Polyhedral combinatorics of the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si9.gif" display="inline" overflow="scroll"><mml:mi>K</mml:mi></mml:math>-partitioning problem with representative variables

Alès, Zacharie; Knippel, Arnaud; Pauchet, Alexandre

doi:10.1016/j.dam.2016.04.002

Cited by 15 publications

(24 citation statements)

References 19 publications

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“…Although this combination of ILP model and optimization solver is sufficient to conduct this task, it can still be time-consuming. One way to deal with this issue is to strengthen the underlying ILP model through the cutting plane approach, as Ales et al [2] do. We adopt the 2-partition and the 2-chorded cycle valid inequalities proposed by Ales et al Namely, we use these tight valid inequalities (only) during the root relaxation phase, before proceeding to the construction of the search tree.…”

Section: Enumerating All Optimal Solutionsmentioning

confidence: 99%

Multiplicity and diversity: analysing the optimal solution space of the correlation clustering problem on complete signed graphs

Arinik

Figueiredo

Labatut

2020

Journal of Complex Networks

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In order to study real-world systems, many applied works model them through signed graphs, i.e. graphs whose edges are labeled as either positive or negative. Such a graph is considered as structurally balanced when it can be partitioned into a number of modules, such that positive (resp. negative) edges are located inside (resp. in-between) the modules. When it is not the case, authors look for the closest partition to such balance, a problem called Correlation Clustering (CC). Due to the complexity of the CC problem, the standard approach is to find a single optimal partition and stick to it, even if other optimal or high scoring solutions possibly exist. In this work, we study the space of optimal solutions of the CC problem, on a collection of synthetic complete graphs. We show empirically that under certain conditions, there can be many optimal partitions of a signed graph. Some of these are very different and thus provide distinct perspectives on the system, as illustrated on a small real-world graph. This is an important result, as it implies that one may have to find several, if not all, optimal solutions of the CC problem, in order to properly study the considered system.

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Section: Enumerating All Optimal Solutionsmentioning

confidence: 99%

Multiplicity and diversity: analysing the optimal solution space of the correlation clustering problem on complete signed graphs

Arinik

Figueiredo

Labatut

2020

Journal of Complex Networks

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“…In (9), 3 | a ij a ik a jk = − 1} denotes the set of node triples that define an unbalanced triangle. The expression in inequality (9) denotes the sum of frustration states for the three edges (i, j), (i, k), (j, k) making an unbalanced triangle.…”

Section: Unbalanced Triangle Constraintsmentioning

confidence: 99%

“…This number determines how many edges should be removed to make the network k ‐balanced. For a more general formulation of partitioning graph vertices into k sets, one may refer to where numerical results for graphs with up to 20 nodes are provided.…”

Section: Other Formulations and Extensionsmentioning

confidence: 99%

A modeling and computational study of the frustration index in signed networks

Aref

Mason

2019

Networks

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Computing the frustration index of a signed graph is a key step toward solving problems in many fields including social networks, political science, physics, chemistry, and biology. The frustration index determines the distance of a network from a state of total structural balance. Although the definition of the frustration index goes back to the 1950s, its exact algorithmic computation, which is closely related to classic NP-hard graph problems, has only become a focus in recent years. We develop three new binary linear programming models to compute the frustration index exactly and efficiently as the solution to a global optimization problem. Solving the models with prioritized branching and valid inequalities in Gurobi, we can compute the frustration index of real signed networks with over 15 000 edges in less than a minute on inexpensive hardware. We provide extensive performance analysis for both random and real signed networks and show that our models outperform all existing approaches by large factors. Based on resolution time, algorithm output, and effective branching factor we highlight the superiority of our models to both exact and heuristic methods in the literature. KEYWORDS 0-1 integer linear programming, balance theory, binary programming, frustration index, line index of balance, optimization, signed graph, signed networks Networks. 2020;75:95-110.wileyonlinelibrary.com/journal/net

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“…Integer programming formulations for clustering (or grouping) problems have been studied extensively in the literature. Those seeking an extensive overview are referred to Ales et al (2016).…”

Section: Fig 1 Sample Stgp Instance Solution 3 Integer Programming Formulationsmentioning

confidence: 99%

The sport teams grouping problem

et al. 2017

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Polyhedral combinatorics of the K-partitioning problem with representative variables

Cited by 15 publications

References 19 publications

Multiplicity and diversity: analysing the optimal solution space of the correlation clustering problem on complete signed graphs

Multiplicity and diversity: analysing the optimal solution space of the correlation clustering problem on complete signed graphs

A modeling and computational study of the frustration index in signed networks

The sport teams grouping problem

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