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118
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
Complete mining of frequent patterns from graphs: Mining graph data
 Machine Learning
, 2003
"... Abstract. Basket Analysis, which is a standard method for data mining, derives frequent itemsets from database. However, its mining ability is limited to transaction data consisting of items. In reality, there are many applications where data are described in a more structural way, e.g. chemical com ..."
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Cited by 70 (4 self)
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Abstract. Basket Analysis, which is a standard method for data mining, derives frequent itemsets from database. However, its mining ability is limited to transaction data consisting of items. In reality, there are many applications where data are described in a more structural way, e.g. chemical compounds and Web browsing history. There are a few approaches that can discover characteristic patterns from graphstructured data in the field of machine learning. However, almost all of them are not suitable for such applications that require a complete search for all frequent subgraph patterns in the data. In this paper, we propose a novel principle and its algorithm that derive the characteristic patterns which frequently appear in graphstructured data. Our algorithm can derive all frequent induced subgraphs from both directed and undirected graph structured data having loops (including selfloops) with labeled or unlabeled nodes and links. Its performance is evaluated through the applications to Web browsing pattern analysis and chemical carcinogenesis analysis.
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...
Solving Difficult Instances of Boolean Satisfiability in the Presence of Symmetry
, 2002
"... Research in algorithms for Boolean satisfiability (SAT) and their implementations [45, 41, 10] has recently outpaced benchmarking efforts. Most of the classic DIMACS benchmarks [21] can now be solved in seconds on commodity PCs. More recent benchmarks [54] take longer to solve due of their large siz ..."
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Cited by 44 (17 self)
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Research in algorithms for Boolean satisfiability (SAT) and their implementations [45, 41, 10] has recently outpaced benchmarking efforts. Most of the classic DIMACS benchmarks [21] can now be solved in seconds on commodity PCs. More recent benchmarks [54] take longer to solve due of their large size, but are still solved in minutes. Yet, small and difficult SAT instances must exist if P##NP. To this end, our work articulates SAT instances that are unusually difficult for their size, including satisfiable instances derived from Very Large Scale Integration (VLSI) routing problems. With an efficient implementation to solve the graph automorphism problem [39, 50, 51], we show that in structured SAT instances difficulty may be associated with large numbers of symmetries.
Symmetry Breaking for Boolean Satisfiability: . . .
"... Boolean Satisfiability solvers improved dramatically over the last seven years [14, 13] and are commonly used in applications such as bounded model checking, planning, and FPGA routing. However, a number of practical SAT instances remain difficult to solve. Recent work pointed out that symmetries i ..."
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Cited by 40 (9 self)
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Boolean Satisfiability solvers improved dramatically over the last seven years [14, 13] and are commonly used in applications such as bounded model checking, planning, and FPGA routing. However, a number of practical SAT instances remain difficult to solve. Recent work pointed out that symmetries in the search space are often to blame [1]. The framework of symmetrybreaking (SBPs) [5], together with further improvements [1], was then used to achieve empirical speedups. For symmetrybreaking to be successful in practice, its overhead must be less than the complexity reduction it brings. In this work we show how logic minimization helps to improve this tradeoff and achieve much better empirical results. We also contribute detailed new studies of SBPs and their efficiency as well as new general constructions of SBPs.
Solving Difficult SAT Instances in the Presence of Symmetry
, 2002
"... Research in algorithms for Boolean satisfiability and their efficient implementations [26, 8] has recently outpaced benchmarking efforts. Most of the classic DIMACS benchmarks from the early 1990s [12] can be solved in seconds on commodity PCs. More recent benchmarks take longer to solve primarily b ..."
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Cited by 38 (1 self)
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Research in algorithms for Boolean satisfiability and their efficient implementations [26, 8] has recently outpaced benchmarking efforts. Most of the classic DIMACS benchmarks from the early 1990s [12] can be solved in seconds on commodity PCs. More recent benchmarks take longer to solve primarily because of their large size, but are still solved in minutes [28]. However, small and difficult SAT instances must exist because Boolean satisfiability is NPcomplete. Our work articulates a number of SAT instances that are unusually difficult for their size, including satisfiable instances from global routing and detailed routing for FPGAs [22]. Using an efficient implementation to solve the graph automorphism problem [21, 23, 25], we show that in structured SAT instances difficulty is sometimes associated with large numbers of symmetries. We propose a new, improved construction of symmetrybreaking clauses [11] and apply them to empirically demonstrate very significant speedups over current state of the art in Boolean satisfiability. Our techniques are formulated as preprocessing and can be applied to an arbitrary SAT solver without modifying its source code. We also show that considerations of symmetry may lead to more efficient reductions to SAT in the routing domain and potentially other applications.
Automated Generation of Search Tree Algorithms for Hard Graph Modification Problems
 Algorithmica
, 2004
"... We present a framework for an automated generation of exact search tree algorithms for NPhard problems. The purpose of our approach is twofoldrapid development and improved upper bounds. ..."
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Cited by 24 (10 self)
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We present a framework for an automated generation of exact search tree algorithms for NPhard problems. The purpose of our approach is twofoldrapid development and improved upper bounds.
Exploiting orbits in symmetric ilp
 Mathematical Programming
"... This Article is brought to you for free and open access by Research Showcase. It has been accepted for inclusion in Tepper School of Business by an authorized administrator of Research Showcase. For more information, please contact kbehrman@andrew.cmu.edu. Mathematical Programming manuscript No. (wi ..."
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Cited by 22 (0 self)
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This Article is brought to you for free and open access by Research Showcase. It has been accepted for inclusion in Tepper School of Business by an authorized administrator of Research Showcase. For more information, please contact kbehrman@andrew.cmu.edu. Mathematical Programming manuscript No. (will be inserted by the editor)
E cient Detection of Network Motifs
 IEEE/ACM Transactions on Computational Biology and Bioinformatics
, 2006
"... © 2006 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other ..."
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Cited by 21 (0 self)
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© 2006 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from the IEEE.
A Note on Large Graphs of Diameter Two and Given Maximum Degree
"... Let vt(d; 2) be the largest order of a vertextransitive graph of degree d and diameter two. It is known that vt(d; 2) = d 2 + 1 for d = 1; 2; 3, and 7; for the remaining values of d we have vt(d; 2) d 2 \Gamma 1. The only known general lower bound on vt(d; 2), valid for all d, seems to be vt(d ..."
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Cited by 19 (5 self)
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Let vt(d; 2) be the largest order of a vertextransitive graph of degree d and diameter two. It is known that vt(d; 2) = d 2 + 1 for d = 1; 2; 3, and 7; for the remaining values of d we have vt(d; 2) d 2 \Gamma 1. The only known general lower bound on vt(d; 2), valid for all d, seems to be vt(d; 2) b d+2 2 cd d+2 2 e. Using voltage graphs, we construct a family of vertextransitive nonCayley graphs which shows that vt(d; 2) 8 9 (d + 1 2 ) 2 for all d of the form d = (3q \Gamma 1)=2 where q is a prime power congruent with 1 (mod 4). The construction generalizes to all prime powers and yields large highly symmetric graphs for other degrees as well. In particular, for d = 7 we obtain as a special case the HoffmanSingleton graph, and for d = 11 and d = 13 we have new largest graphs of diameter two and degree d on 98 and 162 vertices, respectively. 1 Introduction The wellknown degree/diameter problem asks for determining the largest possible number n(d; k) of vertic...