Combinatorics Advances pp 267-276 | Cite as

# Vector-Weighted Matchings

Chapter

## 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
for the
where

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

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

*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.## Keywords

Bipartite Graph Assignment Problem Linear Order Efficient Solution Preference Function
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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© Kluwer Academic Publishers 1995