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Constraint satisfaction problems in clausal form: Autarkies and minimal unsatisfiability
 ELECTRONIC COLLOQUIUM ON COMPUTATIONAL COMPLEXITY (ECCC
, 2007
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A new method to construct lower bounds for van der Waerden numbers
 THE ELECTRONIC JOURNAL OF COMBINATORICS
, 2007
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Finding lean induced cycles in binary hypercubes
, 2009
"... Induced (chordfree) cycles in binary hypercubes have many applications in computer science. The state of the art for computing such cycles relies on genetic algorithms, which are, however, unable to perform a complete search. In this paper, we propose an approach to finding a special class of i ..."
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Induced (chordfree) cycles in binary hypercubes have many applications in computer science. The state of the art for computing such cycles relies on genetic algorithms, which are, however, unable to perform a complete search. In this paper, we propose an approach to finding a special class of induced cycles we call lean, based on an efficient propositional SAT encoding. Lean induced cycles dominate a minimum number of hypercube nodes. Such cycles have been identified in Systems Biology as candidates for stable trajectories of gene regulatory networks. The encoding enabled us to compute lean induced cycles for hypercubes up to dimension 7. We also classify the induced cycles by the number of nodes they fail to dominate, using a custombuilt AllSAT solver. We demonstrate how clause filtering can reduce the number of blocking clauses by two orders of magnitude.
Internal Symmetry
"... We have been studying the internal symmetries within an individual solution of a constraint satisfaction problem [1]. Such internal symmetries can be compared with solution symmetries which map between different solutions of the same problem. We show that we can take advantage of both types of symm ..."
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We have been studying the internal symmetries within an individual solution of a constraint satisfaction problem [1]. Such internal symmetries can be compared with solution symmetries which map between different solutions of the same problem. We show that we can take advantage of both types of symmetry when solving constraint satisfaction solutions within two benchmark domains. By identifying internal symmetries and breaking solution symmetries, we are able to increase the size of problems which have been solved.
DoubleWheel Graphs Are Graceful
 PROCEEDINGS OF THE TWENTYTHIRD INTERNATIONAL JOINT CONFERENCE ON ARTIFICIAL INTELLIGENCE
"... We present the first polynomial time construction procedure for generating graceful doublewheel graphs. A graph is graceful if its vertices can be labeled with distinct integer values from {0,..., e}, where e is the number of edges, such that each edge has a unique value corresponding to the absolu ..."
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We present the first polynomial time construction procedure for generating graceful doublewheel graphs. A graph is graceful if its vertices can be labeled with distinct integer values from {0,..., e}, where e is the number of edges, such that each edge has a unique value corresponding to the absolute difference of its endpoints. Graceful graphs have a range of practical application domains, including in radio astronomy, Xray crystallography, cryptography, and experimental design. Various families of graphs have been proven to be graceful, while others have only been conjectured to be. In particular, it has been conjectured that socalled doublewheel graphs are graceful. A doublewheel graph consists of two cycles of N nodes connected to a common hub. We prove this conjecture by providing the first construction for graceful doublewheel graphs, for any N> 3, using a framework that combines streamlined constraint reasoning with insights from human computation. We also use this framework to provide a polynomial time construction for diagonally ordered magic squares.
Improving the Use of Cyclic Zippers in Finding Lower Bounds for van der Waerden Numbers
"... For integers k and l, each greater than 1, suppose that p is a prime with p ≡ 1 (mod k) and that the kthpower classes mod p induce a coloring of the integer segment [0, p − 1] that admits no monochromatic occurrence of l consecutive members of an arithmetic progression. Such a coloring can lead to ..."
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For integers k and l, each greater than 1, suppose that p is a prime with p ≡ 1 (mod k) and that the kthpower classes mod p induce a coloring of the integer segment [0, p − 1] that admits no monochromatic occurrence of l consecutive members of an arithmetic progression. Such a coloring can lead to a coloring of [0, (l − 1)p] that is similarly free of monochromatic lprogressions, and, hence, can give directly a lower bound for the van der Waerden number W (k, l). P. R. Herwig, M. J. H. Heule, P. M. van Lambalgen, and H. van Maaren have devised a technique for splitting and “zipping ” such a coloring of [0, p−1] to yield a coloring of [0, 2p−1] which, for even values of k, is sometimes extendable to a coloring of [0, 2(l − 1)p] where both new colorings still admit no monochromatic lprogressions. Here we derive a fast procedure for checking whether such a zipped coloring remains free of monochromatic lprogressions, effectively reducing a quadratictime check to a lineartime check. Using this procedure we find some new lower bounds for van der Waerden numbers.