Approximation Schemes for Deal Splitting and Covering Integer Programs with Multiplicity Constraints
We consider the problem of splitting an order for R goods, R ≥ 1, among a set of sellers, each having bounded amounts of the goods, so as to minimize the total cost of the deal. In deal splitting with packages (DSP), the sellers offer packages containing combinations of the goods; in deal splitting with price tables (DST), the buyer can generate such combinations using price tables. Our problems, which often occur in online reverse auctions, generalize covering integer programs with multiplicity constraints (CIP), where we must fill up an R-dimensional bin by selecting (with bounded number of repetitions) from a set of R-dimensional items, such that the overall cost is minimized. Thus, both DSP and DST are NP-hard, already for a single good, and hard to approximate for arbitrary number of goods.
In this paper we focus on finding efficient approximations, and exact solutions, for DSP and DST instances where the number of goods is some fixed constant. In particular, we show that when R is fixed both DSP and DST can be optimally solved in pseudo-polynomial time, and develop polynomial time approximation schemes (PTAS) for several subclasses of instances of practical interest. Our results include a PTAS for CIP in fixed dimension, and a more efficient (combinatorial) scheme for CIP ∞ , where the multiplicity constraints are omitted. Our approximation scheme for CIP ∞ is based on a non-trivial application of the fast scheme for the fractional covering problem, proposed recently by Fleischer [Fl-04].
KeywordsInteger Program Approximation Scheme Knapsack Problem Integral Solution Polynomial Time Approximation Scheme
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