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

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

ISBN-10: 3540102671

ISBN-13: 9783540102670

ISBN-10: 3642515762

ISBN-13: 9783642515767

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Additional info for Assignment and Matching Problems: Solution Methods with FORTRAN-Programs

Example text

I t can be shown (cf. 2) where again ~n denotes the set of all symmetrie permutations iii n E ~n . 3). b) The algorithm The algorithm for solving BMP which we are presenting here can be viewed elther as a transposition of the SMP-procedure to the bottleneck case or as a generalization of the LBAP-procedure to the nonbipartite case. Again successively shortest augmenting paths are computed and the matehing is augrnented along these paths. Let M be a matching and Then 1PM (s) S E V be an unmatched node with respect to M.

An alternating path with respect to a matching M is a path the edges cf which are al ternately in M and not. AZternating tl'ees are defined analogously. A vertex which does not meet an edge in M is called unsaturated with respect to M. An alternating path connecting two unsaturated vertices is called an augmenting path because simply changing the role of rnatching and norunatching edges on the path resul ts in a new matching having larger cardinality. It was first shown by BERGE [1] that a matching solves CMP iff it does not admit an augmenting path.

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. Par the complete graph Kn with n vertices we define edge weights c ij in the following way: c. := lj with c := max {CU max Cij-Cmin i f e. n(cmax-cmin) else I e .. E E} and c min lJ From the optimal solution M for (SMP 3 ) Then C (M) Let lj Mfor := E E tor i, j E V min { c ij e.

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


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