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51
Can you beat treewidth?
, 2007
"... It is wellknown that constraint satisfaction problems (CSP) can be solved in time n O(k) if the treewidth of the primal graph of the instance is at most k and n is the size of the input. We show that no algorithm can be significantly better than this treewidthbased algorithm, even if we restrict t ..."
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Cited by 38 (8 self)
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It is wellknown that constraint satisfaction problems (CSP) can be solved in time n O(k) if the treewidth of the primal graph of the instance is at most k and n is the size of the input. We show that no algorithm can be significantly better than this treewidthbased algorithm, even if we restrict the problem to some special class of primal graphs. Formally, let G be an arbitrary class of graphs and assume that there is an algorithm A solving binary CSP for instances whose primal graph is in G. We prove that if the running time of A is f(G)n o(k/logk) , where k is the treewidth of the primal graph G and f is an arbitrary function, then the Exponential Time Hypothesis fails. We prove the result also in the more general framework of the homomorphism problem for boundedarity relational structures. For this problem, the treewidth of the core of the lefthand side structure plays the same role as the treewidth of the primal graph above.
Tractable Hypergraph Properties for Constraint Satisfaction and Conjunctive Queries
, 2010
"... 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 ..."
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Cited by 31 (4 self)
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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 (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 polynomialtime solvable or fixedparameter 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) polynomialtime 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 fixedparameter 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
Algorithms for propositional model counting.
 In Proc. of the 14th International Conference on Logic for Programming, Artificial Intelligence, and Reasoning (LPAR’07),
, 2007
"... Abstract We present algorithms for the propositional model counting problem #SAT. The algorithms utilize tree decompositions of certain graphs associated with the given CNF formula; in particular we consider primal, dual, and incidence graphs. We describe the algorithms coherently for a direct comp ..."
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Cited by 29 (10 self)
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Abstract We present algorithms for the propositional model counting problem #SAT. The algorithms utilize tree decompositions of certain graphs associated with the given CNF formula; in particular we consider primal, dual, and incidence graphs. We describe the algorithms coherently for a direct comparison and with sufficient detail for making an actual implementation reasonably easy. We discuss several aspects of the algorithms including worstcase time and space requirements.
Size bounds and query plans for relational joins
, 2008
"... Relational joins are at the core of relational algebra, which in turn is the core of the standard database query language SQL. As their evaluation is expensive and very often dominated by the output size, it is an important task for database query optimisers to compute estimates on the size of joins ..."
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Cited by 23 (0 self)
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Relational joins are at the core of relational algebra, which in turn is the core of the standard database query language SQL. As their evaluation is expensive and very often dominated by the output size, it is an important task for database query optimisers to compute estimates on the size of joins and to find good execution plans for sequences of joins. We study these problems from a theoretical perspective, both in the worstcase model, and in an averagecase model where the database is chosen according to a known probability distribution. In the former case, our first key observation is that the worstcase size of a query is characterised by the fractional edge cover number of its underlying hypergraph, a combinatorial parameter previously known to provide an upper bound. We complete the picture by proving a matching lower bound, and by showing that there exist queries for which the joinproject plan suggested by the fractional edge cover approach may be substantially better than any join plan that does not use intermediate projections. On the other hand, we show that in the averagecase model, every joinproject plan can be turned into a plan containing no projections in such a way that the expected time to evaluate the plan increases only by a constant factor independent of the size of the database. Not surprisingly, the key combinatorial parameter in this context is the maximum density of the underlying hypergraph. We show how to make effective use
Communication steps for parallel query processing.
 In Proceedings of the 32nd ACM Symposium on Principles of Database Systems, PODS,
, 2013
"... ABSTRACT We consider the problem of computing a relational query q on a large input database of size n, using a large number p of servers. The computation is performed in rounds, and each server can receive only O(n/p 1−ε ) bits of data, where ε ∈ [0, 1] is a parameter that controls replication. We ..."
