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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.
MeasureAdaptive StateSpace Construction
, 2000
"... Measureadaptive statespace construction is the process of exploiting symmetry in highlevel model and performance measure specifications to automatically construct reduced statespace Markov models that support the evaluation of the performance measure. This paper describes a new reward variable s ..."
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Cited by 16 (1 self)
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Measureadaptive statespace construction is the process of exploiting symmetry in highlevel model and performance measure specifications to automatically construct reduced statespace Markov models that support the evaluation of the performance measure. This paper describes a new reward variable specification technique, which, combined with recently developed statespace construction techniques, will allow us to build tools capable of measureadaptive statespace construction. That is, these tools will automatically adapt the size of the state space to constraints derived from the system model and the userspecified reward variables. The work described in this paper extends previous work in two directions. First, standard reward variable definitions are extended to allow symmetry in the reward variable to be identified and exploited. Then, symmetric reward variables are further extended to include the set of pathbased reward variables described in earlier work. In addition to the theory, several examples are introduced to demonstrate these new techniques.
On the structure and classification of SOMAs: generalizations of mutually orthogonal Latin squares
 Electronic Journal of Combinatorics
, 1999
"... Let k 0 and n 2 be integers. A SOMA, or more specifically a SOMA(k;n), is an n \Theta n array A, whose entries are ksubsets of a knset\Omega\Gamma such that each element of\Omega occurs exactly once in each row and exactly once in each column of A, and no 2subset of\Omega is contained in more ..."
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Cited by 12 (3 self)
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Let k 0 and n 2 be integers. A SOMA, or more specifically a SOMA(k;n), is an n \Theta n array A, whose entries are ksubsets of a knset\Omega\Gamma such that each element of\Omega occurs exactly once in each row and exactly once in each column of A, and no 2subset of\Omega is contained in more than one entry of A. A SOMA(k;n) can be constructed by superposing k mutually orthogonal Latin squares of order n with pairwise disjoint symbolsets, and so a SOMA(k;n) can be seen as a generalization of k mutually orthogonal Latin squares of order n. In this paper we first study the structure of SOMAs, concentrating on how SOMAs can decompose. We then report on the use of computational group theory and graph theory in the discovery and classification of SOMAs. In particular, we discover and classify SOMA(3; 10)s with certain properties, and discover two SOMA(4; 14)s (SOMAs with these parameters were previously unknown to exist). Some of the newly discovered SOMA(3; 10)s come from superpos...
Structural Symmetries and Model Checking
, 1998
"... We present a fully automatic framework for identifying symmetries in structural descriptions of digital circuits and CTL* formulas and using them in a model checker. We show how the set of subformulas of a formula can be partitioned into equivalence classes so that truth values for only one subfor ..."
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Cited by 10 (0 self)
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We present a fully automatic framework for identifying symmetries in structural descriptions of digital circuits and CTL* formulas and using them in a model checker. We show how the set of subformulas of a formula can be partitioned into equivalence classes so that truth values for only one subformula in any class need be evaluated for model checking. We unify and extend the theories developed by Clarke et al [CEFJ96] and Emerson and Sistla [ES96] for symmetries in Kripke structures. We formalize the notion of structural symmetries in netlist descriptions of digital circuits and CTL* formulas. We show how they relate to symmetries in the corresponding Kripke structures. We also show how such symmetries can automatically be extracted by constructing a suitable directed labeled graph and computing its automorphism group. We present a novel fast algorithm for solving the graph automorphism problem for directed labeled graphs.
CROSS RATIO GRAPHS
 J. LONDON MATH. SOC. (2) 64 (2001) 257–272
, 2001
"... A family of arctransitive graphs is studied. The vertices of these graphs are ordered pairs of distinct points from a finite projective line, and adjacency is defined in terms of the cross ratio. A uniform description of the graphs is given, their automorphism groups are determined, the problem of ..."
