The Optimal Partitioning of Graphs

Christofides, Nicos; Brooker, Philip A. H.

doi:10.1137/0130006

Cited by 28 publications

(14 citation statements)

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“…This graph has 40 nodes and ml = m2 = 20. Two sets C1 and C2 of edge costs are given in [5]. The underlying graph is 3-regular.…”

Section: Computational Resultsmentioning

confidence: 99%

“…We also point out that the bound from theorem 5.1 in quite competitive with the subgradient improved bound from lemma 5.3 for the variants V2 and V3. The optimal solution values are from [5]. Table 6 Three variants of a 40-node graph.…”

Section: Computational Resultsmentioning

confidence: 99%

“…m I = m2 = n/2, and improves the eigenvalue bound from [10] for this special case. The papers [5,29] describe branch and bound approaches to solve the partitioning problem in the case k = 2 and for general weighted graphs. Both methods seems to work only for extremely thin graphs (average degree not more than 4).…”

Section: Overview Of Previous Related Researchmentioning

confidence: 99%

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A projection technique for partitioning the nodes of a graph

Rendl

Wolkowicz

1995

Ann Oper Res

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“…This graph has 40 nodes and ml = m2 = 20. Two sets C1 and C2 of edge costs are given in [5]. The underlying graph is 3-regular.…”

Section: Computational Resultsmentioning

confidence: 99%

Section: Computational Resultsmentioning

confidence: 99%

See 1 more Smart Citation

A projection technique for partitioning the nodes of a graph

Rendl

Wolkowicz

1995

Ann Oper Res

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“…Christofides and Brooker [2] consider the case of bipartitioning the nodes with unit weights, while Rendl and Wolkowicz [7] consider the special case with unit node weights and a specific number of nodes in each class. Graph partitioning problems with no limit on the number of classes (free partitions) have been studied by Faigle et al [3] and Lukes [5].…”

Section: Partition the Set Of Nodes In At Most P Classes Such That Thmentioning

confidence: 99%

The optimal graph partitioning problem

Holm

Sørensen

1993

OR Spektrum

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In this paper we consider the problem of partitioning the set of nodes in a graph in at mostp classes, such that the sum of node weights in any class is not greater than the class capacity b, and such that the sum of edge weights, for edges connecting nodes in the same class, is maximal. This problem can be formulated as a MILP, which turns out to be completely symmetrical with respect to the p classes, and the gap between the relaxed LP solution and the optimal solution is the largest one possible. These two properties make it very difficult to solve even smaller problems. In this paper it is shown how it is possible to preassign certain nodes to certain classes, thus reducing both the symmetric nature of the formulation, the number of variables and constraints and the gap. It is also shown how the gap can be reduced even further by introducing combinatorial cuts. Computational results based on the two formulations of the problem and combinatorial cuts are presented.Zusammenfassung. In dieser Arbeit behandeln wir das Problem der Partionierung der Knotenmenge eines Graphen in h6chstens p Klassen, so dab die Summe der Knotengewichte jeder Klasse nicht gr66er als eine Klassenkapazit~it b ist und die Summe der Kantengewichte fiar alle Kanten, die Knoten einer Klasse verbinden, maximal wird. Dieses Problem kann als MILP formuliert werden, es erweist sich als vollst~indig symmetrisch beziiglich der p Klassen, und die Lticke zwischen der L6sung der LP-Relaxation und der Optimall6sung ist so gro6 wie m6glich. Aufgrund dieser beiden Eigenschaften ist schon die L6sung kleiner Probleme schwierig. Wir zeigen in dieser Arbeit M6glichkeiten, bestimmte Knoten vorweg gewissen Klassen zuzuordnen, um damit sowohl die Symmetrie-Eigenschaft der Formulierung als auch die Anzahl der Variablen und Nebenbedingungen und damit die Ltieke zu reduzieren. Weiterhin zeigen wir, wie die Lticke durch Einfiihrung kombinatorischer Schnitte weiter reduziert werden kann. Die Ergebnisse numerischer Untersuchungen werden pr~isentiert.

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“…The first exact algorithm using branch and bound techniques for solving the optimal bipartition problem (i. e. the special case where p = 2) can be found in Christofides and Brooker (1976). Much more recently, Hansen and Roucairol (1987) showed that, after reformulating the bipartition problem as a quadratic pseudoboolean program with a cardinality constraint and applying lagrangean relaxation, lower bounds significantly sharper than those used in Christofides and Brooker could be derived, resulting in improved tree search procedures.…”

Section: Previous Attempts At Solving (Gpp) and (Wgpp)mentioning

confidence: 99%