The partition problem

Chopra, Sunil; Rao, M. R.

doi:10.1007/bf01581239

Cited by 231 publications

(250 citation statements)

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“…Using Ausgrid's training data as a template, we generated a series of random test cases 3 . Each test case had between 100 and 500 students, with between 4 and 12 student types, and between 56 and 98 classes.…”

Section: Computational Resultsmentioning

confidence: 99%

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A Lagrangian Relaxation-Based Heuristic to Solve Large Extended Graph Partitioning Problems

Czibula

Zinder

2016

WALCOM: Algorithms and Computation

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Abstract. The paper is concerned with the planning of training sessions in large organisations requiring periodic retraining of their staff. The allocation of students must take into account student preferences as well as the desired composition of study groups. The paper presents a bicriteria Quadratic Multiple Knapsack formulation of the considered practical problem, and a novel solution procedure based on Lagrangian relaxation. The paper presents the results of computational experiments aimed at testing the optimisation procedure on real world data originating from Australia's largest electricity distributor. Results are compared and validated against a Genetic Algorithm based matheuristic.

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Section: Computational Resultsmentioning

confidence: 99%

“…Chopra and Rao [3] discuss several forms of the graph partitioning problem as well as IP models for each. The authors do not assume a complete graph, allowing them to take advantage of the graph structure when clustering.…”

Section: Introductionmentioning

confidence: 99%

A Lagrangian Relaxation-Based Heuristic to Solve Large Extended Graph Partitioning Problems

Czibula

Zinder

2016

WALCOM: Algorithms and Computation

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“…There are n k+1 inequalities of type (14). Constraints (13) and (14) are also used in [34] to strengthen the SDP relaxation for the graph equipartition problem, and in [8] for a polyhedral setting of the GPP. By adding constraints (13) and/or (14) to GPP RS , we obtain stronger relaxations that are more computationally demanding than GPP RS .…”

Section: It Is Clear That Formentioning

confidence: 99%

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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“…For fixed parameters θ, it specializes to the NP-hard minimum cost clique partition problem (Grötschel and Wakabayashi, 1989;Chopra and Rao, 1993) that is also known as correlation clustering (Bansal et al, 2004;Demaine et al, 2006). …”

Section: Introductionmentioning

confidence: 99%

Convexification of Learning from Constraints

Shcherbatyi

Andres

2016

Lecture Notes in Computer Science

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Regularized empirical risk minimization with constrained labels (in contrast to fixed labels) is a remarkably general abstraction of learning. For common loss and regularization functions, this optimization problem assumes the form of a mixed integer program (MIP) whose objective function is non-convex. In this form, the problem is resistant to standard optimization techniques. We construct MIPs with the same solutions whose objective functions are convex. Specifically, we characterize the tightest convex extension of the objective function, given by the Legendre-Fenchel biconjugate. Computing values of this tightest convex extension is NP-hard. However, by applying our characterization to every function in an additive decomposition of the objective function, we obtain a class of looser convex extensions that can be computed efficiently. For some decompositions, common loss and regularization functions, we derive a closed form.

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The partition problem

Cited by 231 publications

References 10 publications

A Lagrangian Relaxation-Based Heuristic to Solve Large Extended Graph Partitioning Problems

A Lagrangian Relaxation-Based Heuristic to Solve Large Extended Graph Partitioning Problems

An Efficient Semidefinite Programming Relaxation for the Graph Partition Problem

Convexification of Learning from Constraints

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