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Branch and Cut for Partitioning a Graph into a Cycle of Clusters

Leon Eifler, Jakob Witzig, Ambros Gleixner

TL;DR

This paper provides a problem-specific binary programming formulation and compares it to a formulation based on the reformulation-linearization technique (RLT), and develops primal heuristics and separation routines for both formulations.

Abstract

In this paper we study formulations and algorithms for the cycle clustering problem, a partitioning problem over the vertex set of a directed graph with nonnegative arc weights that is used to identify cyclic behavior in simulation data generated from nonreversible Markov state models. Here, in addition to partitioning the vertices into a set of coherent clusters, the resulting clusters must be ordered into a cycle such as to maximize the total net flow in the forward direction of the cycle. We provide a problem-specific binary programming formulation and compare it to a formulation based on the reformulation-linearization technique (RLT). We present theoretical results on the polytope associated with our custom formulation and develop primal heuristics and separation routines for both formulations. In computational experiments on simulation data from biology we find that branch and cut based on the problem-specific formulation outperforms the one based on RLT.

Branch and Cut for Partitioning a Graph into a Cycle of Clusters

TL;DR

This paper provides a problem-specific binary programming formulation and compares it to a formulation based on the reformulation-linearization technique (RLT), and develops primal heuristics and separation routines for both formulations.

Abstract

In this paper we study formulations and algorithms for the cycle clustering problem, a partitioning problem over the vertex set of a directed graph with nonnegative arc weights that is used to identify cyclic behavior in simulation data generated from nonreversible Markov state models. Here, in addition to partitioning the vertices into a set of coherent clusters, the resulting clusters must be ordered into a cycle such as to maximize the total net flow in the forward direction of the cycle. We provide a problem-specific binary programming formulation and compare it to a formulation based on the reformulation-linearization technique (RLT). We present theoretical results on the polytope associated with our custom formulation and develop primal heuristics and separation routines for both formulations. In computational experiments on simulation data from biology we find that branch and cut based on the problem-specific formulation outperforms the one based on RLT.
Paper Structure (10 sections, 4 theorems, 23 equations, 1 figure, 2 tables)

This paper contains 10 sections, 4 theorems, 23 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. Formulations
  3. Valid Inequalities
  4. Triangle Inequalities
  5. Partition Inequalities
  6. Subtour and Path Inequalitities
  7. Primal Heuristics
  8. Computational Experiments
  9. Acknowledgments.
  10. Disclosure of Interests.

Key Result

theorem thmcountertheorem

Let $m\geq 4$ and $i,j,k \in V\xspace$ with $(i,j),(j,k),(i,k) \in A\xspace$, then is a valid, facet-defining inequality for $CCP$.

Figures (1)

  • Figure 1: Illustration of an extended subtour inequality (left) and a path inequality (right) in a $5$-cluster problem. Not all of the displayed variables can be set to one.

Theorems & Definitions (6)

  • definition thmcounterdefinition
  • theorem thmcountertheorem
  • theorem thmcountertheorem
  • theorem thmcountertheorem
  • theorem thmcountertheorem
  • proof