[Paper Review] Distance geometry approach for special graph coloring problems

Rosiane Rodrigues, Bruno Dias+2 more|arXiv (Cornell University)|Jun 15, 2016

graph theory and CDMA systems2 references4 citations

TL;DR

This paper proposes a distance geometry approach to solve special graph coloring problems with distance constraints, modeling vertex coloring as embedding vertices on the real line with prescribed inter-vertex distances. It introduces branch-prune-and-bound algorithms based on DMDGP techniques, establishing feasibility conditions for specific graph classes and demonstrating efficient solution of T-coloring variants through constraint propagation and optimization.

ABSTRACT

One of the most important combinatorial optimization problems is graph coloring. There are several variations of this problem involving additional constraints either on vertices or edges. They constitute models for real applications, such as channel assignment in mobile wireless networks. In this work, we consider some coloring problems involving distance constraints as weighted edges, modeling them as distance geometry problems. Thus, the vertices of the graph are considered as embedded on the real line and the coloring is treated as an assignment of positive integers to the vertices, while the distances correspond to line segments, where the goal is to find a feasible intersection of them. We formulate different such coloring problems and show feasibility conditions for some problems. We also propose implicit enumeration methods for some of the optimization problems based on branch-and-prune methods proposed for distance geometry problems in the literature. An empirical analysis was undertaken, considering equality and inequality constraints, uniform and arbitrary set of distances, and the performance of each variant of the method considering the handling and propagation of the set of distances involved.

Motivation & Objective

• To model graph coloring problems with distance constraints as distance geometry problems by embedding vertices on the real line.
• To identify forbidden graph structures that make certain distance coloring problems infeasible.
• To develop exact and approximate algorithms based on branch-and-prune methods for solving these constrained coloring problems.
• To evaluate the performance of different distance handling strategies—equality vs. inequality, uniform vs. arbitrary sets—on computational instances.
• To establish theoretical feasibility conditions for specific graph classes, such as odd cycles, even cycles, and trees, under distance constraints.

Proposed method

• Models vertex coloring with distance constraints as a distance geometry problem (DG), where vertices are embedded on the real line and color assignments correspond to integer positions.
• Adapts the branch-and-prune (BP) algorithm from the Discretizable Molecular Distance Geometry Problem (DMDGP) to solve constrained coloring problems.
• Introduces branch-prune-and-bound (BPB) algorithms that combine branch-and-bound with constraint propagation to improve search efficiency.
• Applies domain reduction and constraint propagation techniques to manage sets of distances, including both equality and inequality constraints.
• Uses implicit enumeration via BPB to explore the solution space while pruning infeasible branches early based on distance violations.
• Employs computational experiments on randomly generated graphs to compare performance across different distance set types and constraint types.

Experimental results

Research questions

• RQ1 Which graph classes admit feasible solutions for distance-constrained coloring problems, and which contain structural obstructions to feasibility?
• RQ2 How do equality and inequality constraints on distances affect the solvability and computational complexity of the coloring problem?
• RQ3 What is the impact of uniform versus arbitrary distance sets on the performance of branch-prune-and-bound algorithms?
• RQ4 Can constraint propagation and domain reduction significantly improve the efficiency of solving distance-constrained coloring problems?
• RQ5 How do different graph structures—such as odd cycles, even cycles, and trees—affect the number of prunes and CPU time in the BPB algorithm?

Key findings

• Odd cycles of odd length (e.g., 3, 5, 7) are infeasible for certain distance coloring problems when constraints are strict, as shown by the absence of valid solutions in the experiments.
• For even cycles and trees, feasible solutions were consistently found, with low pruning counts and short CPU times, indicating structural feasibility.
• The BPB algorithm with inequality constraints and arbitrary distance sets required significantly more pruning (e.g., up to 11 million prunes in one instance) and longer CPU times (e.g., 10,800 seconds), indicating higher computational cost.
• In contrast, equality constraints with uniform distance sets led to faster convergence, with CPU times under 10 seconds in most tested cases.
• The number of prunes and bounds varied widely depending on the graph type and distance set: for example, Tree instance 4 with 21 vertices and 3 distance constraints required 863,702 prunes and 3,463,966 bounds in 13.998 seconds.
• The method successfully identified infeasibility in specific graph classes (e.g., odd cycles) through structural analysis, confirming theoretical feasibility conditions.