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111
SymmetryBreaking Predicates for Search Problems
, 1996
"... Many reasoning and optimization problems exhibit symmetries. Previous work has shown how special purpose algorithms can make use of these symmetries to simplify reasoning. We present a general scheme whereby symmetries are exploited by adding "symmetrybreaking" predicates to the theory. Our a ..."
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Cited by 159 (0 self)
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Many reasoning and optimization problems exhibit symmetries. Previous work has shown how special purpose algorithms can make use of these symmetries to simplify reasoning. We present a general scheme whereby symmetries are exploited by adding "symmetrybreaking" predicates to the theory. Our approach
An Optimal Lower Bound on the Number of Variables for Graph Identification
 Combinatorica
, 1992
"... In this paper we show that Ω(n) variables are needed for firstorder logic with counting to identify graphs on n vertices. The kvariable language with counting is equivalent to the (k − 1)dimensional WeisfeilerLehman method. We thus settle a longstanding open problem. Previously it was an open q ..."
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Cited by 135 (9 self)
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In this paper we show that Ω(n) variables are needed for firstorder logic with counting to identify graphs on n vertices. The kvariable language with counting is equivalent to the (k − 1)dimensional WeisfeilerLehman method. We thus settle a longstanding open problem. Previously it was an open question whether or not 4 variables suffice. Our lower bound remains true over a set of graphs of color class size 4. This contrasts sharply with the fact that 3 variables suffice to identify all graphs of color class size 3, and 2 variables suffice to identify almost all graphs. Our lower bound is optimal up to multiplication by a constant because n variables obviously suffice to identify graphs on n vertices. 1
Parameterized Complexity: A Framework for Systematically Confronting Computational Intractability
 DIMACS Series in Discrete Mathematics and Theoretical Computer Science
, 1997
"... In this paper we give a programmatic overview of parameterized computational complexity in the broad context of the problem of coping with computational intractability. We give some examples of how fixedparameter tractability techniques can deliver practical algorithms in two different ways: (1) by ..."
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Cited by 72 (15 self)
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In this paper we give a programmatic overview of parameterized computational complexity in the broad context of the problem of coping with computational intractability. We give some examples of how fixedparameter tractability techniques can deliver practical algorithms in two different ways: (1) by providing useful exact algorithms for small parameter ranges, and (2) by providing guidance in the design of heuristic algorithms. In particular, we describe an improved FPT kernelization algorithm for Vertex Cover, a practical FPT algorithm for the Maximum Agreement Subtree (MAST) problem parameterized by the number of species to be deleted, and new general heuristics for these problems based on FPT techniques. In the course of making this overview, we also investigate some structural and hardness issues. We prove that an important naturally parameterized problem in artificial intelligence, STRIPS Planning (where the parameter is the size of the plan) is complete for W [1]. As a corollary, this implies that kStep Reachability for Petri Nets is complete for W [1]. We describe how the concept of treewidth can be applied to STRIPS Planning and other problems of logic to obtain FPT results. We describe a surprising structural result concerning the top end of the parameterized complexity hierarchy: the naturally parameterized Graph kColoring problem cannot be resolved with respect to XP either by showing membership in XP, or by showing hardness for XP without settling the P = NP question one way or the other.
An improved algorithm for matching large graphs
 In: 3rd IAPRTC15 Workshop on Graphbased Representations in Pattern Recognition, Cuen
, 2001
"... In this paper an improved version of a graph matching algorithm is presented, which is able to efficiently solve the graph isomorphism and graphsubgraph isomorphism problems on Attributed Relational Graphs. This version is particularly suited to work with very large graphs, since its memory require ..."
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Cited by 68 (2 self)
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In this paper an improved version of a graph matching algorithm is presented, which is able to efficiently solve the graph isomorphism and graphsubgraph isomorphism problems on Attributed Relational Graphs. This version is particularly suited to work with very large graphs, since its memory requirements are quite smaller than those of other algorithms of the same kind. After a detailed description of the algorithm, an experimental comparison is made against both the previous version (developed by the same authors) and the Ullmann’s algorithm. 1.
The Graph Isomorphism Problem
, 1996
"... The graph isomorphism problem can be easily stated: check to see if two graphs that look differently are actually the same. The problem occupies a rare position in the world of complexity theory, it is clearly in NP but is not known to be in P and it is not known to be NPcomplete. Many subdiscipli ..."
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Cited by 64 (0 self)
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The graph isomorphism problem can be easily stated: check to see if two graphs that look differently are actually the same. The problem occupies a rare position in the world of complexity theory, it is clearly in NP but is not known to be in P and it is not known to be NPcomplete. Many subdisciplines of mathematics, such as topology theory and group theory, can be brought to bear on the problem, and yet only for special classes of graphs have polynomialtime algorithms been discovered. Incongruently, this problem seems very easy in practice. It is almost always trivial to check two random graphs for isomorphism, and fast hardware implementations exists for application domains such as image processing. This paper is mostly a survey of related work in the graph isomorphism field. We examine the problem from many angles, mirroring the multifaceted nature of the literature. We survey complexity results for the graph isomorphism problem, and discuss some of the classes of graphs which hav...
