Assignment and Matching Problems: Solution Methods with by Rainer E. Burkard, Ulrich Derigs (auth.)

By Rainer E. Burkard, Ulrich Derigs (auth.)

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LJ From the optimal solution M for tain the optimal solution M M = M and C(M) Let G = SMP 3 with edge weights c .. we oblJ for SMP 2 in the fOllowing way: IMI = (V,E) be the underlying graph for (SMP1)' Then we obtain the graph G = (V,E) E for (SMP 2) by (e ij I c ij :': 0) c. lJ and for e .. E E. 1J Then the optimal solution Mfor SMP 2 is optimal for SMP 1 , toD. 39 Hence we can restriet ourselves to problems of the kind SMP. The formulation of SMP as a linear program is known from the result of EDMONDS [4]: Far anode i E V we define 6(i) the Set of all edges that are incident with i.

E M .. l] ;p li) ;Pli) and =i ;p I j ) = (jJ Ein i can be obtained as follows: if cij+c Jl < cii+Cji else and ;Plj) b) Description cf the algorithrn The algorithm presented here is an adaption of the LSAP-algorithrn to the nonbipartite case. We assume G = (V,E) to be complete with IV\ even and the weigths c ij for e ij E E to be nonnegative integer/real nurnbers. 41 Let M be a matehing in G. A path P is called an altepnaring path if the edges of P are alternately in M and not in M. An alternating path P is called an augmenting path if it joins two unsaturated vertices.

Matehing) . c. matching is called perfeet if every node 15 incident to a matching edge. We denote the set of all perfeet matchings by ~p' It is obvious that not every graph allows aperfeet matching. Let integer/real edge weights~Cij be given. c. c. matching} We will now demonstrate hcw these problems can be transformed into the standard problem (SMP) rnin where the underlying graph number and the edge weights G (V,E) is complete, lvi is an even Cij are nonnegative. 38 If IV! e. V := V 0 {v} and we define n := lVI.

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Categories: Operations Research