Ibm Research Report Approximation Algorithms for the Multiple Knapsack Problem with Assign- Ment Restrictions

Motivated by a real world application, we study the multiple knapsack problem with assignment restrictions (MKAR): We are given a set of items N = f1; : : : ; ng and a set of knapsacks M = f1; : : : ;mg. Each item j 2 N has a positive real weight wj and each knapsack i 2 M has a positive real capacity ci associated with it. In addition, for each item j 2 N a set Aj M of knapsacks that can hold item j is speci ed. In a feasible assignment of items to knapsacks, for each knapsack i 2 M , we need to choose a subset Si of items in N to be assigned to knapsack i, such that (i) Each item is assigned to at most one knapsack (ii) Assignment restrictions are satis ed and (iii) For each knapsack, its capacity constraint is satis ed. We consider two objectives (i) Maximize assigned weight P i2M P j2Si wj and (ii) minimize utilized capacity P i:Si 6=; ci Our results include two 13 approximation algorithms and two 1 2 approximation algorithms for the single objective problem of maximizing assigned weight. For the bi-criteria problem which considers both the objectives, we present two algorithms with performance ratios (1 3 ; 2) and (1 2 ; 3) respectively.