Packing items into several bins facilitates approximating the separable assignment problem

( ( 1 - 1 e k ) β ) -approximation for k-SAP if the single bin problem admits a β-approximation. ( 1 - 1 e k ) -approximation for k-SAP if the single bin problem admits an FPTAS.For k 2, our algorithm beats the upper bound of ( 1 - 1 e ) known for other special cases of SAP. We consider a variant of the separable assignment problem (SAP). In the classic version of SAP, we are given a set of bins and a set of items to pack into the bins together with a profit p i, j for assigning item i to bin j. Each bin j has a separate packing constraint, i.e., only certain subsets of the items fit into bin j. The objective is to find an assignment of a subset of the items to the bins such that the packing constraints of all bins are satisfied, no item is assigned to more than one bin, and the total profit is maximized. As an important special case, this problem contains the maximum generalized assignment problem (GAP).It is known that, given a β-approximation algorithm for the single bin subproblem (i.e., the problem of finding the most profitable packing for a single bin), it is possible to obtain a ( ( 1 - 1 e ) β ) -approximation for SAP using randomized rounding. If the single bin subproblem admits an FPTAS, one can obtain a ( 1 - 1 e ) -approximation. This is best possible in the sense that there exist special cases of SAP which do not admit polynomial-time approximation algorithms with an approximation factor better than ( 1 - 1 e ) unless NP DTIME ( n O ( log log n ) ) .In this paper, we consider the case of SAP where each item may be assigned at most k 1 times (but at most once to each bin) and present a ( ( 1 - 1 e k ) β ) -approximation algorithm for this case under the assumption that the single bin subproblem admits a β-approximation algorithm. If the single bin subproblem admits an FPTAS, we obtain a ( 1 - 1 e k ) -approximation, which shows that, for k 2, the problem admits approximation algorithms that beat the upper bound of ( 1 - 1 e ) known for other special cases of SAP.