Results 1  10
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161
The computational structure of monotone monadic SNP and constraint satisfaction: A study through Datalog and group theory
 SIAM J. Comput
, 1998
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Conjunctive Queries over Trees
, 2004
"... We study the complexity and expressive power of conjunctive queries over unranked labeled trees, where the tree structures are represented using "axis relations" such as "child", "descendant", and "following" (we consider a superset of the XPath axes) as well as unary relations for node labels. (Cyc ..."
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Cited by 63 (7 self)
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We study the complexity and expressive power of conjunctive queries over unranked labeled trees, where the tree structures are represented using "axis relations" such as "child", "descendant", and "following" (we consider a superset of the XPath axes) as well as unary relations for node labels. (Cyclic) conjunctive queries over trees occur in a wide range of data management scenarios related to XML, the Web, and computational linguistics. We establish a framework for characterizing structures representing trees for which conjunctive queries can be evaluated e# ciently. Then we completely chart the tractability frontier of the problem for our axis relations, i.e., we find all subsetmaximal sets of axes for which query evaluation is in polynomial time. All polynomialtime results are obtained immediately using the proof techniques from our framework. Finally, we study the expressiveness of conjunctive queries over trees and compare it to the expressive power of fragments of XPath. We show that for each conjunctive query, there is an equivalent acyclic positive query (i.e., a set of acyclic conjunctive queries), but that in general this query is not of polynomial size.
Duality and polynomial testing of tree homomorphisms
 Trans. Amer. Math. Soc
, 1996
"... Abstract. Let H be a fixed digraph. We consider the Hcolouring problem, i.e., the problem of deciding which digraphs G admit a homomorphism to H. We are interested in a characterization in terms of the absence in G of certain treelike obstructions. Specifically, we say that H has tree duality if, ..."
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Cited by 53 (16 self)
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Abstract. Let H be a fixed digraph. We consider the Hcolouring problem, i.e., the problem of deciding which digraphs G admit a homomorphism to H. We are interested in a characterization in terms of the absence in G of certain treelike obstructions. Specifically, we say that H has tree duality if, for all digraphs G, G is not homomorphic to H if and only if there is an oriented tree which is homomorphic to G but not to H. Weprovethatif Hhas tree duality then the Hcolouring problem is polynomial. We also generalize tree duality to bounded treewidth duality and prove a similar result. We relate these duality concepts to the notion of the Xproperty studied by Gutjahr, Welzl, and Woeginger. We then focus on the case when H itself is an oriented tree. In fact, we are particularly interested in those trees that have exactly one vertex of degree three and all other vertices of degree one or two. Such trees are called triads. We have shown in a companion paper that there exist oriented triads H for which the Hcolouring problem is NPcomplete. We contrast these with several families of oriented triads H which have tree duality, or bounded treewidth duality, and hence polynomial Hcolouring problems. If P � = NP, then no oriented triad H with an NPcomplete Hcolouring problem can have bounded treewidth duality; however no proof of this is known, for any oriented triad H. We prove that none of the oriented triads H with NPcomplete Hcolouring problems given in the companion paper has tree duality. 1.
Testing Subgraphs in Directed Graphs
 Proc. of the 35 th Annual Symp. on Theory of Computing (STOC
, 2003
"... Let H be a fixed directed graph on h vertices, let G be a directed graph on n vertices and suppose that at least #n edges have to be deleted from it to make it Hfree. We show that in this case G contains at least f(#, H)n copies of H. This is proved by establishing a directed version of Sz ..."
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Cited by 51 (17 self)
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Let H be a fixed directed graph on h vertices, let G be a directed graph on n vertices and suppose that at least #n edges have to be deleted from it to make it Hfree. We show that in this case G contains at least f(#, H)n copies of H. This is proved by establishing a directed version of Szemeredi's regularity lemma, and implies that for every H there is a onesided error property tester whose query complexity is bounded by a function of # only for testing the property PH of being Hfree.
Acyclic and Oriented Chromatic Numbers of Graphs
 J. Graph Theory
, 1997
"... . The oriented chromatic number o ( ~ G) of an oriented graph ~ G = (V; A) is the minimum number of vertices in an oriented graph ~ H for which there exists a homomorphism of ~ G to ~ H . The oriented chromatic number o (G) of an undirected graph G is the maximum of the oriented chromatic n ..."
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Cited by 40 (13 self)
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. The oriented chromatic number o ( ~ G) of an oriented graph ~ G = (V; A) is the minimum number of vertices in an oriented graph ~ H for which there exists a homomorphism of ~ G to ~ H . The oriented chromatic number o (G) of an undirected graph G is the maximum of the oriented chromatic numbers of all the orientations of G. This paper discusses the relations between the oriented chromatic number and the acyclic chromatic number and some other parameters of a graph. We shall give a lower bound for o (G) in terms of a (G). An upper bound for o (G) in terms of a (G) was given by Raspaud and Sopena. We also give an upper bound for o (G) in terms of the maximum degree of G. We shall show that this upper bound is not far from being optimal. Keywords. Oriented chromatic number, Acyclic chromatic number. 1
Constraint Satisfaction with Countable Homogeneous Templates
 IN PROCEEDINGS OF CSL’03
, 2003
"... For a fixed countable homogeneous structure we study the computational problem whether a given finite structure of the same relational signature homomorphically maps to . This problem is known as the constraint satisfaction problem CSP( ) for and was intensively studied for finite . We show that ..."