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Cited by 10 (8 self)
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A family of arctransitive graphs is studied. The vertices of these graphs are ordered pairs of distinct points from a finite projective line, and adjacency is defined in terms of the cross ratio. A uniform description of the graphs is given, their automorphism groups are determined, the problem of isomorphism between graphs in the family is solved, some combinatorial properties are explored, and the graphs are characterised as a certain class of arctransitive graphs. Some of these graphs have arisen as examples in studies of arctransitive graphs with complete quotients and arctransitive graphs which ‘almost cover’ a 2arc transitive graph.
Sudoku, gerechte designs, resolutions, affine space, spreads, reguli, and Hamming codes
 Amer. Math. Monthly
, 2008
"... Solving a Sudoku puzzle involves putting the symbols 1,..., 9 into the cells of a 9 × 9 grid partitioned into 3 × 3 subsquares, in such a way that each symbol occurs just once in each row, column, or subsquare. Such a solution is a special case of a gerechte design, in which an n×n grid is partition ..."
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Cited by 7 (1 self)
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Solving a Sudoku puzzle involves putting the symbols 1,..., 9 into the cells of a 9 × 9 grid partitioned into 3 × 3 subsquares, in such a way that each symbol occurs just once in each row, column, or subsquare. Such a solution is a special case of a gerechte design, in which an n×n grid is partitioned into n regions with n squares in each, and each of the symbols 1,..., n occurs once in each row, column, or region. Gerechte designs originated in statistical design of agricultural experiments, where they ensure that treatments are fairly exposed to localised variations in the field containing the experimental plots. In this paper we consider several related topics. In the first section, we define gerechte designs and some generalizations, and explain a computational technique for finding and classifying them. The second section looks at the statistical background, explaining how such designs are used for designing agricultural experiments, and what additional properties statisticians would like them to have. In the third section, we focus on a special class of Sudoku solutions which we call “symmetric”. They turn out to be related to some important topics in finite geometry over the 3element field, and to This research partially supported by NSF Grant Number DMS0510625. 1 errorcorrecting codes. We explain all of these connections, and use them to classify the symmetric Sudoku solutions (there are just two, up to the appropriate notion of equivalence). In the final section, we construct some further Sudoku solutions with desirable statistical properties, and briefly consider some generalizations. 1 Gerechte designs 1.1
Efficient Exhaustive Listings of Reversible One Dimensional Cellular Automata
"... Algebra From a rectangular structure R, using the bijection d from equation (55) above and denoting by R(s; t) the unique rectangle on the pair (s; t) guaranteed by (52), define ffl : S \Theta S ! S (63) (s; t) 7! u where fug = (d \Gamma1 (s)) 2 " (d \Gamma1 (t)) 1 ffi : S \Theta S ! S (64) ( ..."
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Cited by 5 (2 self)
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Algebra From a rectangular structure R, using the bijection d from equation (55) above and denoting by R(s; t) the unique rectangle on the pair (s; t) guaranteed by (52), define ffl : S \Theta S ! S (63) (s; t) 7! u where fug = (d \Gamma1 (s)) 2 " (d \Gamma1 (t)) 1 ffi : S \Theta S ! S (64) (s; t) 7! u where fug = d(R(s; t)) as binary operations on S.
Switching of edges in strongly regular graphs. I. A family of partial difference sets on 100 vertices
 ELECTRON. J. COMBIN., 10(1):RESEARCH PAPER
, 2003
"... We present 15 new partial difference sets over 4 nonabelian groups of order 100 and 2 new strongly regular graphs with intransitive automorphism groups. The strongly regular graphs and corresponding partial difference sets have the following parameters: (100,22,0,6), (100,36,14,12), (100,45,20,2 ..."
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Cited by 5 (1 self)
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We present 15 new partial difference sets over 4 nonabelian groups of order 100 and 2 new strongly regular graphs with intransitive automorphism groups. The strongly regular graphs and corresponding partial difference sets have the following parameters: (100,22,0,6), (100,36,14,12), (100,45,20,20), (100,44,18,20). The existence of strongly regular graphs with the latter set of parameters was an open question. Our method is based on combination of Galois correspondence between permutation groups and association schemes, classical Seidel's switching of edges and essential use of computer algebra packages. As a byproduct, a few new amorphic association schemes with 3 classes on 100 points are discovered.