Semidefinite programming relaxations for the graph partitioning problem

Wolkowicz, Henry; Qing, Zhao

doi:10.1016/s0166-218x(99)00102-x

Cited by 78 publications

(121 citation statements)

References 14 publications

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“…In Section 2, we study vector lifting SDP relaxations of the GPP. In particular, in Section 2.1 we simplify the technical approach from the paper by Wolkowicz and Zhao [47] to derive the Wolkowicz-Zhao relaxation of the GPP. Then, the Wolkowicz-Zhao bound is improved by adding nonnegativity constraints (we refer further to this relaxation as the improved Wolkowicz-Zhao relaxation).…”

Section: Main Results and Outlinementioning

confidence: 99%

“…In this section we study the Wolkowicz-Zhao relaxation [47], the improved Wolkowicz-Zhao relaxation, and the Zhao-Karisch-Rendl-Wolkowicz relaxation [48].…”

Section: Vector Lifting Sdp Relaxations Of the Gppmentioning

confidence: 99%

“…We simplify the technical approach from [47] to derive the Wolkowicz-Zhao relaxation. Further, we impose on the derived SDP relaxation nonnegativity constraints and obtain the improved Wolkowicz-Zhao relaxation.…”

Section: The Improved Wolkowicz-zhao Relaxationmentioning

confidence: 99%

“…In particular, besides the SDP formulation of the Donath-Hoffman bound we only know of the Wolkowicz-Zhao relaxation for the general GPP [47]. The Wolkowicz-Zhao relaxation is based on vector lifting and its matrix variable has order kn.…”

Section: Introductionmentioning

confidence: 99%

“…For the GBP there is a SDP relaxation with matrix variable of order n. This SDP relaxation is introduced by Karisch, Rendl, and Clausen [35] and it is also used in [17,25] to derive approximation algorithms for the GBP. Of course the Wolkowicz-Zhao relaxation [47] also provides a bound for the GBP.…”

Section: Introductionmentioning

confidence: 99%

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An Efficient Semidefinite Programming Relaxation for the Graph Partition Problem

Sotirov

2014

INFORMS Journal on Computing

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We derive a new semidefinite programming relaxation for the general graph partition problem (GPP). Our relaxation is based on matrix lifting with matrix variable having order equal to the number of vertices of the graph. We show that this relaxation is equivalent to the Frieze-Jerrum relaxation [A. Frieze and M. Jerrum. Improved approximation algorithms for max k-cut and max bisection. Algorithmica, 18(1):67-81, 1997] for the maximum k-cut problem with an additional constraint that involves the restrictions on the subset sizes. Since the new relaxation does not depend on the number of subsets k into which the graph should be partitioned we are able to compute bounds for large k.We compare theoretically and numerically the new relaxation with other SDP relaxations for the GPP. The results show that our relaxation provides competitive bounds and is solved significantly faster than any other known SDP bound for the general GPP.

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Section: Main Results and Outlinementioning

confidence: 99%

“…In this section we study the Wolkowicz-Zhao relaxation [47], the improved Wolkowicz-Zhao relaxation, and the Zhao-Karisch-Rendl-Wolkowicz relaxation [48].…”

Section: Vector Lifting Sdp Relaxations Of the Gppmentioning

confidence: 99%

Section: The Improved Wolkowicz-zhao Relaxationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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An Efficient Semidefinite Programming Relaxation for the Graph Partition Problem

Sotirov

2014

INFORMS Journal on Computing

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Realignment in the National Football League: Did they do it right?

Mitchell¹

2003

Naval Research Logistics

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Abstract:The National Football League (NFL) in the United States expanded to 32 teams in 2002 with the addition of a team in Houston. At that point, the league was realigned into eight divisions, each containing four teams. We describe a branch-and-cut algorithm for minimizing the sum of intradivisional travel distances. We consider first the case where any team can be assigned to any division. We also consider imposing restrictions, such as aligning the AFC (American Football Conference) and the NFC (National Football Conference) separately or maintaining traditional rivalries. We show that the alignment chosen by the NFL does not minimize the sum of intradivisional travel distances, but that it is close to optimal for an alignment that aligns the NFC and AFC separately and imposes some additional restrictions.

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Optimizing Connected Components Graph Partitioning With Minimum Size Constraints Using Integer Programming and Spectral Clustering Techniques

Cordero,

Miniguano–Trujillo,

Recalde

et al. 2024

Networks

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In this work, a graph partitioning problem in a fixed number of connected components is considered. Given an undirected graph with costs on the edges, the problem consists of partitioning the set of nodes into a fixed number of subsets with minimum size, where each subset induces a connected subgraph with minimal edge cost. The problem naturally surges in applications where connectivity is essential, such as cluster detection in social networks, political districting, sports team realignment, and energy distribution. Mixed Integer Programming formulations together with a variety of valid inequalities are demonstrated and computationally tested. An assisted column generation approach by spectral clustering is also proposed for this problem with additional valid inequalities. Finally, the methods are tested for several simulated instances, and computational results are discussed. Overall, the proposed column generation technique enhanced by spectral clustering offers a promising approach to solve clustering and partitioning problems.

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Semidefinite programming relaxations for the graph partitioning problem

Cited by 78 publications

References 14 publications

An Efficient Semidefinite Programming Relaxation for the Graph Partition Problem

An Efficient Semidefinite Programming Relaxation for the Graph Partition Problem

Realignment in the National Football League: Did they do it right?

Optimizing Connected Components Graph Partitioning With Minimum Size Constraints Using Integer Programming and Spectral Clustering Techniques

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