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Algebraic structures in combinatorial problems
 TECHNICAL REPORT, TECHNISCHE UNIVERSITAT DRESDEN
, 2001
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Pure Nash Equilibria: Hard and Easy Games
"... In this paper we investigate complexity issues related to pure Nash equilibria of strategic games. We show that, even in very restrictive settings, determining whether a game has a pure Nash Equilibrium is NPhard, while deciding whether a game has a strong Nash equilibrium is Stcomplete. We then s ..."
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Cited by 81 (5 self)
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In this paper we investigate complexity issues related to pure Nash equilibria of strategic games. We show that, even in very restrictive settings, determining whether a game has a pure Nash Equilibrium is NPhard, while deciding whether a game has a strong Nash equilibrium is Stcomplete. We then study practically relevant restrictions that lower the complexity. In particular, we are interested in quantitative and qualitative restrictions of the way each player's move depends on moves of other players. We say that a game has small neighborhood if the &quot; utility function for each player depends only on (the actions of) a logarithmically small number of other players, The dependency structure of a game G can he expressed by a graph G(G) or by a hypergraph II(G). Among other results, we show that if jC has small neighborhood and if II(G) has botmdecl hypertree width (or if G(G) has bounded treewidth), then finding pure Nash and Pareto equilibria is feasible in polynomial time. If the game is graphical, then these problems are LOGCFLcomplete and thus in the class _NC ~ of highly parallelizable problems. 1 Introduction and Overview of Results The theory of strategic games and Nash equilibria has important applications in economics and decision making [31, 2]. Determining whether Nash equilibria exist, and effectively computing
CSP dichotomy holds for digraphs with no sources and no sinks (a positive answer to the conjecture of BangJensen and Hell)
"... ... of graph homomorphisms) a CSP dichotomy for digraphs with no sources or sinks. The conjecture states that the constraint satisfaction problem for such a digraph is tractable if each component of its core is a circle and is NPcomplete otherwise. In this paper we prove this conjecture, and, as a ..."
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Cited by 45 (7 self)
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... of graph homomorphisms) a CSP dichotomy for digraphs with no sources or sinks. The conjecture states that the constraint satisfaction problem for such a digraph is tractable if each component of its core is a circle and is NPcomplete otherwise. In this paper we prove this conjecture, and, as a consequence, a conjecture of BangJensen, Hell and MacGillivray from 1995 classifying hereditarily hard digraphs. Further, we show that the CSP dichotomy for digraphs with no sources or sinks agrees with the algebraic characterization conjectured by Bulatov, Jeavons and Krokhin in 2005.
The Complexity Of Maximal Constraint Languages
, 2001
"... Many combinatorial search problems can be expressed as "constraint satisfaction problems" using an appropriate "constraint language", that is, a set of relations over some fixed finite set of values. It is wellknown that there is a tradeoff between the expressive power of a con ..."
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Cited by 38 (9 self)
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Many combinatorial search problems can be expressed as "constraint satisfaction problems" using an appropriate "constraint language", that is, a set of relations over some fixed finite set of values. It is wellknown that there is a tradeoff between the expressive power of a constraint language and the complexity of the problems it can express. In the present paper we systematically study the complexity of all maximal constraint languages, that is, languages whose expressive power is just weaker than that of the language of all constraints. Using the algebraic invariance properties of constraints, we exhibit a strong necessary condition for tractability of such a constraint language. Moreover, we show that, at least for small sets of values, this condition is also sufficient.
On Database Theory and XML
 ACM SIGMOD Special Section on Advanced XML Data Processing
"... Over the years, the connection between database theory and database practice has weakened. We argue here that the new challenges posed by XML and its applications are strengthening this connection today. We illustrate three examples of theoretical problems arising from XML applications, based on ou ..."
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Cited by 34 (1 self)
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Over the years, the connection between database theory and database practice has weakened. We argue here that the new challenges posed by XML and its applications are strengthening this connection today. We illustrate three examples of theoretical problems arising from XML applications, based on our own research. 1 On Database Theory The eld of relational databases is the product of a theoretician, E.F. Codd, from the early 70s. Relational databases had to struggle for a while against the industry proposal CODASYL [58], but then became universally adopted and today we have both a strong industry and a
ourishing research eld.
An Algebraic Approach To MultiSorted Constraints
 Proceedings of 9th International Conference on Principles and Practice of Constraint Programming
, 2003
"... We describe a common framework for the Constraint Satisfaction Problem and the Conjunctive Query Evaluation Problem, encompassing a generalised form of these problems in which different variables may take values from different sets. The framework we develop allows us to specify natural subclasses of ..."
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Cited by 23 (7 self)
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We describe a common framework for the Constraint Satisfaction Problem and the Conjunctive Query Evaluation Problem, encompassing a generalised form of these problems in which different variables may take values from different sets. The framework we develop allows us to specify natural subclasses of these two problems using algebraic techniques, and to establish when these subclasses are tractable. We show that a range of tractable classes can be obtained by combining recently identified tractable subclasses of the usual constraint satisfaction problem over a single set of values. We also systematically develop an algebraic structural theory for the general problem, which provides the prerequisites for further use of the powerful algebraic machinery.
