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Reformulations in mathematical programming: Automatic symmetry detection and exploitation
 Mathematical Programming
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Symmetry in mathematical programming
 MIXED INTEGER NONLINEAR PROGRAMMING. VOLUME IMA
"... Symmetry is mainly exploited in mathematical programming in order to reduce the computation times of enumerative algorithms. The most widespread approach rests on: (a) finding symmetries in the problem instance; (b) reformulating the problem so that it does not allow some of the symmetric optima; ( ..."
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Symmetry is mainly exploited in mathematical programming in order to reduce the computation times of enumerative algorithms. The most widespread approach rests on: (a) finding symmetries in the problem instance; (b) reformulating the problem so that it does not allow some of the symmetric optima; (c) solving the modified problem. Sometimes (b) and (c) are performed concurrently: the solution algorithm generates a sequence of subproblems, some of which are recognized to be symmetrically equivalent and either discarded or treated differently. We review symmetrybased analyses and methods for Linear Programming, Integer Linear Programming, MixedInteger Linear Programming and Semidefinite Programming. We then discuss a method (introduced in [35]) for automatically detecting symmetries of general (nonconvex) Nonlinear and MixedInteger Nonlinear Programming problems and a reformulation based on adjoining symmetry breaking constraints to the original formulation. We finally present a new theoretical and computational study of the formulation symmetries of the Kissing Number Problem.
On the impact of symmetrybreaking constraints on spatial BranchandBound for circle packing in a square
, 2011
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"... On the impact of symmetrybreaking constraints on spatial BranchandBound for circle packing in a squareI ..."
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On the impact of symmetrybreaking constraints on spatial BranchandBound for circle packing in a squareI
Exploratory Equivalence in Graphs: Definition and Algorithms
, 2014
"... Motivated by improving the efficiency of pattern matching on graphs, we define a new kind of equivalence on graph vertices. Since it can be used in various graph algorithms that explore graphs, we call it exploratory equivalence. The equivalence is based on graph automorphisms. Because many similar ..."
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Motivated by improving the efficiency of pattern matching on graphs, we define a new kind of equivalence on graph vertices. Since it can be used in various graph algorithms that explore graphs, we call it exploratory equivalence. The equivalence is based on graph automorphisms. Because many similar equivalences exist (some also based on automorphisms), we argue that this one is novel. For each graph, there are many possible exploratory equivalences, but for improving the efficiency of the exploration, some are better than others. To this end, we define a goal function that models the reduction of the search space in such algorithms. We describe two greedy algorithms for the underlying optimization problem. One is based directly on the definition using a straightforward greedy criterion, whereas the second one uses several practical speedups and a different greedy criterion. Finally, we demonstrate the huge impact of exploratory equivalence on a real application, i.e., graph grammar parsing.