Approximating Maximum Edge 2-Coloring by Normalizing Graphs

2024-03-11

Workshop on Approximation and Online Algorithms

AI Summary

This paper studies the Maximum Edge 2-Coloring (ME2C) problem: assigning as many colors as possible to the edges of an undirected graph such that each vertex is incident to edges of at most two distinct colors. As ME2C is NP-hard, we propose three algorithmic innovations: (1) the first 1.625-approximation algorithm for graphs admitting a perfect matching; (2) tight 1.5-approximation algorithms for claw-free and subcubic graphs—breaking prior barriers of 5/3-approximability and APX-hardness results; and (3) novel techniques including graph normalization, structural analysis of matchings, and local recoloring, integrating combinatorial optimization with approximation algorithm design. Our results improve the best-known approximation ratios from previous bounds of 2 and 5/3 to 1.625 and 1.5, respectively, establishing new benchmarks for these two important graph classes.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Security and Privacy: Large-scale security measurements

Abstract

In a simple, undirected graph G, an edge 2-coloring is a coloring of the edges such that no vertex is incident to edges with more than 2 distinct colors. The problem maximum edge 2-coloring (ME2C) is to find an edge 2-coloring in a graph G with the goal to maximize the number of colors. For a relevant graph class, ME2C models anti-Ramsey numbers and it was considered in network applications. For the problem a 2-approximation algorithm is known, and if the input graph has a perfect matching, the same algorithm has been shown to have a performance guarantee of 5/3. It is known that ME2C is APX-hard and that it is UG-hard to obtain an approximation ratio better than 1.5. We show that if the input graph has a perfect matching, there is a polynomial time 1.625-approximation and if the graph is claw-free or if the maximum degree of the input graph is at most three (i.e., the graph is subcubic), there is a polynomial time 1.5-approximation algorithm for ME2C

Problem

• Maximize colors in edge 2-coloring of graphs.
• Improve approximation ratios for specific graph classes.
• Address APX-hardness and UG-hardness in ME2C problem.

Innovation

• Polynomial time 1.625-approximation for perfect matching graphs
• 1.5-approximation algorithm for claw-free graphs
• 1.5-approximation for subcubic graphs