Abstract
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].
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Keywords
- Integer Program
- Approximation Scheme
- Knapsack Problem
- Integral Solution
- Polynomial Time Approximation Scheme
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References
Akbar, M.M., Manning, E.G., Shoja, G.C., Khan, S.: Heuristic Solutions for the Multiple-Choice Multi-Dimension Knapsack Problem. In: Int. Conference on Computational Science, vol. (2), pp. 659–668 (2001)
Bichler, M., Kalagnanam, J., Lee, H.S., Lee, J.: Winner Determination Algorithms for Electronic Auctions: A Framework Design. In: EC-Web, pp. 37–46 (2002)
Chandra, A.K., Hirschberg, D.S., Wong, C.K.: Approximate Algorithms for Some Generalized Knapsack Problems. Theoretical Computer Science 3, 293–304 (1976)
Chekuri, C., Khanna, S.: A PTAS for the Multiple Knapsack Problem. In: Proc. of SODA, pp. 213–222 (2000)
Chvátal, V.: A Greedy Heuristic for the Set Covering Problem. Math. Oper. Res. 4, 233–235 (1979)
Dobson, G.: Worst-case Analysis of Greedy for Integer Programming with Nonnegative Data. Math. of Operations Research 7, 515–531 (1982)
Frieze, A.M., Clarke, M.R.B.: Approximation Algorithms for the m-dimensional 0-1 knapsack problem: worst-case and probabilistic analyses. European J. of Operational Research 15(1), 100–109 (1984)
Feige, U.: A threshold of ln n for approximating set cover. In: Proc. of 28th Symposium on Theory of Computing, pp. 314–318 (1996)
Fleischer, L.: A Fast Approximation Scheme for Fractional Covering Problems with Variable Upper Bounds. In: Proc. of SODA, pp. 994–1003 (2004)
Garey, M.R., Johnson, D.S.: Computers and intractability: A Guide to the Theory of NP-Completeness. W.H. Freeman, New York (1979)
Ibaraki, T.: Approximate algorithms for the multiple-choice continuous knapsack problems. J. of Operations Research Society of Japan 23, 28–62 (1980)
Ibaraki, T., Hasegawa, T., Teranaka, K., Iwase, J.: The Multiple Choice Knapsack Problem. J. Oper. Res. Soc. Japan 21, 59–94 (1978)
Kolliopoulos, S.G.: Approximating covering integer programs with multiplicity constraints. Discrete Applied Math. 129(2–3), 461–473 (2003)
Kolliopoulos, S.G., Young, N.E.: Tight Approximation Results for General Covering Integer Programs. In: Proc. of FOCS, pp. 522–528 (2001)
Kothari, A., Parkes, D., Suri, S.: Approximately-Strategyproof and Tractable Multi-Unit Auctions. In: Proc. of ACM-EC (2003)
Lawler, E.L.: Combinatorial Optimization: Networks and Metroids. Holt, Reinhart and Winston (1976)
Lueker, G.S.: Two NP-complete problems in nonnegative integer programming. Report # 178, Computer science Lab., Princeton Univ. (1975)
Rajagopalan, S., Vazirani, V.V.: Primal-Dual RNC Approximation Algorithms for Set Cover and Covering Integer Programs. SIAM J. Comput. 28(2), 525–540 (1998)
Shachnai, H., Shmueli, O., Sayegh, R.: Approximation Schemes for Deal Splitting and Covering Integer Programs with Multiplicity Constraints, full version, http://www.cs.technion.ac.il/~hadas/PUB/ds.ps
Shachnai, H., Tamir, T.: Approximation Schemes for Generalized 2-dimensional Vector Packing with Application to Data Placement. In: Arora, S., Jansen, K., Rolim, J.D.P., Sahai, A. (eds.) RANDOM 2003 and APPROX 2003. LNCS, vol. 2764, pp. 165–177. Springer, Heidelberg (2003)
Srinivasan, A.: An Extension of the Lovász Local Lemma, and its Applications to Integer Programming. In: Proc. of SODA, pp. 6–15 (1996)
Srinivasan, A.: Improved Approximation Guarantees for Packing and Covering Integer Programs. SIAM J. Comput. 29(2), 648–670 (1999)
Shmueli, O., Golany, B., Sayegh, R., Shachnai, H., Perry, M., Gradovitch, N., Yehezkel, B.: Negotiation Platform. International Patent Application WO 02077759 (2001-2002)
Vazirani, V.V.: Approximation Algorithms. Springer, Heidelberg (2001)
Wolfstetter, E.: Auctions: An Introduction. J. of Economic Surveys 10, 367–420 (1996)
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Shachnai, H., Shmueli, O., Sayegh, R. (2005). Approximation Schemes for Deal Splitting and Covering Integer Programs with Multiplicity Constraints. In: Persiano, G., Solis-Oba, R. (eds) Approximation and Online Algorithms. WAOA 2004. Lecture Notes in Computer Science, vol 3351. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-31833-0_11
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