The Backboard Wiring Problem: A Placement Algorithm

Steinberg, Leon

doi:10.1137/1003003

Cited by 323 publications

(103 citation statements)

References 4 publications

Supporting

Mentioning

100

Contrasting

Unclassified

Order By: Relevance

“…They solved to optimality several famous instances, including the problem of size n = 30 posed by Nugent et al [57], and problems of size n = 30, 32 posed by Krarup and Pruzan [47]. Optimal solutions of the problems of size n = 36 posed by Steinberg [66] are reported in [8,58]. For a detailed survey on recent developments regarding the QAP problem, see Anstreicher [2].…”

Section: The Quadratic Assignment Problemmentioning

confidence: 99%

SDP Relaxations for Some Combinatorial Optimization Problems

Sotirov

2011

International Series in Operations Research &Amp; Management Science

View full text Add to dashboard Cite

In this chapter we present recent developments on solving various combinatorial optimization problems by using semidefinite programming (SDP). We present several SDP relaxations of the quadratic assignment problem and the traveling salesman problem. Further, we show the equivalence of several known SDP relaxations of the graph equipartition problem, and present recent results on the bandwidth problem. NotationThe space of p × q real matrices is denoted by R p×q , the space of k × k symmetric matrices is denoted by S k , and the space of k×k symmetric positive semidefinite matrices by S + k . We will sometimes also use the notation X 0 instead of X ∈ S + k , if the order of the matrix is clear from the context. For index sets α, β ⊂ {1, . . . , n}, we denote the submatrix that contains the rows of A indexed by α and the columns indexed by β as A(α, β). If α = β, the principal submatrix A(α, α) of A is abbreviated as A(α). The ith column of a matrix C is denoted by C :,i .We use I n to denote the identity matrix of order n, and e i to denote the ith standard basis vector. Similarly, J n and u n denote the n×n all-ones matrix and all-ones n-vector respectively, and 0 n×n is the zero matrix of order n. We will omit subscripts if the order is clear from the context. The set of n × n permutation matrices is denoted by Π n . We set E ij = e i e T j . The 'vec' operator stacks the columns of a matrix, while the 'diag' operator maps an n × n matrix to the n-vector given by its diagonal. The adjoint operator of 'diag' we denote by 'Diag'. The trace operator is denoted by 'tr'.The Kronecker product A ⊗ B of matrices A ∈ R p×q and B ∈ R r×s is defined as the pr × qs matrix composed of pq blocks of size r × s, with block ij given by a ij B, i = 1, . . . , p, j = 1, . . . , q.

show abstract

Section: The Quadratic Assignment Problemmentioning

confidence: 99%

SDP Relaxations for Some Combinatorial Optimization Problems

Sotirov

2011

International Series in Operations Research &Amp; Management Science

View full text Add to dashboard Cite

show abstract

“…These problems are: Ste36a, Ste36c (Steinberg, 1961), Tho40 (Thonemann & Bolte, 1994), Sko49 (Skorin-Kapov, 1990), Wil50 (Wilhelm & Ward, 1987), Sko56, Sko64, Sko72, Sko81, Sko90, Sko100a-f (Skorin-Kapov, 1990), Wil100 (Wilhelm & Ward, 1987), and Tho150 (Thonemann & Bolte, 1994). Each problem was solved twenty times.…”

Section: Results For the First Set Of Problemsmentioning

confidence: 99%

Tabu Search and Hybrid Genetic Algorithms for Quadratic Assignment Problems

Drezner¹

2008

Tabu Search

View full text Add to dashboard Cite

“…From the practical point of view, it is because of the diversified applications of the QAP. The QAP has been applied in many fields such as backboard wiring [7], typewriter keyboards and control panels design [8], scheduling [9], numerical analysis [10], storage-and-retrieval [11], and many others. More advances in theoretical aspects, solution methods and applications of the QAP can be found in [5,12,13,14,15,16,17,18].…”

Section: Introductionmentioning

confidence: 99%

Solving the quadratic assignment problem by means of general purpose mixed integer linear programming solvers

2012

View full text Add to dashboard Cite

The Quadratic Assignment Problem (QAP) can be solved by linearization, where one formulates the QAP as a mixed integer linear programming (MILP) problem. On the one hand, most of these linearization are tight, but hardly exploited within a reasonable computing time because of their size. On the other hand, Kaufman and Broeckx formulation [1] is the smallest of these linearizations, but very weak. In this paper, we analyze how Kaufman and Broeckx formulation can be tightened to obtain better QAP-MILP formulations. As we show in our numerical experiments, these tightened formulations remain small but computationally effective in order to solve the QAP by means of general purpose MILP solvers.

show abstract

The Backboard Wiring Problem: A Placement Algorithm

Cited by 323 publications

References 4 publications

SDP Relaxations for Some Combinatorial Optimization Problems

SDP Relaxations for Some Combinatorial Optimization Problems

Tabu Search and Hybrid Genetic Algorithms for Quadratic Assignment Problems

Solving the quadratic assignment problem by means of general purpose mixed integer linear programming solvers

Contact Info

Product

Resources

About