Estimating bounds for quadratic assignment problems associated with Hamming and Manhattan distance matrices based on semidefinite programming

Hans Mittelmann, Jiming Peng

Research output: Contribution to journalArticle

12 Scopus citations

Abstract

Quadratic assignment problems (QAPs) with a Hamming distance matrix for a hypercube or a Manhattan distance matrix for a rectangular grid arise frequently from communications and facility locations and are known to be among the hardest discrete optimization problems. In this paper we consider the issue of how to obtain lower bounds for those two classes of QAPs based on semidefinite programming (SDP). By exploiting the data structure of the distance matrix B, we first show that for any permutation matrix X, the matrix Y = αE ? XBXT is positive semidefinite, where α is a properly chosen parameter depending only on the associated graph in the underlying QAP and E = eeT. S is a rank-1 matrix whose elements have value 1. This results in a natural way to approximate the original QAPs via SDP relaxation based on the matrix-splitting technique. Our new SDP relaxations have a smaller size compared with other SDP relaxations in the literature and can be solved efficiently by most open source SDP solvers. Experimental results show that for the underlying QAPs of size up to n = 200, strong bounds can be obtained effectively.

Original languageEnglish (US)
Pages (from-to)3408-3426
Number of pages19
JournalSIAM Journal on Optimization
Volume20
Issue number6
DOIs
StatePublished - Dec 1 2010

Keywords

  • Lower bound
  • Quadratic assignment problem
  • Relaxation
  • Semidefinite programming
  • Singular value decomposition

ASJC Scopus subject areas

  • Software
  • Theoretical Computer Science

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