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

Anjos, Miguel F.; Ghaddar, Bissan; Hupp, Lena; Liers, Frauke; Wiegele, Angelika

doi:10.1007/978-3-642-38189-8_15

Cited by 19 publications

(38 citation statements)

References 46 publications

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“…These constraints ensure that the graph with adjacency matrix Y = (y ij ) has no independent set (Q) of size k + 1. Anjos et al [3] use a bundle method to solve (FJ) with triangle and independent set inequalities in each node of a branch-and-bound tree. Their approach resulted in a significantly faster max-k-cut solver than the one presented in [26].…”

Section: Sdp Relaxations For the Max-k-cut Problemmentioning

confidence: 99%

“…Ghaddar, Anjos, and Liers [26] developed a branch-and-cut algorithm that is based on a SDP relaxation for the minimum k-partition problem. They were able to solve to optimality dense instances with up to 60 vertices and some special instances with up to 100 vertices, and for different values of k. Anjos et al [3] improved the algorithm from [26] and developed a more efficient solver.…”

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confidence: 99%

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New bounds for the max-k-cut and chromatic number of a graph

Dam

Sotirov

2016

Linear Algebra and its Applications

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We consider several semidefinite programming relaxations for the max-k-cut problem, with increasing complexity. The optimal solution of the weakest presented semidefinite programming relaxation has a closed form expression that includes the largest Laplacian eigenvalue of the graph under consideration. This is the first known eigenvalue bound for the max-k-cut when k > 2 that is applicable to any graph. This bound is exploited to derive a new eigenvalue bound on the chromatic number of a graph. For regular graphs, the new bound on the chromatic number is the same as the well-known Hoffman bound; however, the two bounds are incomparable in general.We prove that the eigenvalue bound for the max-k-cut is tight for several classes of graphs. We investigate the presented bounds for specific classes of graphs, such as walk-regular graphs, strongly regular graphs, and graphs from the Hamming association scheme.

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Section: Sdp Relaxations For the Max-k-cut Problemmentioning

confidence: 99%

mentioning

confidence: 99%

New bounds for the max-k-cut and chromatic number of a graph

Dam

Sotirov

2016

Linear Algebra and its Applications

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“…Because of the strength of the SDP, many researchers have used this formulation to design approximations [8,14] and exact methods [2,16]. In particular, [14] extends the max-cut approximation of [17] to the max-k-cut.…”

Section: Semidefinite Programming Formulationmentioning

confidence: 99%

“…In particular, [14] extends the max-cut approximation of [17] to the max-k-cut. In [2] the bundleBC algorithm is proposed to solve max-k-cut problems with 60 vertices by combining the SDP branchand-cut method of [16] with the principles of the Biq Mac algorithm [45]. In [2] the authors show that their method achieves a dramatic speedup in comparison to [16], especially when k = 3.…”

Section: Semidefinite Programming Formulationmentioning

confidence: 99%

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Improving the linear relaxation of maximum k-cut with semidefinite-based constraints

Sousa

Anjos

Digabel

2019

EURO Journal on Computational Optimization

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We consider the maximum k-cut problem that involves partitioning the vertex set of a graph into k subsets such that the sum of the weights of the edges joining vertices in different subsets is maximized. The associated semidefinite programming (SDP) relaxation is known to provide strong bounds, but it has a high computational cost. We use a cutting plane algorithm that relies on the early termination of an interior point method, and we study the performance of SDP and linear programming (LP) relaxations for various values of k and instance types. The LP relaxation is strengthened using combinatorial facet-defining inequalities and SDP-based constraints. Our computational results suggest that the LP approach, especially with the addition of SDP-based constraints, outperforms the SDP relaxations for graphs with positive-weight edges and k ≥ 7.

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Exact Solution Methods for the k-Item Quadratic Knapsack Problem

Létocart

Wiegele

2016

Lecture Notes in Computer Science

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The purpose of this paper is to solve the 0-1 k-item quadratic knapsack problem (kQKP ), a problem of maximizing a quadratic function subject to two linear constraints. We propose an exact method based on semidefinite optimization. The semidefinite relaxation used in our approach includes simple rank one constraints, which can be handled efficiently by interior point methods. Furthermore, we strengthen the relaxation by polyhedral constraints and obtain approximate solutions to this semidefinite problem by applying a bundle method. We review other exact solution methods and compare all these approaches by experimenting with instances of various sizes and densities.Keywords: quadratic programming, 0-1 knapsack, k-cluster, semidefinite programming either the value of the problem is equal to max i=1,...,n c ii (for k = 1), or the domain of (kQKP ) is empty (for k > k max ).(kQKP ) is an NP-hard problem as it includes two classical NP-hard subproblems, the k-cluster problem [6] by dropping constraint (1), and the quadratic knapsack problem [20] by dropping constraint (2). Even more, the work of Bhaskara et. al [4] indicates that approximating k-cluster within a polynomial factor might be a harder problem than Unique Games. Rader and Woeginger [16] state negative results concerning the approximability of QKP if negative cost coefficients are present.Applications of (kQKP ) cover those found in previous references for k-cluster or classical quadratic knapsack problems (e.g., task assignment problems in a client-server architecture with limited memory), but also multivariate linear regression and portfolio selection. Specific heuristic and exact methods including branch-and-bound and branch-and-cut with surrogate relaxations have been designed for these applications (see, e.g., [3,5,9,19,23]).The purpose of this paper is twofold.

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

Cited by 19 publications

References 46 publications

New bounds for the max-k-cut and chromatic number of a graph

New bounds for the max-k-cut and chromatic number of a graph

Improving the linear relaxation of maximum k-cut with semidefinite-based constraints

Exact Solution Methods for the k-Item Quadratic Knapsack Problem

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