Robbers, Marshals, and Guards: Game Theoretic and Logical Characterizations of Hypertree Width

In a previous paper (J. Comput. System Sci. 64 (2002) 519), 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 if and only if 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 (J. Comput. System Sci. 61 (2000) 302) 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] = GF(k)(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 hypertree width to nonrecursive stratified Datalog and show that the k-guarded fragment GF(k)(FO) of first-order logic has the same expressive power as nonrecursive stratified Datalog of hypertree width at most k.