An important question in the study of constraint satisfaction problems (CSP) is understanding how the graph or hypergraph describing the incidence structure of the constraints influences the complexity of the problem. For binary CSP instances (i.e., where each constraint involves only two variables), the situation is well understood: the complexity of the problem essentially depends on the treewidth of the graph of the constraints [19, 24] . However, this is not the correct answer if constraints with unbounded number of variables are allowed, and in particular, for CSP instances arising from query evaluation problems in database theory. Formally, if H is a class of hypergraphs, then let CSP(H) be CSP restricted to instances whose hypergraph is in H. Our goal is to characterize those classes of hypergraphs for which CSP(H) is polynomial-time solvable or fixed-parameter tractable, parameterized by the number of variables. In the applications related to database query evaluation, we usually assume that the number of variables is much smaller than the size of the instance, thus parameterization by the number of variables is a meaningful question. The most general known property of H that makes CSP(H) polynomial-time solvable is bounded fractional hypertree width. Here we introduce a new hypergraph measure called submodular width, and show that bounded submodular width of H (which is a strictly more general property than bounded fractional hypertree width) implies that CSP(H) is fixedparameter tractable. In a matching hardness result, we show that if H has unbounded submodular width, then CSP(H) is not fixed-parameter tractable (and hence not polynomialtime solvable), unless the Exponential Time Hypothesis (ETH) fails. The algorithmic result uses tree decompositions in a novel way: instead of using a single decomposition depending on the hypergraph, the instance is split into a set of † Research supported by ERC Advanced Grant DMMCA. * The full version of the paper is available at http://arxiv.org/abs/0911.0801 Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. instances (all on the same set of variables as the original instance), and then the new instances are solved by choosing a different tree decomposition for each of them. The reason why this strategy works is that the splitting can be done in such a way that the new instances are "uniform" with respect to the number extensions of partial solutions, and therefore the number of partial solutions can be described by a submodular function. For the hardness result, we prove via a series of combinatorial results that if a hypergraph H has large submodular width, then a 3SAT instance can be efficiently simulated by a CSP instance whose hypergraph is H. To prove these combinatorial results, we need to develop a theory of (multicommodity) flows on hypergraphs and vertex separators in the case when the function b(S) defining the cost of separator S is submodular.
The situation is different for fixedparameter tractability: Marx gave a complete classification of the fixed-parameter tractable restrictions CSP(C, −), for classes C of structures of unbounded arity REF .
Tractable hypergraph properties for constraint satisfaction and conjunctive queries
STOC '10
Computer Science Mathematics