[Paper Review] On the polyhedron of the K-partitioning problem with representative variables

Zacharie Alès, Arnaud Knippel+1 more|arXiv (Cornell University)|Nov 23, 2014

VLSI and FPGA Design Techniques1 references3 citations

TL;DR

This paper introduces a mixed-integer linear formulation for the K-partitioning problem using edge variables and representative variables to fix the number of clusters and break symmetry. It characterizes facet-defining inequalities—including 2-chorded cycle, 2-partition, clique, strengthened triangle, and paw inequalities—demonstrating that these significantly improve the linear relaxation bound, especially for complete graphs with diverse edge weights.

ABSTRACT

The K-partitioning problem consists of partitioning the vertices of a graph in K sets so as to minimize a function of the edge weights. We introduce a linear mixed integer formulation with edge variables and representative variables. We consider the corresponding polyhedron and show which inequalities are facet-defining. We study several families of facet-defining inequalities and provide experimental results showing that they improve significantly the linear relaxation of our formulation.

Motivation & Objective

• Address the challenge of improving exact solution methods for the K-partitioning problem by strengthening the linear relaxation.
• Develop a mixed-integer linear formulation that fixes the number of clusters K and reduces symmetry through representative variables.
• Characterize which inequalities are facet-defining for the polyhedron of the K-partitioning problem with representative variables.
• Provide a polyhedral analysis to enhance bounds in branch-and-bound algorithms for exact solution of graph partitioning problems.
• Demonstrate the practical impact of facet-defining inequalities on relaxation quality using experimental results on complete graphs.

Proposed method

• Propose a mixed-integer linear formulation using edge variables (x_{u,v}) and representative variables (x_u) to indicate the smallest-indexed vertex in each cluster.
• Define the polyhedron P_{n,K} as the convex hull of all feasible K-partitions under this formulation.
• Apply transformations (e.g., T(C, S, T)) to analyze coefficient relations in hyperplanes that contain the polyhedron, proving facet-defining conditions.
• Characterize trivial inequalities, 2-chorded cycle inequalities, 2-partition inequalities, and general clique inequalities as facet-defining under specific structural conditions.
• Strengthen standard triangle inequalities by adding representative variable constraints, showing they become facet-defining in most cases.
• Introduce and analyze paw inequalities for four-vertex subsets, identifying necessary and sufficient conditions for facet-defining status.

Experimental results

Research questions

• RQ1 Which inequalities in the proposed formulation are facet-defining for the K-partitioning polyhedron P_{n,K}?
• RQ2 How do representative variables improve the polyhedral structure compared to standard edge-only formulations?
• RQ3 Under what conditions do strengthened triangle inequalities become facet-defining?
• RQ4 What are the necessary and sufficient conditions for paw inequalities to define facets of P_{n,K}?
• RQ5 To what extent do the identified facet-defining inequalities improve the linear relaxation bound in practice?

Key findings

• The 2-chorded cycle inequalities are facet-defining for P_{n,K} when the cycle length is at least 4 and the representative condition is satisfied.
• The 2-partition inequalities are facet-defining if the two clusters are non-empty and the representative constraints are properly enforced.
• General clique inequalities are facet-defining when the clique size is at least 3 and the representative constraints are respected.
• Strengthened triangle inequalities—derived by combining standard triangle inequalities with representative variable constraints—become facet-defining in most cases, significantly improving relaxation bounds.
• Paw inequalities are facet-defining if and only if d < b < c and a < b, with the face F_P being non-facial when c < b or K = n−2.
• Experimental results show that the inclusion of these facet-defining inequalities leads to a substantial improvement in the linear relaxation value for complete graphs with various edge weight distributions.