4,022 research outputs found
Minimum cost b-matching problems with neighborhoods
In this paper, we deal with minimum cost b-matching problems on graphs where the nodes are assumed to belong to non-necessarily convex regions called neighborhoods, and the costs are given by the distances between points of the neighborhoods. The goal in the proposed problems is twofold: (i) finding a b-matching in the graph and (ii) determining a point in each neighborhood to be the connection point among the edges defining the b-matching. Different variants of the minimum cost b-matching problem are considered depending on the criteria to match neighborhoods: perfect, maximum cardinality, maximal and the a-b-matching problems. The theoretical complexity of solving each one of these problems is analyzed. Different mixed integer non-linear programming formulations are proposed for each one of the considered problems and then reformulated as Second Order Cone formulations. An extensive computational experience shows the efficiency of the proposed formulations to solve the problems under study
Optimal Recombination in Genetic Algorithms
This paper surveys results on complexity of the optimal recombination problem
(ORP), which consists in finding the best possible offspring as a result of a
recombination operator in a genetic algorithm, given two parent solutions. We
consider efficient reductions of the ORPs, allowing to establish polynomial
solvability or NP-hardness of the ORPs, as well as direct proofs of hardness
results
Hypergraphic LP Relaxations for Steiner Trees
We investigate hypergraphic LP relaxations for the Steiner tree problem,
primarily the partition LP relaxation introduced by Koenemann et al. [Math.
Programming, 2009]. Specifically, we are interested in proving upper bounds on
the integrality gap of this LP, and studying its relation to other linear
relaxations. Our results are the following. Structural results: We extend the
technique of uncrossing, usually applied to families of sets, to families of
partitions. As a consequence we show that any basic feasible solution to the
partition LP formulation has sparse support. Although the number of variables
could be exponential, the number of positive variables is at most the number of
terminals. Relations with other relaxations: We show the equivalence of the
partition LP relaxation with other known hypergraphic relaxations. We also show
that these hypergraphic relaxations are equivalent to the well studied
bidirected cut relaxation, if the instance is quasibipartite. Integrality gap
upper bounds: We show an upper bound of sqrt(3) ~ 1.729 on the integrality gap
of these hypergraph relaxations in general graphs. In the special case of
uniformly quasibipartite instances, we show an improved upper bound of 73/60 ~
1.216. By our equivalence theorem, the latter result implies an improved upper
bound for the bidirected cut relaxation as well.Comment: Revised full version; a shorter version will appear at IPCO 2010
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