number of rounds