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Closure properties of constraints
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A Game-Theoretic Approach to Constraint Satisfaction
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Constraint Satisfaction, Bounded Treewidth, and Finite-Variable Logics
CP '02 Proceedings of the 8th International Conference on Principles and Practice of Constraint Programming
Properties of acyclic database schemes
STOC '81 Proceedings of the thirteenth annual ACM symposium on Theory of computing
Optimal implementation of conjunctive queries in relational data bases
STOC '77 Proceedings of the ninth annual ACM symposium on Theory of computing
The complexity of satisfiability problems
STOC '78 Proceedings of the tenth annual ACM symposium on Theory of computing
A Parametrized Algorithm for Matroid Branch-Width
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Parameterized Complexity Theory (Texts in Theoretical Computer Science. An EATCS Series)
Parameterized Complexity Theory (Texts in Theoretical Computer Science. An EATCS Series)
Approximating clique-width and branch-width
Journal of Combinatorial Theory Series B
On the Optimality of the Dimensionality Reduction Method
FOCS '06 Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science
European Journal of Combinatorics - Special issue on Eurocomb'03 - graphs and combinatorial structures
The complexity of homomorphism and constraint satisfaction problems seen from the other side
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Partitioning multi-dimensional sets in a small number of "Uniform" parts
European Journal of Combinatorics
Improved approximation for directed cut problems
Proceedings of the thirty-ninth annual ACM symposium on Theory of computing
Submodular function minimization
Mathematical Programming: Series A and B
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VLDB '81 Proceedings of the seventh international conference on Very Large Data Bases - Volume 7
Hypertree width and related hypergraph invariants
European Journal of Combinatorics
On the Optimality of Planar and Geometric Approximation Schemes
FOCS '07 Proceedings of the 48th Annual IEEE Symposium on Foundations of Computer Science
Finding Branch-Decompositions and Rank-Decompositions
SIAM Journal on Computing
On tree width, bramble size, and expansion
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Uniform Constraint Satisfaction Problems and Database Theory
Complexity of Constraints
An O(n)-approximation algorithm for directed sparsest cut
Information Processing Letters
Approximating fractional hypertree width
ACM Transactions on Algorithms (TALG)
Tractable hypergraph properties for constraint satisfaction and conjunctive queries
Proceedings of the forty-second ACM symposium on Theory of computing
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Constraint satisfaction with succinctly specified relations
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Complexity of K-tree structured constraint satisfaction problems
AAAI'90 Proceedings of the eighth National conference on Artificial intelligence - Volume 1
On the possibility of faster SAT algorithms
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
Tractable Structures for Constraint Satisfaction with Truth Tables
Theory of Computing Systems
Complexity of conservative constraint satisfaction problems
ACM Transactions on Computational Logic (TOCL)
The structure of tractable constraint satisfaction problems
MFCS'06 Proceedings of the 31st international conference on Mathematical Foundations of Computer Science
Approximating rank-width and clique-width quickly
WG'05 Proceedings of the 31st international conference on Graph-Theoretic Concepts in Computer Science
ICALP'07 Proceedings of the 34th international conference on Automata, Languages and Programming
Parameterized Complexity
Theoretical Computer Science
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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 (that is, 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 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. 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.