Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Troika algorithm: approximate optimization for accurate clique partitioning and clustering of weighted networks

Samin Aref, Boris Ng

TL;DR

The paper introduces Troika, a branch-and-cut algorithm that approximates clique partitioning on weighted graphs by branching on node triples and solving LP relaxations with strengthened transitivity constraints. Troika provides guaranteed proximity to global optima within a user-specified optimality gap and often outperforms a state-of-the-art IP solver (Gurobi) in both solution quality and/or time across multiple benchmark families. It also demonstrates practical applicability to modularity-based community detection and portfolio analysis by converting modularity maximization into CP instances and analyzing temporal structure in stock correlations. The work combines graph pre-processing, variable fixing, and implied branching to achieve scalable performance on small-to-mid-sized networks (up to thousands of edges) and offers insights for future improvements in CP approximations and related clustering tasks.

Abstract

Clique partitioning is a fundamental network clustering task, with applications in a wide range of computational sciences. It involves identifying an optimal partition of the nodes for a real-valued weighted graph according to the edge weights. An optimal partition is one that maximizes the sum of within-cluster edge weights over all possible node partitions. This paper introduces a novel approximation algorithm named Troika to solve this NP-hard problem in small to mid-sized networks for instances of theoretical and practical relevance. Troika uses a branch-and-cut scheme for branching on node triples to find a partition that is within a user-specified optimality gap tolerance. Troika offers advantages over alternative methods like integer programming solvers and heuristics for clique partitioning. Unlike existing heuristics, Troika returns solutions within a guaranteed proximity to global optimality. And our results indicate that Troika is faster than using the state-of-the-art integer programming solver Gurobi for most benchmark instances. Besides its advantages for solving the clique partitioning problem, we demonstrate the applications of Troika in community detection and portfolio analysis. Troika returns partitions with higher proximity to optimal compared to eight modularity-based community detection algorithms. When used on networks of correlations among stocks, Troika reveals the dynamic changes in the structure of portfolio networks including downturns from the 2008 financial crisis and the reaction to the COVID-19 pandemic. Our comprehensive results based on benchmarks from the literature and new real and random networks point to Troika as a reliable and accurate method for solving clique partitioning instances with up to 5000 edges on standard hardware.

Troika algorithm: approximate optimization for accurate clique partitioning and clustering of weighted networks

TL;DR

The paper introduces Troika, a branch-and-cut algorithm that approximates clique partitioning on weighted graphs by branching on node triples and solving LP relaxations with strengthened transitivity constraints. Troika provides guaranteed proximity to global optima within a user-specified optimality gap and often outperforms a state-of-the-art IP solver (Gurobi) in both solution quality and/or time across multiple benchmark families. It also demonstrates practical applicability to modularity-based community detection and portfolio analysis by converting modularity maximization into CP instances and analyzing temporal structure in stock correlations. The work combines graph pre-processing, variable fixing, and implied branching to achieve scalable performance on small-to-mid-sized networks (up to thousands of edges) and offers insights for future improvements in CP approximations and related clustering tasks.

Abstract

Clique partitioning is a fundamental network clustering task, with applications in a wide range of computational sciences. It involves identifying an optimal partition of the nodes for a real-valued weighted graph according to the edge weights. An optimal partition is one that maximizes the sum of within-cluster edge weights over all possible node partitions. This paper introduces a novel approximation algorithm named Troika to solve this NP-hard problem in small to mid-sized networks for instances of theoretical and practical relevance. Troika uses a branch-and-cut scheme for branching on node triples to find a partition that is within a user-specified optimality gap tolerance. Troika offers advantages over alternative methods like integer programming solvers and heuristics for clique partitioning. Unlike existing heuristics, Troika returns solutions within a guaranteed proximity to global optimality. And our results indicate that Troika is faster than using the state-of-the-art integer programming solver Gurobi for most benchmark instances. Besides its advantages for solving the clique partitioning problem, we demonstrate the applications of Troika in community detection and portfolio analysis. Troika returns partitions with higher proximity to optimal compared to eight modularity-based community detection algorithms. When used on networks of correlations among stocks, Troika reveals the dynamic changes in the structure of portfolio networks including downturns from the 2008 financial crisis and the reaction to the COVID-19 pandemic. Our comprehensive results based on benchmarks from the literature and new real and random networks point to Troika as a reliable and accurate method for solving clique partitioning instances with up to 5000 edges on standard hardware.
Paper Structure (29 sections, 16 equations, 10 figures, 3 tables)

This paper contains 29 sections, 16 equations, 10 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Mathematical Background
  3. Problem Statement
  4. Integer Programming Formulations
  5. The Troika algorithm
  6. The branch and cut implementation
  7. Search termination criteria
  8. Results on solution quality and time
  9. Comparisons on ABR benchmarks
  10. Comparisons on equicut benchmarks
  11. Comparisons on correlation benchmarks
  12. Comparisons on clusedit benchmarks
  13. Comparisons on Barabási Albert Graphs
  14. Applicability of Troika in community detection
  15. The modularity maximization problem
  16. ...and 14 more sections

Figures (10)

  • Figure 1: Two comparative performance measures for the three method Troika, Combo, and Gurobi IP on the ABR benchmark dataset: (a) Extent of sub-optimality, (b) solve time.
  • Figure 2: Two comparative performance measures for the three method Troika, Combo, and Gurobi IP on the Equicut benchmark dataset: (a) Extent of sub-optimality, (b) solve time.
  • Figure 3: Two comparative performance measures for the three method Troika, Combo, and Gurobi IP on the Correlation benchmark dataset: (a) Extent of sub-optimality, (b) solve time.
  • Figure 4: Two comparative performance measures for the three method Troika, Combo, and Gurobi IP on the Clusedit benchmark dataset: (a) Extent of sub-optimality, (b) solve time.
  • Figure 5: Extent of sub-optimality and solve time for the three method Troika, Combo, and Gurobi IP on the BA instances.
  • ...and 5 more figures