A branch-and-cut algorithm based on semidefinite programming for the minimum k-partition problem

Ghaddar, Bissan; Anjos, Miguel F.; Liers, Frauke

doi:10.1007/s10479-008-0481-4

Cited by 45 publications

(66 citation statements)

References 24 publications

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“…Their SBC algorithm combines the SDP relaxation proposed by Eisenblätter [20] with valid inequalities for the k-partition polytope and with a novel iterative clustering heuristic (ICH) that finds feasible solutions using the SDP optimal solution. The computational results reported in [24] show that ICH consistently provides feasible solutions that are better than those obtained using the hyperplane rounding techniques of Goemans and Williamson (for k = 2) and of Frieze and Jerrum (for k ≥ 3). Ghaddar et al presented results showing that SBC computes globally optimal solutions for dense graphs with up to 60 nodes, for (sparse) grid graphs with up to 100 nodes, and for different values of k ≥ 3.…”

Section: Introductionmentioning

confidence: 92%

“…Note that if we fix k = 2 in (SMkC), we obtain the SDP relaxation used for max-cut by Goemans and Williamson [25]. The relaxation (SMkC) was first used by Frieze and Jerrum [23]; it is the basis of the SBC algorithm of Ghaddar et al [24]. In that algorithm, the SDP relaxation was further tightened by adding valid inequalities.…”

Section: Problem Description Formulations and Relaxationsmentioning

confidence: 99%

“…In each iteration, a vertex is added to a clique of size smaller than k + 1 that contributes the smallest amount to the lefthand side of the corresponding clique inequality. The heuristic is described in detail in [24]. For a graph with n vertices, this procedure generates n clique inequalities.…”

Section: Adding Valid Inequalitiesmentioning

confidence: 99%

“…It was proposed by Ghaddar et al [24] and used in their SBC algorithm. ICH works by aggregating information from X * corresponding to subgraphs of G. Specifically, ICH sums the X * i j values on the edges between each of the n k subsets of k vertices, then sorts the resulting list of values, and places a subset of k vertices all in the same partition (or all in different partitions) when the sum is one of the largest values (or one of the smallest).…”

Section: Lower Bound Heuristicmentioning

confidence: 99%

“…Another exception is the SDP-based branch-and-cut algorithm for the minimum k-partition problem proposed by Ghaddar, Anjos and Liers [24]. Their SBC algorithm combines the SDP relaxation proposed by Eisenblätter [20] with valid inequalities for the k-partition polytope and with a novel iterative clustering heuristic (ICH) that finds feasible solutions using the SDP optimal solution.…”

Section: Introductionmentioning

confidence: 99%

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Solving k-Way Graph Partitioning Problems to Optimality: The Impact of Semidefinite Relaxations and the Bundle Method

Anjos

Ghaddar²,

Hupp

et al. 2013

Facets of Combinatorial Optimization

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This paper is concerned with computing global optimal solutions for maximum k-cut problems. We improve on the SBC algorithm of Ghaddar, Anjos and Liers in order to compute such solutions in less time. We extend the design principles of the successful BiqMac solver for maximum 2-cut to the general maximum k-cut problem. As part of this extension, we investigate different ways of choosing variables for branching. We also study the impact of the separation of clique inequalities within this new framework and observe that it frequently reduces the number of subproblems considerably. Our computational results suggest that the proposed approach achieves a drastic speedup in comparison to SBC, especially when k = 3. We also made a comparison with the orbitopal fixing approach of Kaibel, Peinhardt and Pfetsch. The results suggest that while their performance is better for sparse instances and larger values of k, our proposed approach is superior for smaller k and for dense instances of medium size. Furthermore, we used CPLEX for solv- ing the ILP formulation underlying the orbitopal fixing algorithm and conclude that especially on dense instances the new algorithm outperforms CPLEX by far.

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Section: Introductionmentioning

confidence: 92%

Section: Problem Description Formulations and Relaxationsmentioning

confidence: 99%

Section: Adding Valid Inequalitiesmentioning

confidence: 99%

Section: Lower Bound Heuristicmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Solving k-Way Graph Partitioning Problems to Optimality: The Impact of Semidefinite Relaxations and the Bundle Method

Anjos

Ghaddar²,

Hupp

et al. 2013

Facets of Combinatorial Optimization

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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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