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Cited by 22 (4 self)
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ABSTRACT We consider the problem of computing a relational query q on a large input database of size n, using a large number p of servers. The computation is performed in rounds, and each server can receive only O(n/p 1−ε ) bits of data, where ε ∈ [0, 1] is a parameter that controls replication. We examine how many global communication steps are needed to compute q. We establish both lower and upper bounds, in two settings. For a single round of communication, we give lower bounds in the strongest possible model, where arbitrary bits may be exchanged; we show that any algorithm requires ε ≥ 1−1/τ * , where τ * is the fractional vertex cover of the hypergraph of q. We also give an algorithm that matches the lower bound for a specific class of databases. For multiple rounds of communication, we present lower bounds in a model where routing decisions for a tuple are tuplebased. We show that for the class of treelike queries there exists a tradeoff between the number of rounds and the space exponent ε. The lower bounds for multiple rounds are the first of their kind. Our results also imply that transitive closure cannot be computed in O(1) rounds of communication.
Constraint satisfaction with bounded treewidth revisited
 In CP’06
, 2006
"... We consider the constraint satisfaction problem (CSP) parameterized by the treewidth of primal, dual, and incidence graphs, combined with several other basic parameters such as domain size and arity. We determine all combinations of the considered parameters that admit fixedparameter tractability. ..."
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Cited by 21 (6 self)
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We consider the constraint satisfaction problem (CSP) parameterized by the treewidth of primal, dual, and incidence graphs, combined with several other basic parameters such as domain size and arity. We determine all combinations of the considered parameters that admit fixedparameter tractability. Key words: Constraint satisfaction, parameterized complexity, treewidth 1
Upper and Lower Bounds on the Cost of a MapReduce Computation
, 2013
"... In this paper we study the tradeoff between parallelism and communication cost in a mapreduce computation. For any problem that is not “embarrassingly parallel,” the finer we partition the work of the reducers so that more parallelism can be extracted, the greater will be the total communication be ..."
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Cited by 20 (1 self)
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In this paper we study the tradeoff between parallelism and communication cost in a mapreduce computation. For any problem that is not “embarrassingly parallel,” the finer we partition the work of the reducers so that more parallelism can be extracted, the greater will be the total communication between mappers and reducers. We introduce a model of problems that can be solved in a single round of mapreduce computation. This model enables a generic recipe for discovering lower bounds on communication cost as a function of the maximum number of inputs that can be assigned to one reducer. We use the model to analyze the tradeoff for three problems: finding pairs of strings at Hamming distance d, finding triangles and other patterns in a larger graph, and matrix multiplication. For finding strings of Hamming distance 1, we have upper and lower bounds that match exactly. For triangles and many other graphs, we have upper and lower bounds that are the same to within a constant factor. For the problem of matrix multiplication, we have matching upper and lower bounds for oneround mapreduce algorithms. We are also able to explore tworound mapreduce algorithms for matrix multiplication and show that these never have more communication, for a given reducer size, than the best oneround algorithm, and often have significantly less.
The structure of tractable constraint satisfaction problems
 In MFCS 2006
, 2006
"... Abstract We give a survey of recent results on the complexity of constraint satisfaction problems. Our main emphasis is on tractable structural restrictions. 1 ..."
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Cited by 18 (0 self)
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Abstract We give a survey of recent results on the complexity of constraint satisfaction problems. Our main emphasis is on tractable structural restrictions. 1
Hypertree Width and Related Hypergraph Invariants
, 2006
"... We study the notion of hypertree width of hypergraphs. We prove that, up to a constant factor, hypertree width is the same as a number of other hypergraph invariants that resemble graph invariants such as bramble number, branch width, linkedness, and the minimum number of cops required to win Seymou ..."
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Cited by 18 (3 self)
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We study the notion of hypertree width of hypergraphs. We prove that, up to a constant factor, hypertree width is the same as a number of other hypergraph invariants that resemble graph invariants such as bramble number, branch width, linkedness, and the minimum number of cops required to win Seymour and Thomas’s robber and cops game. 1