A threshold of ln n for approximating set cover (preliminary version)
STOC '96 Proceedings of the twenty-eighth annual ACM symposium on Theory of computing
Primal-Dual RNC Approximation Algorithms for Set Cover and Covering Integer Programs
SIAM Journal on Computing
An extension of the Lovász local lemma, and its applications to integer programming
Proceedings of the seventh annual ACM-SIAM symposium on Discrete algorithms
Improved Approximation Guarantees for Packing and Covering Integer Programs
SIAM Journal on Computing
A PTAS for the multiple knapsack problem
SODA '00 Proceedings of the eleventh annual ACM-SIAM symposium on Discrete algorithms
Approximation algorithms
Computers and Intractability: A Guide to the Theory of NP-Completeness
Computers and Intractability: A Guide to the Theory of NP-Completeness
Heuristic Solutions for the Multiple-Choice Multi-dimension Knapsack Problem
ICCS '01 Proceedings of the International Conference on Computational Science-Part II
Winner Determination Algorithms for Electronic Auctions: A Framework Design
EC-WEB '02 Proceedings of the Third International Conference on E-Commerce and Web Technologies
Approximately-strategyproof and tractable multi-unit auctions
Proceedings of the 4th ACM conference on Electronic commerce
Tight Approximation Results for General Covering Integer Programs
FOCS '01 Proceedings of the 42nd IEEE symposium on Foundations of Computer Science
Approximating covering integer programs with multiplicity constraints
Discrete Applied Mathematics
A fast approximation scheme for fractional covering problems with variable upper bounds
SODA '04 Proceedings of the fifteenth annual ACM-SIAM symposium on Discrete algorithms
Dynamic programming based algorithms for set multicover and multiset multicover problems
Theoretical Computer Science
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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].