Vector-Weighted Matchings

Abstract

The theorem of efficiency shows that every efficient solution of a multiple objective linear program (MOLP) is the optimal solution of a parametric opti-mization problem. For the parametric optimization one uses weighted sums

$$ {\lambda _1}{c_1}({x_1},...,{x_n}) + ... + {\lambda _k}{c_k}({x_1},...,{c_n}) $$

for the k objective functions c _i (x ₁,…, x _n), i = 1,…, k where the weights have the properties that and λ _i > 0 and λ _i = 1,…, k = 1. To get every efficient solution one has to organize effectively a search of these vectors (1, 0,…, 0) which de¬livers different efficient solutions. We decompose the poly tope given by the unit vectors (1,0,…,0),…,(0,…,0,1) into the poly topes which correspond to the efficient solutions. This decomposition can be performed by an algorithm using hyperplanes. Our methods will be examined for vector-weighted matchings but of course apply to every multiobjective linear program. We consider a simple bipartite graph G where the edges have weights w(e) = [w ₁(e),…,[w _n (e)] of vectors of positive reals. A matching M of G is called efficient if there is no other matching M’ with

$$ w(M) = [{w_1}(M),...,{w_n}(M)] < [{w_1}(M'),...,{w_n}(M')] = w(M') $$

where w _i (M) is the sum of the ith components of the weights of all edges contained in the matching M. The problem to find the efficient matchings can be considered as a problem in multicriteria optimization. One can even present this problem as multiple objective linear program (MOLP). The solution of such a problem provides some unexpected difficulties.