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Tractable hypergraph properties for constraint satisfaction and conjunctive queries

<span title="">2010</span>
<i title="ACM Press">
<a target="_blank" rel="noopener" href="https://fatcat.wiki/container/jlc5kugafjg4dl7ozagimqfcbm" style="color: black;">Proceedings of the 42nd ACM symposium on Theory of computing - STOC '10</a>
</i>

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 [Grohe 2007; Marx 2010b] . However, this is not the correct

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... r 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. Note that 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 fixed-parameter 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 polynomial-time 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 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, which can be of independent interest. Tractable hypergraph properties for constraint satisfaction and conjunctive queries A:3 that the two vertices are distinct and adjacent. Therefore, the CSP instance has a solution if and only if G has a k-clique. It is easy to see that Boolean Conjunctive Query can be formulated as the problem of deciding if a CSP instance has a solution: the variables of the CSP instance correspond to the variables appearing in the query and the constraints correspond to the database relations. A distinctive feature of CSP instances obtained this way is that the number of variables is small (as queries are typically small), while the domain of the variables are large (as the database relations usually contain a large number of entries). This has to be contrasted with typical CSP problems from AI, such as 3-colorability and satisfiability, where the domain is small, but the number of variables is large. As our motivation is database-theoretic, in the rest of the paper the reader should keep in mind that we are envisioning scenarios where the number of variables is small and the domain is large. As the examples above show, solving constraint satisfaction problems is NP-hard in general if there are no additional restrictions on the input instances. The main goal of the research on CSP is to identify tractable special cases of the general problem. The theoretical literature on CSP investigates two main types of restrictions. The first type is to restrict the constraint language, that is, the type of constraints that are allowed. This direction includes the classical work of Schaefer [1978] and its many generalizations [Bulatov 2006; 2003; Bulatov et al. 2001; Feder and Vardi 1999; Jeavons et al. 1997 ]. The second type is to restrict the structure induced by the constraints on the variables. The hypergraph of a CSP instance is defined to be a hypergraph on the variables of the instance such that for each constraint c ∈ C there is a hyperedge e c containing exactly the variables that appear in c. If the hypergraph of the CSP instance has very simple structure, then the instance is easy to solve. For example, it is well-known that a CSP instance I with hypergraph H can be solved in time I O(tw(H)) [Freuder 1990 ], where tw(H) denotes the treewidth of H and
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