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31
Constraint solving via fractional edge covers
 In Proceedings of the of the 17th Annual ACMSIAM Symposium on Discrete Algorithms
, 2006
"... Many important combinatorial problems can be modelled as constraint satisfaction problems, hence identifying polynomialtime solvable classes of constraint satisfaction problems received a lot of attention. In this paper, we are interested in structural properties that can make the problem tractable ..."
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Cited by 51 (9 self)
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Many important combinatorial problems can be modelled as constraint satisfaction problems, hence identifying polynomialtime solvable classes of constraint satisfaction problems received a lot of attention. In this paper, we are interested in structural properties that can make the problem tractable. So far, the largest structural class that is known to be polynomialtime solvable is the class of bounded hypertree width instances introduced by Gottlob et al. [20]. Here we identify a new class of polynomialtime solvable instances: those having bounded fractional edge cover number. Combining hypertree width and fractional edge cover number, we then introduce the notion of fractional hypertree width. We prove that constraint satisfaction problems with bounded fractional hypertree width can be solved in polynomial time (provided that a the tree decomposition is given in the input). We also prove that certain parameterized constraint satisfaction, homomorphism, and embedding problems are fixedparameter tractable on instances having bounded fractional hypertree width. 1.
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.
The complexity of conservative valued CSPs
 in: Proceedings of the 23rd ACMSIAM Symposium on Discrete Algorithms (SODA'12), 2012
"... We study the complexity of valued constraint satisfaction problems (VCSP). A problem from VCSP is characterised by a constraint language, a fixed set of cost functions over a finite domain. An instance of the problem is specified by a sum of cost functions from the language and the goal is to minimi ..."
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Cited by 13 (6 self)
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We study the complexity of valued constraint satisfaction problems (VCSP). A problem from VCSP is characterised by a constraint language, a fixed set of cost functions over a finite domain. An instance of the problem is specified by a sum of cost functions from the language and the goal is to minimise the sum. Under the unique games conjecture, the approximability of finitevalued VCSPs is wellunderstood, see Raghavendra [FOCS’08]. However, there is no characterisation of finitevalued VCSPs, let alone generalvalued VCSPs, that can be solved exactly in polynomial time, thus giving insights from a combinatorial optimisation perspective. We consider the case of languages containing all possible unary cost functions. In the case of languages consisting of only f0;1gvalued cost functions (i.e. relations), such languages have been called conservative and studied by Bulatov [LICS’03] and
The complexity of finitevalued CSPs
 Institute of Informatics, University of Warsaw, Poland
, 2013
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ENUMERATING HOMOMORPHISMS
, 2009
"... The homomorphism problem for relational structures is an abstract way of formulating constraint satisfaction problems (CSP) and various problems in database theory. The decision version of the homomorphism problem received a lot of attention in literature; in particular, the way the graphtheoretic ..."
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Cited by 8 (1 self)
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The homomorphism problem for relational structures is an abstract way of formulating constraint satisfaction problems (CSP) and various problems in database theory. The decision version of the homomorphism problem received a lot of attention in literature; in particular, the way the graphtheoretical structure of the variables and constraints influences the complexity of the problem is intensively studied. Here we study the problem of enumerating all the solutions with polynomial delay from a similar point of view. It turns out that the enumeration problem behaves very differently from the decision version. We give evidence that it is unlikely that a characterization result similar to the decision version can be obtained. Nevertheless, we show nontrivial cases where enumeration can be done with polynomial delay.
Worstcase Optimal Join Algorithms
 PODS'12
, 2012
"... Efficient join processing is one of the most fundamental and wellstudied tasks in database research. In this work, we examine algorithms for natural join queries over many relations and describe a novel algorithm to process these queries optimally in terms of worstcase data complexity. Our result b ..."
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Cited by 7 (2 self)
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Efficient join processing is one of the most fundamental and wellstudied tasks in database research. In this work, we examine algorithms for natural join queries over many relations and describe a novel algorithm to process these queries optimally in terms of worstcase data complexity. Our result builds on recent work by Atserias, Grohe, and Marx, who gave bounds on the size of a full conjunctive query in terms of the sizes of the individual relations in the body of the query. These bounds, however, are not constructive: they rely on Shearer’s entropy inequality which is informationtheoretic. Thus, the previous results leave open the question of whether there exist algorithms whose running time achieve these optimal bounds. An answer to this question may be interesting to database practice, as we show in this paper that any projectjoin plan is polynomially slower than the optimal bound for some queries. We construct an algorithm whose running time is worstcase optimal for all natural join queries. Our result may be of independent interest, as our algorithm also yields a constructive proof of the general fractional cover bound by Atserias, Grohe, and Marx without using Shearer’s inequality. In addition, we show that this bound is equivalent to a geometric inequality by Bollobás and Thomason, one of whose special cases is the famous LoomisWhitney inequality. Hence, our results algorithmically prove these inequalities as well. Finally, we discuss how our algorithm can be used to compute a relaxed notion of joins.
