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A fast approximation algorithm for solving the complete set packing problem

A fast approximation algorithm for solving the complete set packing problem
A fast approximation algorithm for solving the complete set packing problem
We study the complete set packing problem (CSPP) where the family of feasible subsets may include all possible combinations of objects. This setting arises in applications such as combinatorial auctions (for selecting optimal bids) and cooperative game theory (for finding optimal coalition structures). Although the set packing problem has been well-studied in the literature, where exact and approximation algorithms can solve very large instances with up to hundreds of objects and thousands of feasible subsets, these methods are not extendable to the CSPP since the number of feasible subsets is exponentially large. Formulating the CSPP as an MILP and solving it directly, using CPLEX for example, is impossible for problems with more than 20 objects. We propose a new mathematical formulation for the CSPP that directly leads to an efficient algorithm for finding feasible set packings (upper bounds). We also propose a new formulation for finding tighter lower bounds compared to LP relaxation and develop an efficient method for solving the corresponding large-scale MILP. We test the algorithm with the winner determination problem in spectrum auctions, the coalition structure generation problem in coalitional skill games, and a number of other simulated problems that appear in the literature.
0377-2217
62-70
Nguyen, Tri-Dung
a6aa7081-6bf7-488a-b72f-510328958a8e
Nguyen, Tri-Dung
a6aa7081-6bf7-488a-b72f-510328958a8e

Nguyen, Tri-Dung (2014) A fast approximation algorithm for solving the complete set packing problem. European Journal of Operational Research, 237 (1), 62-70. (doi:10.1016/j.ejor.2014.01.024).

Record type: Article

Abstract

We study the complete set packing problem (CSPP) where the family of feasible subsets may include all possible combinations of objects. This setting arises in applications such as combinatorial auctions (for selecting optimal bids) and cooperative game theory (for finding optimal coalition structures). Although the set packing problem has been well-studied in the literature, where exact and approximation algorithms can solve very large instances with up to hundreds of objects and thousands of feasible subsets, these methods are not extendable to the CSPP since the number of feasible subsets is exponentially large. Formulating the CSPP as an MILP and solving it directly, using CPLEX for example, is impossible for problems with more than 20 objects. We propose a new mathematical formulation for the CSPP that directly leads to an efficient algorithm for finding feasible set packings (upper bounds). We also propose a new formulation for finding tighter lower bounds compared to LP relaxation and develop an efficient method for solving the corresponding large-scale MILP. We test the algorithm with the winner determination problem in spectrum auctions, the coalition structure generation problem in coalitional skill games, and a number of other simulated problems that appear in the literature.

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More information

Accepted/In Press date: 12 January 2014
e-pub ahead of print date: 20 January 2014
Published date: 16 August 2014
Organisations: Centre of Excellence for International Banking, Finance & Accounting, Operational Research

Identifiers

Local EPrints ID: 361100
URI: http://eprints.soton.ac.uk/id/eprint/361100
ISSN: 0377-2217
PURE UUID: c6d5f3aa-1371-41ac-9f0e-4d14c6638611
ORCID for Tri-Dung Nguyen: ORCID iD orcid.org/0000-0002-4158-9099

Catalogue record

Date deposited: 13 Jan 2014 14:01
Last modified: 26 Nov 2021 02:56

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