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Modelling Equidistant Frequency Permutation Arrays: An Application of Constraints to Mathematics
"... Abstract Equidistant Frequency Permutation Arrays are combinatorial objects of interest in coding theory. A frequency permutation array is a type of constant composition code in which each symbol occurs the same number of times in each codeword. The problem is to find a set of codewords such that an ..."
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Abstract Equidistant Frequency Permutation Arrays are combinatorial objects of interest in coding theory. A frequency permutation array is a type of constant composition code in which each symbol occurs the same number of times in each codeword. The problem is to find a set of codewords such that any pair of codewords are a given uniform Hamming distance apart. The equidistant case is of special interest given the result that any optimal constant composition code is equidistant. This paper presents, compares and combines a number of different constraint formulations of this problem class, including a new method of representing permutations with constraints. Using these constraint models, we are able to establish several new results, which are contributing directly to mathematical research in this area. 3 1
An Effective Algorithm for and Phase Transitions of the Directed Hamiltonian Cycle Problem
"... The Hamiltonian cycle problem (HCP) is an important combinatorial problem with applications in many areas. It is among the first problems used for studying intrinsic properties, including phase transitions, of combinatorial problems. While thorough theoretical and experimental analyses have been mad ..."
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The Hamiltonian cycle problem (HCP) is an important combinatorial problem with applications in many areas. It is among the first problems used for studying intrinsic properties, including phase transitions, of combinatorial problems. While thorough theoretical and experimental analyses have been made on the HCP in undirected graphs, a limited amount of work has been done for the HCP in directed graphs (DHCP). The main contribution of this work is an effective algorithm for the DHCP. Our algorithm explores and exploits the close relationship between the DHCP and the Assignment Problem (AP) and utilizes a technique based on Boolean satisfiability (SAT). By combining effective algorithms for the AP and SAT, our algorithm significantly outperforms previous exact DHCP algorithms, including an algorithm based on the awardwinning Concorde TSP algorithm. The second result of the current study is an experimental analysis of phase transitions of the DHCP, verifying and refining a known phase transition of the DHCP. 1.
Design of Parallel Portfolios for SATBased Solving of Hamiltonian Cycle Problems
"... We study portfolios of parallel strategies for Boolean Satisfiability (SAT) based solving of Hamiltonian Cycle Problems (HCPs). The strategies are based on our techniques for relative SAT encoding of permutations with constraints, and exploit: 1) encodings that eliminate half of the ordering Boolean ..."
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We study portfolios of parallel strategies for Boolean Satisfiability (SAT) based solving of Hamiltonian Cycle Problems (HCPs). The strategies are based on our techniques for relative SAT encoding of permutations with constraints, and exploit: 1) encodings that eliminate half of the ordering Boolean variables and twothirds of the transitivity constraints; 2) 12 triangulation heuristics for minimal enumeration of transitivity; 3) 11 heuristics for selecting the first node in the Hamiltonian cycle; 4) inverse transitivity constraints; and 5) exclusivity successor constraints between neighbors. We achieve up to 3 orders of magnitude speedup on random graphs that have Hamiltonian cycles and are in the phase transition region.
MathCheck: A Math Assistant via a Combination of Computer Algebra Systems and SAT Solvers
"... Abstract. We present a method and an associated system, calledMathCheck, that embeds the functionality of a computer algebra system (CAS) within the inner loop of a conflictdriven clauselearning SAT solver. SAT+CAS systems, a la MathCheck, can be used as an assistant by mathematicians to either ..."
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Abstract. We present a method and an associated system, calledMathCheck, that embeds the functionality of a computer algebra system (CAS) within the inner loop of a conflictdriven clauselearning SAT solver. SAT+CAS systems, a la MathCheck, can be used as an assistant by mathematicians to either counterexample or finitely verify open universal conjectures on any mathematical topic (e.g., graph and number theory, algebra, geometry, etc.) supported by the underlying CAS system. Such a SAT+CAS system combines the efficient search routines of modern SAT solvers, with the expressive power of CAS, thus complementing both. The key insight behind the power of the SAT+CAS combination is that the CAS system can help cut down the searchspace of the SAT solver, by providing learned clauses that encode theoryspecific lemmas, as it searches for a counterexample to the input conjecture (just like the T in DPLL(T)). In addition, the combination enables a more efficient encoding of problems than a pure Boolean representation. In this paper, we leverage the graphtheoretic capabilities of an opensource CAS, called SAGE. As case studies, we look at two longstanding open mathematical conjectures from graph theory regarding properties of hypercubes: the first conjecture states that any matching of any ddimensional hypercube can be extended to a Hamiltonian cycle; and the second states that given an edgeantipodal coloring of a hypercube, there always exists a monochromatic path between two antipodal vertices. Previous results have shown the conjectures true up to certain lowdimensional hypercubes, and attempts to extend them have failed until now. Using our SAT+CAS system, MathCheck, we extend these two conjectures to higherdimensional hypercubes. We provide detailed performance analysis and show an exponential reduction in search space via the SAT+CAS combination relative to finite bruteforce search. 1