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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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. Par the complete graph Kn with n vertices we define edge weights c ij in the following way: c.

Y1, Y2,DPLUS ,DMINUS,Ml) INTEGER INTEGER INTEGER REAL BASIS(N),MEM(N),KA(N),KB(N) CC(1),P(l),SM(1),TMA(1),TMB(1) N,TOP,MI(N) YI(N) ,Y2(N) ,DPLUS(N) ,DMINUS(N) *** SCANNING OF NODE NBt rop =N+2 DI :DPLUS(NBI)-Yl(NBI) DMINUS(NBI):SUP D2 :DI-Y2(NBI) TMA(NBI) :0 II DO IF I! GE. t E-OS READ(5,100) N NI-N /2 NI-2*NI IF (NI. NE. I) GOTO 55 DO 54 J-I,I! N) GOTO 58 DO 56 J=I2,N IND=P(J)+I SM(J)-CC(IND) WRITE(6,220) (SM(J),J-I,N) CONTINUE WRITE(6,225) WRITE(6,230) «I,NMATCH(I)),I=l,N) WRITE(6,235) ZFW FORMAT(I2) FORMAT(10I2) FORMAT(lHl,IOX,20HSUM MATCHING PROBLEM,///) FORMAT(/,IOX,18HCOST-MATRIX C(I,J) ,/f) FORMAT(IIX,8(3X,I2)) FORMAT(/I/, 10X, 16HOPTIMAL MATCHING, / f) FORMAT(IIX,I5,2X,2H--,1X,II) FORMAT(/,IOX,2SHCOST OF THE OPTIMAL MATCHING,2X,II0) STOP \/RITE(6,8001) FORMAT(40HOERROR : THE NUMBER OF NODES IS NOT EVEN) STOP END 59 SUM MATCRING PROBLEM COST-MATRIX C (I, J) 33 33 55 46 29 68 38 37 57 95 46 30 38 28 ..

Then the linear description of SMP runs as follows: min r subject to x .. 1) eijEE (4. 2) X .. 1) for all R E J1. 3) for all e .. E E 1) SMP is equivalent to the SO called symmetrie assignment problem (SLSAP) . A permutation ~ E ~n is called symmetrie if

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