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Cited by 33 (15 self)
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For a fixed countable homogeneous structure we study the computational problem whether a given finite structure of the same relational signature homomorphically maps to . This problem is known as the constraint satisfaction problem CSP( ) for and was intensively studied for finite . We show that  as in the case of finite  the computational complexity of CSP( ) for countable homogeneous is determinded by the clone of polymorphisms of . To this end we prove the following theorem which is of independent interest: The primitive positive definable relations over an !categorical structure are precisely the relations that are invariant under the polymorphisms of .
Counting graph homomorphisms
 In:Topics in Discrete Math
, 2006
"... Counting homomorphisms between graphs (often with weights) comes up in a wide variety of areas, including extremal graph theory, properties of graph products, partition functions in statistical physics and property testing of large graphs. In this paper we survey recent developments in the study of ..."
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Cited by 32 (8 self)
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Counting homomorphisms between graphs (often with weights) comes up in a wide variety of areas, including extremal graph theory, properties of graph products, partition functions in statistical physics and property testing of large graphs. In this paper we survey recent developments in the study of homomorphism numbers, including the characterization of the homomorphism numbers in terms of the semidefiniteness of “connection matrices”, and some applications of this fact in extremal graph theory. We define a distance of two graphs in terms of similarity of their global structure, which also reflects the closeness of (appropriately scaled) homomorphism numbers into the two graphs. We use homomorphism numbers to define convergence of a sequence of graphs, and show that a graph sequence is convergent if and only if it is Cauchy in this distance. Every convergent graph sequence has a limit in the form of a symmetric measurable function in two variables. We use these notions of distance and graph limits to give a general theory for parameter testing. The convergence can also be characterized in terms of mappings of the graphs into fixed small graphs, which is strongly connected to important parameters like ground state energy in statistical physics, and to weighted maximum cut problems in computer science. 1
On the Maximum Average Degree and the Oriented Chromatic Number of a Graph
 Discrete Math
, 1995
"... The oriented chromatic number o(H) of an oriented graph H is defined as the minimum order of an oriented graph H 0 such that H has a homomorphism to H 0 . The oriented chromatic number o(G) of an undirected graph G is then defined as the maximum oriented chromatic number of its orientations. In ..."
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Cited by 30 (15 self)
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The oriented chromatic number o(H) of an oriented graph H is defined as the minimum order of an oriented graph H 0 such that H has a homomorphism to H 0 . The oriented chromatic number o(G) of an undirected graph G is then defined as the maximum oriented chromatic number of its orientations. In this paper we study the links between o(G) and mad(G) defined as the maximum average degree of the subgraphs of G. 1 Introduction and statement of results For every graph G we denote by V (G), with vG = jV (G)j, its set of vertices and by E(G), with e G = jE(G)j, its set of arcs or edges. A homomorphism from a graph G to a graph On leave of absence from the Institute of Mathematics, Novosibirsk, 630090, Russia. With support from Engineering and Physical Sciences Research Council, UK, grant GR/K00561, and from the International Science Foundation, grant NQ4000. y This work was partially supported by the Network DIMANET of the European Union and by the grant 960101614 of the Russian F...
Probabilistic data exchange
 In Proc. ICDT
, 2010
"... The work reported here lays the foundations of data exchange in the presence of probabilistic data. This requires rethinking the very basic concepts of traditional data exchange, such as solution, universal solution, and the certain answers of target queries. We develop a framework for data exchange ..."
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Cited by 28 (5 self)
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The work reported here lays the foundations of data exchange in the presence of probabilistic data. This requires rethinking the very basic concepts of traditional data exchange, such as solution, universal solution, and the certain answers of target queries. We develop a framework for data exchange over probabilistic databases, and make a case for its coherence and robustness. This framework applies to arbitrary schema mappings, and finite or countably infinite probability spaces on the source and target instances. After establishing this framework and formulating the key concepts, we study the application of the framework to a concrete and practical setting where probabilistic databases are compactly encoded by means of annotations formulated over random Boolean variables. In this setting, we study the problems of testing for the existence of solutions and universal solutions, materializing such solutions, and evaluating target queries (for unions of conjunctive queries) in both the exact sense and the approximate sense. For each of the problems, we carry out a complexity analysis based on properties of the annotation, in various classes of dependencies. Finally, we show that the framework and results easily and completely generalize to allow not only the data, but also the schema mapping itself to be probabilistic.
Level of repair analysis and minimum cost homomorphisms of graphs
 Discrete Appl. Math
"... This paper is dedicated to the memory of Lillian Barros Abstract. Level of Repair Analysis (LORA) is a prescribed procedure for defence logistics support planning. For a complex engineering system containing perhaps thousands of assemblies, subassemblies, components, etc. organized into several lev ..."
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Cited by 27 (11 self)
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This paper is dedicated to the memory of Lillian Barros Abstract. Level of Repair Analysis (LORA) is a prescribed procedure for defence logistics support planning. For a complex engineering system containing perhaps thousands of assemblies, subassemblies, components, etc. organized into several levels of indenture and with a number of possible repair decisions, LORA seeks to determine an optimal provision of repair and maintenance facilities to minimize overall lifecycle costs. For a LORA problem with two levels of indenture with three possible repair decisions, which is of interest in UK and US military and which we call LORABR, Barros (1998) and Barros and Riley (2001) developed certain branchandbound heuristics. The surprising result of this paper is that LORABR is, in fact, polynomialtime solvable. To obtain this result, we formulate the general LORA problem as an optimization homomorphism problem on bipartite graphs, and reduce a generalization of LORABR, LORAM, to the maximum weight independent set problem on a bipartite graph. We prove that the general LORA problem is NPhard by using an important result on list homomorphisms of graphs. We introduce the minimum cost graph homomorphism problem, provide partial results and pose an open problem. Finally, we show that our result for LORABR can be applied to prove that an extension of the maximum weight independent set problem on bipartite graphs is polynomial time solvable.