Partition Search for Nonbinary Constraint Satisfaction
 Information Sciences
, 2007
"... Previous algorithms for unrestricted constraint satisfaction use reduction search, which inferentially removes values from domains in order to prune the backtrack search tree. This paper introduces partition search, which uses an efficient join mechanism instead of removing values from domains. Anal ..."
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Cited by 19 (0 self)
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Previous algorithms for unrestricted constraint satisfaction use reduction search, which inferentially removes values from domains in order to prune the backtrack search tree. This paper introduces partition search, which uses an efficient join mechanism instead of removing values from domains. Analytical prediction of quantitative performance of partition search appears to be intractable and evaluation therefore has to be by experimental comparison with reduction search algorithms that represent the state of the art. Instead of working only with available reduction search algorithms, this paper introduces enhancements such as semijoin reduction preprocessing using Bloom filtering.
Colouring, constraint satisfaction, and complexity
"... Constraint satisfaction problems have enjoyed much attention since the early seventies, and in the last decade have become also a focus of attention amongst theoreticians. Graph colourings are a special class of constraint satisfaction problems; they offer a microcosm of many of the considerations t ..."
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Cited by 18 (1 self)
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Constraint satisfaction problems have enjoyed much attention since the early seventies, and in the last decade have become also a focus of attention amongst theoreticians. Graph colourings are a special class of constraint satisfaction problems; they offer a microcosm of many of the considerations that occur in constraint satisfaction. From the point of view of theory, they are well known to exhibit a dichotomy of complexity the kcolouring problem is polynomial time solvable when k ≤ 2, and NPcomplete when k ≥ 3. Similar dichotomy has been proved for the class of graph homomorphism problems, which are intermediate problems between graph colouring and constraint satisfaction
Full Constraint Satisfaction Problems
"... Feder and Vardi have conjectured that all constraint satisfaction problems to a fixed structure(constraint language) are polynomial or NPcomplete. This socalled Dichotomy Conjecture remains open, although it has been proved in a number of special cases. Most recently, Bulatovhas verified the conje ..."
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Cited by 17 (8 self)
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Feder and Vardi have conjectured that all constraint satisfaction problems to a fixed structure(constraint language) are polynomial or NPcomplete. This socalled Dichotomy Conjecture remains open, although it has been proved in a number of special cases. Most recently, Bulatovhas verified the conjecture for conservative structures, i.e., structures which contain all possible unary relations.We explore three different implications of Bulatov's result. Firstly, the above dichotomy can be extended to socalled inclusive structures, corresponding to conservative constraintsatisfaction problems in which each variable comes with its own domain. (This has also been independently observed by Bulatov.) We prove a more general version, extending the dichotomyto socalled threeinclusive structures, i.e., structures which contain, with any unary relation R,all unary relations R0 for subsets R0 ` R with at most three elements.For the constraint satisfaction problems in this generalization we must restrict the instances to socalled 1full structures, in which each variable is involved in a unary constraint. This leadsto our second focus, which is on restrictions to more general kinds of `full ' input structures. For any set W of positive integers, we consider a restriction to Wfull input structures, i.e.,structures in which, for each w 2 W, any w variables are involved in a wary constraint. Weidentify a class of structures (the socalled Wsetfull structures) for which the restriction to Wfull input structures does not change the complexity of the constraint satisfaction problem,and hence the family of these restricted problems also exhibits dichotomy. The general family of threeinclusive constraint satisfaction problems restricted to Wfull input structures containsexamples which we cannot seem to prove either polynomial or NPcomplete. Nevertheless, we are able to use our result on the dichotomy for threeinclusive constraint satisfaction problems,to deduce the fact that all threeinclusive constraint satisfaction problems restricted to Wfullinput structures are NPcomplete or `quasipolynomial ' (of order nO(log n)).Our third focus deals with bounding the number of occurrences of a variable, which we
Weighted hypertree decompositions and optimal query plans
 In Proc. of PODS’04
, 2004
"... Hypertree width [22, 25] is a measure of the degree of cyclicity of hypergraphs. A number of relevant problems from different areas, e.g., the evaluation of conjunctive queries in database theory or the constraint satisfaction in AI, are tractable when their underlying hypergraphs have bounded hyper ..."
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Cited by 10 (3 self)
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Hypertree width [22, 25] is a measure of the degree of cyclicity of hypergraphs. A number of relevant problems from different areas, e.g., the evaluation of conjunctive queries in database theory or the constraint satisfaction in AI, are tractable when their underlying hypergraphs have bounded hypertree width. However, in practical contexts like the evaluation of database queries, we have more information besides the structure of queries. For instance, we know the number of tuples in relations, the selectivity of attributes and so on. In fact, all commercial queryoptimizers are based on quantitative methods and do not care about structural properties. In this paper, we define the notion of weighted hypertree decomposition, in order to combine structural decomposition methods with quantitative approaches. Weighted hypertree decompositions are equipped with cost functions, that can be used for modelling many