A spectral approach to bandwidth and separator problems in graphs

Helmberg, Christoph; Rendl, Franz; Mohar, Bojan; Poljak, Svatopluk

doi:10.1080/03081089508818381

Cited by 36 publications

(50 citation statements)

References 12 publications

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“…Several lower bounds for the bandwidth of a graph are established in [41,40,62]. In those papers, the basic idea to obtain lower bounds on σ ∞ (G) is connecting the bandwidth minimization problem with the following graph partition problem.…”

Section: The Bandwidth Problem In Graphsmentioning

confidence: 99%

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SDP Relaxations for Some Combinatorial Optimization Problems

Sotirov

2011

International Series in Operations Research &Amp; Management Science

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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.

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Section: The Bandwidth Problem In Graphsmentioning

confidence: 99%

“…as noted in [41,40,62]. Helmberg et al [40] use the following relaxation of MC to compute a bound for OPT MC :…”

Section: The Bandwidth Problem In Graphsmentioning

confidence: 99%

SDP Relaxations for Some Combinatorial Optimization Problems

Sotirov

2011

International Series in Operations Research &Amp; Management Science

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“…The min-cut problem can also be seen as a special case of the quadratic assignment problem (QAP), as noted already by Helmberg et al [12]. This idea is further exploited by van Dam and Sotirov [19] where the authors use the well known SDP relaxation for the QAP [17], as the SDP relaxation for the min-cut problem.…”

Section: Theoremmentioning

confidence: 89%

“…We prove that the eigenvalue bound from [12] equals the optimal value of the SDP relaxation from Theorem 5, with matrix variable of order 2n. In [16] it is proven that the same eigenvalue bound is equal to the optimal solution of an SDP relaxation with matrix variable of order 3n.…”

Section: Resultsmentioning

confidence: 99%

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The min-cut and vertex separator problem

Rendl¹,

Sotirov

2017

Comput Optim Appl

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We consider graph three-partitions with the objective of minimizing the number of edges between the first two partition sets while keeping the size of the third block small. We review most of the existing relaxations for this min-cut problem and focus on a new class of semidefinite relaxations, based on matrices of order 2n + 1 which provide a good compromise between quality of the bound and computational effort to actually compute it. Here, n is the order of the graph. Our numerical results indicate that the new bounds are quite strong and can be computed for graphs of medium size (n ≈ 300) with reasonable effort of a few minutes of computation time. Further, we exploit those bounds to obtain bounds on the size of the vertex separators. A vertex separator is a subset of the vertex set of a graph whose removal splits the graph into two disconnected subsets. We also present an elegant way of convexifying nonconvex quadratic problems by using semidefinite programming. This approach results with bounds that can be computed with any standard convex quadratic programming solver.

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Matrix Relaxations in Combinatorial Optimization

Rendl

2011

Mixed Integer Nonlinear Programming

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A spectral approach to bandwidth and separator problems in graphs

Cited by 36 publications

References 12 publications

SDP Relaxations for Some Combinatorial Optimization Problems

SDP Relaxations for Some Combinatorial Optimization Problems

The min-cut and vertex separator problem

Matrix Relaxations in Combinatorial Optimization

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