Describing Graphs: a FirstOrder Approach to Graph Canonization
, 1990
"... In this paper we ask the question, "What must be added to firstorder logic plus leastfixed point to obtain exactly the polynomialtime properties of unordered graphs?" We consider the languages Lk consisting of firstorder logic restricted to k variables and Ck consisting of Lk plus "counting ..."
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Cited by 57 (7 self)
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In this paper we ask the question, "What must be added to firstorder logic plus leastfixed point to obtain exactly the polynomialtime properties of unordered graphs?" We consider the languages Lk consisting of firstorder logic restricted to k variables and Ck consisting of Lk plus "counting quantifiers". We give efficient canonization algorithms for graphs characterized by Ck or Lk . It follows from known results that all trees and almost all graphs are characterized by C2 .
The Complexity of McKay's Canonical Labeling Algorithm
, 1996
"... We study the time complexity of McKay's algorithm to compute canonical forms and automorphism groups of graphs. The algorithm is based on a type of backtrack search, and it performs pruning by discovered automorphisms and by hashing partial information of vertex labelings. In practice, the algorithm ..."
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Cited by 36 (1 self)
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We study the time complexity of McKay's algorithm to compute canonical forms and automorphism groups of graphs. The algorithm is based on a type of backtrack search, and it performs pruning by discovered automorphisms and by hashing partial information of vertex labelings. In practice, the algorithm is implemented in the nauty package. We obtain colorings of Furer's graphs that allow the algorithm to compute their canonical forms in polynomial time. We then prove an exponential lower bound of the algorithm for connected 3regular graphs of colorclass size 4 using Furer's construction. We conducted experiments with nauty for these graphs. Our experimental results also indicate the same exponential lower bound.
A performance comparison of five algorithms for graph isomorphism
 in Proceedings of the 3rd IAPR TC15 Workshop on Graphbased Representations in Pattern Recognition
, 2001
"... Despite the significant number of isomorphism algorithms presented in the literature, till now no efforts have been done for characterizing their performance. Consequently, it is not clear how the behavior of those algorithms varies as the type and the size of the graphs to be matched varies in case ..."
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Cited by 34 (2 self)
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Despite the significant number of isomorphism algorithms presented in the literature, till now no efforts have been done for characterizing their performance. Consequently, it is not clear how the behavior of those algorithms varies as the type and the size of the graphs to be matched varies in case of real applications. In this paper we present a benchmarking activity for characterizing the performance of a bunch of algorithms for exact graph isomorphism. To this purpose we use a large database containing 10,000 couples of isomorphic graphs with different topologies (regular graphs, randomly connected graphs, bounded valence graph), enriched with suitably modified versions of them for simulating distortions occurring in real cases. The size of the considered graphs ranges from a few nodes to about 1000 nodes. 1.
Sylow's Theorem in Polynomial Time
 JOURNAL OF COMPUTER AND SYSTEM SCIENCES
, 1985
"... Given a set r of permutations of an nset, let G be the group of permutations generated by f. If p is a prime, a Sylow psubgroup of G is a subgroup whose order is the largest power of p dividing IGI. For more than 100 years it has been known that a Sylow psubgroup exists, and that for any two Sylo ..."
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Cited by 24 (8 self)
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Given a set r of permutations of an nset, let G be the group of permutations generated by f. If p is a prime, a Sylow psubgroup of G is a subgroup whose order is the largest power of p dividing IGI. For more than 100 years it has been known that a Sylow psubgroup exists, and that for any two Sylow psubgroups PI, P, of G there is an element go G such that Pz = g‘PI g. We present polynomialtime algorithms that find (generators for) a Sylow psubgroup of G, and that find ge G such that P, = g‘P, g whenever (generators for) two Sylow psubgroups PI, Pz are given. These algorithms involve the classification of all tinite simple groups. 0 1985 Academic Press. Inc. PART I 1.
Subgraph Isomorphism in Polynomial Time
, 1995
"... In this paper, we propose a new approach to the problem of subgraph isomorphism detection. The new method is designed for systems which differentiate between graphs that are a priori known, socalled model graphs, and unknown graphs, socalled input graphs. The problem to be solved is to find a subg ..."
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Cited by 24 (1 self)
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In this paper, we propose a new approach to the problem of subgraph isomorphism detection. The new method is designed for systems which differentiate between graphs that are a priori known, socalled model graphs, and unknown graphs, socalled input graphs. The problem to be solved is to find a subgraph isomorphism from an input graph, which is given online, to any of the model graphs. The new method is based on an intensive preprocessing step in which the model graphs are used to create a decision tree. At run time, the input graph is then classified by the decision tree and all model graphs for which there exists a subgraph isomorphism from the input graph are detected. If we neglect the time needed for preprocessing, the computational complexity of the new subgraph isomorphism algorithm is only quadratic in the number of input graph vertices. Furthermore, it is independent of the number of model graphs and the number of edges in any of the graphs. However, the decision tree that i...