The tractability of CSP classes defined by forbidden patterns
 J. Artif. Intell. Res. (JAIR
"... The constraint satisfaction problem (CSP) is a general problem central to computer science and artificial intelligence. Although the CSP is NPhard in general, considerable effort has been spent on identifying tractable subclasses. The main two approaches consider structural properties (restrictions ..."
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Cited by 3 (2 self)
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The constraint satisfaction problem (CSP) is a general problem central to computer science and artificial intelligence. Although the CSP is NPhard in general, considerable effort has been spent on identifying tractable subclasses. The main two approaches consider structural properties (restrictions on the hypergraph of constraint scopes) and relational properties (restrictions on the language of constraint relations). Recently, some authors have considered hybrid properties that restrict the constraint hypergraph and the relations simultaneously. Our key contribution is the novel concept of a CSP pattern and classes of problems defined by forbidden patterns (which can be viewed as forbidding generic subproblems). We describe the theoretical framework which can be used to reason about classes of problems defined by forbidden patterns. We show that this framework generalises certain known hybrid tractable classes. Although we are not close to obtaining a complete characterisation concerning the tractability of general forbidden patterns, we prove a dichotomy in a special case: classes of problems that arise when we can only forbid binary negative patterns (generic subproblems in which only disallowed tuples are specified). In this case we show that all (finite sets of) forbidden patterns define either polynomialtime solvable or NPcomplete classes of instances.
Structural Decomposition Methods and What They are Good For
"... This paper reviews structural problem decomposition methods, such as tree and path decompositions. It is argued that these notions can be applied in two distinct ways: Either to show that a problem is efficiently solvable when a width parameter is fixed, or to prove that the unrestricted (or some wi ..."
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Cited by 3 (1 self)
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This paper reviews structural problem decomposition methods, such as tree and path decompositions. It is argued that these notions can be applied in two distinct ways: Either to show that a problem is efficiently solvable when a width parameter is fixed, or to prove that the unrestricted (or some widthparameter free) version of a problem is tractable by using a widthnotion as a mathematical tool for directly solving the problem at hand. Examples are given for both cases. As a new showcase for the latter usage, we report some recent results on the Partner Units Problem, a form of configuration problem arising in an industrial context. We use the notion of a path decomposition to identify and solve a tractable class of instances of this problem with practical relevance.
Constraint Satisfaction problems and global cardinality constraints
"... In a constraint satisfaction problem (CSP) the goal is to find an assignment of a given set of variables subject to specified constraints. A global cardinality constraint is an additional requirement that prescribes how many variables must be assigned a certain value. We study the complexity of the ..."
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Cited by 2 (0 self)
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In a constraint satisfaction problem (CSP) the goal is to find an assignment of a given set of variables subject to specified constraints. A global cardinality constraint is an additional requirement that prescribes how many variables must be assigned a certain value. We study the complexity of the problem CCSP(Γ), the constraint satisfaction problem with global cardinality constraints that allows only relations from the set Γ. The main result of this paper characterizes sets Γ that give rise to problems solvable in polynomial time, and states that the remaining such problems are NPcomplete.
Constraint Satisfaction Problems: Convexity Makes AllDifferent Constraints Tractable
 PROCEEDINGS OF THE TWENTYSECOND INTERNATIONAL JOINT CONFERENCE ON ARTIFICIAL INTELLIGENCE
, 2011
"... We examine the complexity of constraint satisfaction problems that consist of a set of AllDiff constraints. Such CSPs naturally model a wide range of realworld and combinatorial problems, like scheduling, frequency allocations and graph coloring problems. As this problem is known to be NPcomplete, ..."
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We examine the complexity of constraint satisfaction problems that consist of a set of AllDiff constraints. Such CSPs naturally model a wide range of realworld and combinatorial problems, like scheduling, frequency allocations and graph coloring problems. As this problem is known to be NPcomplete, we investigate under which further assumptions it becomes tractable. We observe that a crucial property seems to be the convexity of the variable domains and constraints. Our main contribution is an extensive study of the complexity of Multiple AllDiff CSPs for a set of natural parameters, like maximum domain size and maximum size of the constraint scopes. We show that, depending on the parameter, convexity can make the problem tractable while it is provably intractable in general.