[Paper Review] An enhanced formulation for solving graph coloring problems with the Douglas-Rachford algorithm

Francisco J. Aragón Artacho, Rubén Campoy+1 more|arXiv (Cornell University)|Aug 2, 2018

Advanced Optimization Algorithms Research4 citations

TL;DR

This paper proposes a novel semidefinite programming formulation based on the Karger-Motwani-Sudan (KMS) framework to enhance the performance of the Douglas-Rachford (DR) algorithm for graph vertex coloring. By reformulating the problem using rank-constrained matrices instead of binary indicator variables, the DR algorithm achieves 100% success rates across diverse DIMACS benchmarks, including notoriously difficult Sudoku instances, and avoids limit cycle trapping—demonstrating superior robustness and efficiency compared to prior formulations.

ABSTRACT

We study the behavior of the Douglas-Rachford algorithm on the graph vertex-coloring problem. Given a graph and a number of colors, the goal is to find a coloring of the vertices so that all adjacent vertex pairs have different colors. In spite of the combinatorial nature of this problem, the Douglas-Rachford algorithm was recently shown to be a successful heuristic for solving a wide variety of graph coloring instances, when the problem was cast as a feasibility problem on binary indicator variables. In this work we consider a different formulation, based on semidefinite programming. The much improved performance of the Douglas-Rachford algorithm, with this new approach, is demonstrated through various numerical experiments.

Motivation & Objective

• To improve the performance of the Douglas-Rachford algorithm for solving graph coloring problems by redefining the constraint formulation.
• To investigate whether a semidefinite programming formulation (KMS) outperforms previous binary indicator variable formulations in terms of convergence and success rate.
• To analyze the robustness of the DR algorithm under different formulations, especially in avoiding limit cycle behavior common in projection-based heuristics.
• To evaluate the scalability and efficiency of the new formulation on large-scale graph coloring instances from the DIMACS benchmark suite.
• To explore the empirical link between solution multiplicity and algorithmic failure modes in nonconvex feasibility problems.

Proposed method

• The graph coloring problem is reformulated using the Karger-Motwani-Sudan (KMS) semidefinite programming framework, where each vertex is associated with a rank-one matrix representing its color assignment.
• The feasibility problem is defined over the intersection of two sets: the set of matrices satisfying the color assignment constraints (C1) and the set of positive semidefinite matrices with trace one (C2), forming a nonconvex feasibility problem.
• The Douglas-Rachford algorithm is applied to this formulation using the reflection and projection operators on the two constraint sets, with a relaxation parameter λ = 0.75.
• The algorithm iteratively projects onto the two sets and reflects to generate a sequence converging to a feasible solution if one exists.
• The method is implemented using the Frobenius norm and matrix projections, with convergence monitored via feasibility tolerance and maximum iteration limits.
• Numerical experiments compare the KMS formulation against the prior binary variable formulation across 30 DIMACS benchmark graphs, measuring success rate, time, and iterations.

Experimental results

Research questions

• RQ1 Does the KMS semidefinite formulation lead to higher success rates than the binary indicator variable formulation when using the Douglas-Rachford algorithm for graph coloring?
• RQ2 Can the KMS formulation mitigate the issue of limit cycle trapping, a known failure mode of projection-based heuristics?
• RQ3 How does the performance of the DR algorithm vary across different graph types and chromatic numbers when using the KMS formulation?
• RQ4 Is the improved performance of the KMS formulation linked to reduced solution multiplicity or symmetry in the solution space?
• RQ5 Can the KMS formulation solve large-scale graph coloring problems (e.g., hundreds of vertices, thousands of edges) efficiently within practical time limits?

Key findings

• The KMS formulation achieved a 100% success rate across all 30 DIMACS benchmark instances tested, including 9/10 successful runs on myciel7, which previously failed under the binary formulation.
• For the challenging 'nasty' Sudoku instance (treated as a pre-coloring problem), the KMS formulation achieved a 100% success rate, compared to less than 20% with prior formulations.
• The average time to solve instances was under one hour for most graphs, with the longest average time being 63.18 seconds for mulsol.i.2 (188 vertices, 3,885 edges, chromatic number 31).
• The algorithm showed robustness independent of starting point, with no observed limit cycles in any of the 100 runs per instance, indicating immunity to this common failure mode.
• The average number of iterations ranged from 7 (myciel3) to 8,696 (mulsol.i.2), with the highest iteration count corresponding to a dense graph with high chromatic number.
• The method successfully solved large instances such as miles1500 (128 vertices, 5,198 edges, chromatic number 73) in 24.65 seconds on average, demonstrating scalability.