On the exact separation of mixed integer knapsack cuts - Marcos ...

38 R. Fukasawa, M. Goycoolea10.90.80.70.60.50.40.30.20.101 10 100 1000 10000Fig. 2 Performance profile comparing kbb, kbd, kbr, and kboIt is clear from this Figure that the kbb algorithm outperforms cpx and cpp in thisinstance set. Note that this does not necessarily mean that kbb solves every instancefaster than cpx, but rather, that cumulatively, kbb performs better. Moreover, cpx/cppfails to find the optimum solution in 11/8 instances, since it runs out of memory aftercreating too large a branch and bound tree.In Fig. 2 we compare the different versions of our MIKP solver:– kbo: Without domination branching or reduced-cost bound improvement.– kbr: With reduced-cost bound improvement but without domination branching.– kbd: With domination branching but without reduced-cost bound improvement– kbb: With both domination branching and reduced-cost bound improvementIt is interesting to see in Fig. 2 how domination branching and reduced-cost fixinginteract to improve the performance of the algorithm. Domination branching, whenused as a single feature, clearly helps decrease solution times. This is not the caseof reduced-cost fixing, however, which actually makes the algorithm perform worse.What is very surprising, however, is that when both features are used together, theperformance is altogether markedly improved.It makes sense that these two features should complement each other, since, wheneverbounds are changed, more domination conditions can be applied which leads toadditional pruning of the tree. On the other hand, the extra computational effort ofimproving bounds by reduced-cost does not translate in any pruning of the branchand-boundtree. Indeed, in all instances, the size of the branch-and-bound tree for kboand kbr were exactly the same.123