The edge-weighted clique problem: Valid inequalities, facets and polyhedral computations

Macambira, Elder M.; Souza, Cid C. de

doi:10.1016/s0377-2217(99)00262-3

Cited by 47 publications

(26 citation statements)

References 7 publications

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“…Actually, if y can assume any fractional value, then the separation problem for (8.iii) reduces to the Maximum Edge-Weighted Clique problem in undirected graphs. The latter is known to be an NP-hard problem (see [24]), leaving very little hope to solve the separation e ciently. When y is binary and no constraints of type (8).ii are violated, the situation changes drastically.…”

Section: Solution Algorithmmentioning

confidence: 99%

An Exact Decomposition Approach for the Real-Time Train Dispatching Problem

Lamorgese

Mannino

2015

Operations Research

117

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Trains’ movements on a railway network are regulated by official timetables. Deviations and delays occur quite often in practice, demanding fast rescheduling and rerouting decisions in order to avoid conflicts and minimize overall delay. This is the real-time train dispatching problem. In contrast with the classic “holistic” approach, we show how to decompose the problem into smaller subproblems associated with the line and the stations. This decomposition is the basis for a master-slave solution algorithm, in which the master problem is associated with the line and the slave problem is associated with the stations. The two subproblems are modeled as mixed integer linear programs, with specific sets of variables and constraints. Similarly to the classical Benders’ decomposition approach, slave and master communicate through suitable feasibility cuts in the variables of the master. Extensive tests on real-life instances from single and double-track lines in Italy showed significant improvements over current dispatching performances. A decision support system based on this exact approach has been in operation in Norway since February 2014 and represents one of the first operative applications of mathematical optimization to train dispatching.

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Section: Solution Algorithmmentioning

confidence: 99%

An Exact Decomposition Approach for the Real-Time Train Dispatching Problem

Lamorgese

Mannino

2015

Operations Research

117

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“…Obviously, TSF1-DecðG; P; p ¼ 1; KÞ belongs to NP. To prove its completeness, we use a reduction from MEWC, the Maximum EdgeWeighted Clique problem, which is NP-complete [10]. Consider an arbitrary instance I 1 of MEWC: given a complete edge-weighted graph G c = (V c , E c , w), where w : E c !…”

Section: With a Single Slavementioning

confidence: 99%

Scheduling tasks sharing files on heterogeneous master-slave platforms

Giersch

Robert²,

Vivien³

2004

12th Euromicro Conference on Parallel, Distributed and Network-Based Processing, 2004. Proceedings.

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This paper is devoted to scheduling a large collection of independent tasks onto heterogeneous clusters. The tasks depend upon (input) files which initially reside on a master processor. A given file may well be shared by several tasks. The role of the master is to distribute the files to the processors, so that they can execute the tasks. The objective for the master is to select which file to send to which slave, and in which order, so as to minimize the total execution time. The contribution of this paper is twofold. On the theoretical side, we establish complexity results that assess the difficulty of the problem. On the practical side, we design several new heuristics, which are shown to perform as efficiently as the best heuristics in [H. Casanova, A. Legrand, D. Zagorodnov, F. Berman, Heuristics for scheduling parameter sweep applications in Grid environments, in: Ninth Heterogeneous Computing Workshop, IEEE Computer Society Press, Silver Spring, MD, 2000, pp. 349-363; Berman, Using simulation to evaluate scheduling heuristics for a class of applications in Grid environments, Research Report RR-1999-46, LIP, ENS Lyon, France, 1999 although their cost is an order of magnitude lower.

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“…Recently, Macambira and De Souza [1996] presented the following class of facet-defining inequalities for CP n b that generalizes the cut inequalities. …”

Section: Remark 22 the Fact That The Inequalities In Proposition 2mentioning

confidence: 99%

“…. , w n ) ∈ R n , we define the quadratic knapsack polytope QKP n b (w) as the convex hull of those vectors (x, y) ∈ R n(n+1)/2 satisfying the conditions (3)- (6) together with the following knapsack generalization of (2): Tree inequalities were also considered by Macambira and De Souza [1996]. They combined a tree inequality with an (α, β)-inequality for α = 2 and β = 1, and showed that the resulting inequality is facet-defining if the underlying tree is a path.…”

Section: 2mentioning

confidence: 99%

A Lagrangian relaxation approach to the edge-weighted clique problem

Hunting

Faigle

Kern

2001

European Journal of Operational Research

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Abstract. The b-clique polytope CP n b is the convex hull of the node and edge incidence vectors of all subcliques of size at most b of a complete graph on n nodes. Including the Boolean quadric polytope QP n as a special case and being closely related to the quadratic knapsack polytope, it has received considerable attention in the literature. In particular, the max-cut problem is equivalent with optimizing a linear function over QP n n . The problem of optimizing linear functions over CP n b has so far been approached via heuristic combinatorial algorithms and cutting-plane methods.We study the structure of CP n b in further detail and present a new computational approach to the linear optimization problem based on Lucena's suggestion of integrating cutting planes into a Lagrangian relaxation of an integer programming problem. In particular, we show that the separation problem for tree inequalities becomes polynomial in our Lagrangian framework. Finally, computational results are presented.

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The edge-weighted clique problem: Valid inequalities, facets and polyhedral computations

Cited by 47 publications

References 7 publications

An Exact Decomposition Approach for the Real-Time Train Dispatching Problem

An Exact Decomposition Approach for the Real-Time Train Dispatching Problem

Scheduling tasks sharing files on heterogeneous master-slave platforms

A Lagrangian relaxation approach to the edge-weighted clique problem

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