Robbers, marshals, and guards: game theoretic and logical characterizations of hypertree width

In a previous paper [10], the authors introduced the notion of hypertree decomposition and the corresponding concept of hypertree width and showed that the conjunctive queries whose hypergraphs have bounded hypertree-width can be evaluated in polynomial time. Bounded hypertree-width generalizes the notions of acyclicity and bounded treewidth and corresponds to larger classes of tractable queries. In the present paper, we provide natural characterizations of hypergraphs and queries having bounded hypertree-width in terms of game-theory and logic. First we define the Robber and Marshals game, and prove that a hypergraph H has hypertree-width at most k iff k marshals have a winning strategy on H, allowing them to trap a robber who moves along the hyperedges. This game is akin the well-known Robber and Cops game (which characterizes bounded treewidth), except that marshals are more powerful than cops: They can control entire hyperedges instead of just vertices. Kolaitis and Vardi [17] recently gave an elegant characterization of the conjunctive queries having treewidth < k in terms of the k-variable fragment of a certain logic L ( = existential-conjunctive fragment of positive FO). We use the Robber and Marshals game to derive a surprisingly simple and equally elegant characterization of the class HW[k] of queries of hypertree-width at most k in terms of guarded logic. In particular, we show that HW[k] = GFk (L), where GFk(L) denotes the k-guarded fragment of L. In this fragment, conjunctions of k atoms rather than just single atoms are allowed to act as guards. Note that, for the particular case k = 1, our results provide new characterizations of the class of acyclic queries. We extend the notion of bounded hypertreewidth to nonrecursive stratified datalog and show that the k-guarded fragment GFk(FO) of first order logic has the same expressive power as nonrecursive stratified datalog of hypertreewidth ⪇ k.

[1]  Georg Gottlob,et al.  The complexity of acyclic conjunctive queries , 1998, Proceedings 39th Annual Symposium on Foundations of Computer Science (Cat. No.98CB36280).

[2]  Bruno Courcelle,et al.  The Monadic Second-Order Logic of Graphs VII: Graphs as Relational Structures , 1992, Theor. Comput. Sci..

[3]  Ashok K. Chandra,et al.  Optimal implementation of conjunctive queries in relational data bases , 1977, STOC '77.

[4]  Rina Dechter,et al.  Tree Clustering for Constraint Networks , 1989, Artif. Intell..

[5]  Christophe Lecoutre Constraint Networks , 1992 .

[6]  Marc Gyssens,et al.  A Decomposition Methodology for Cyclic Databases , 1982, Advances in Data Base Theory.

[7]  Georg Gottlob,et al.  Datalog LITE: a deductive query language with linear time model checking , 2002, TOCL.

[8]  Oded Shmueli,et al.  Acyclic Hypergraph Projections , 1999, J. Algorithms.

[9]  Mihalis Yannakakis,et al.  Algorithms for Acyclic Database Schemes , 1981, VLDB.

[10]  TutorialMoshe Y. Vardi Constraint Satisfa tion and Database Theory : a , 2000 .

[11]  Moshe Y. Vardi Constraint satisfaction and database theory: a tutorial , 2000, PODS.

[12]  Georg Gottlob,et al.  Hypergraphs in Model Checking: Acyclicity and Hypertree-Width versus Clique-Width , 2001, SIAM J. Comput..

[13]  Egon Wanke Bounded Tree-Width and LOGCFL , 1993, WG.

[14]  Anand Rajaraman,et al.  Conjunctive query containment revisited , 2000, Theor. Comput. Sci..

[15]  Moshe Y. Vardi The complexity of relational query languages (Extended Abstract) , 1982, STOC '82.

[16]  Serge Abiteboul,et al.  Complexity of answering queries using materialized views , 1998, PODS.

[17]  Erich Grädel,et al.  On the Restraining Power of Guards , 1999, Journal of Symbolic Logic.

[18]  GottlobGeorg,et al.  The complexity of acyclic conjunctive queries , 2001 .

[19]  Jörg Flum,et al.  Query evaluation via tree-decompositions , 2001, JACM.

[20]  Jeffrey D. Ullman,et al.  Principles of Database and Knowledge-Base Systems, Volume II , 1988, Principles of computer science series.

[21]  Jeffrey D. Uuman Principles of database and knowledge- base systems , 1989 .

[22]  Thomas Schwentick,et al.  When is the evaluation of conjunctive queries tractable? , 2001, STOC '01.

[23]  Jeffrey D. Ullman,et al.  Information integration using logical views , 1997, Theor. Comput. Sci..

[24]  Marc Gyssens,et al.  Decomposing Constraint Satisfaction Problems Using Database Techniques , 1994, Artif. Intell..

[25]  Georg Gottlob,et al.  A Comparison of Structural CSP Decomposition Methods , 1999, IJCAI.

[26]  Paul D. Seymour,et al.  Graph Minors. II. Algorithmic Aspects of Tree-Width , 1986, J. Algorithms.

[27]  Phokion G. Kolaitis,et al.  Conjunctive-query containment and constraint satisfaction , 1998, PODS.

[28]  Robin Thomas,et al.  Graph Searching and a Min-Max Theorem for Tree-Width , 1993, J. Comb. Theory, Ser. B.

[29]  Bruno Courcelle,et al.  Handle-Rewriting Hypergraph Grammars , 1993, J. Comput. Syst. Sci..

[30]  J. Benthem DYNAMIC BITS AND PIECES , 1997 .

[31]  Georg Gottlob,et al.  Computing LOGCFL Certificates , 1999, ICALP.

[32]  Serge Abiteboul,et al.  Foundations of Databases , 1994 .

[33]  Eugene C. Freuder A sufficient condition for backtrack-bounded search , 1985, JACM.

[34]  Georg Gottlob,et al.  On Tractable Queries and Constraints , 1999, DEXA.