A polynomial case of unconstrained zero-one quadratic optimization

Allemand, Kim; Fukuda, Komei; Liebling, Th. M.; Steiner, Erich

doi:10.1007/s101070100233

Cited by 62 publications

(46 citation statements)

References 6 publications

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“…Convex combinatorial optimization has a very broad expressive power and conveniently captures a variety of problems studied in the operations research and mathematical programming literature including quadratic assignment, inventory management, scheduling, reliability, bargaining games, clustering, and vector partitioning, see [2], [4], [7], [10], [12], [22], [27], [43], and references therein. In Section 3 we discuss some of these applications in detail and demonstrate that, as a consequence of our framework, all admit a simple unified strongly polynomial time algorithm.…”

Section: Introductionmentioning

confidence: 99%

Convex Combinatorial Optimization

Onn

Rothblum

2004

Discrete Comput Geom

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Section: Introductionmentioning

confidence: 99%

Convex Combinatorial Optimization

Onn

Rothblum

2004

Discrete Comput Geom

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“…Among these, Allemand et al [1] develop a polynomial time algorithm for solving unconstrained fixed-ranked homogeneous QBP, i.e. one with the form max z∈{0,1} n {z t Qz} where Q is a symmetric positive definite matrix with a fixed rank.…”

Section: The Winner Determination Problem In Spectrum Auctionsmentioning

confidence: 99%

“…Since there are 2 n such points z, we have 2 n corresponding points P (z) in 2-D. It is very interesting, however, that the convex hall of these points has at most 2n extreme points (Allemand et al [1]). Since we are maximising a convex function {P 2 (z) 2 − P 1 (z)}, the maximum is attained at one of the extreme points.…”

Section: Proofmentioning

confidence: 99%

A fast approximation algorithm for solving the complete set packing problem

Nguyen

2014

European Journal of Operational Research

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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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“…Thus, (P ) is NP-hard in general. Polynomially solvable cases of (P ) are investigated in [1,7,8,19]. A systematic survey of the solution methods for solving (P ) can be found in Chapter 10 of [14].…”

mentioning

confidence: 99%

“…. , 1) T and diag(λ) denotes the diagonal matrix with λ i being its ith diagonal element. The dual problem of (P c ) (or (P )) is…”

mentioning

confidence: 99%

On duality gap in binary quadratic programming

Sun

Liu

et al. 2011

J Glob Optim

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Abstract. We present in this paper new results on the duality gap between the binary quadratic optimization problem and its Lagrangian dual or semidefinite programming relaxation. We first derive a necessary and sufficient condition for the zero duality gap and discuss its relationship with the polynomial solvability of the primal problem. We then characterize the zeroness of the duality gap by the distance, δ, between {−1, 1} n and certain affine subspace C and show that the duality gap can be reduced by an amount proportional to δ 2 . We finally establish the connection between the computation of δ and cell enumeration of hyperplane arrangement in discrete geometry and further identify two polynomially solvable cases of computing δ.

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A polynomial case of unconstrained zero-one quadratic optimization

Cited by 62 publications

References 6 publications

Convex Combinatorial Optimization

Convex Combinatorial Optimization

A fast approximation algorithm for solving the complete set packing problem

On duality gap in binary quadratic programming

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