Approximating maximum properly colored forests via degree bounded independent sets

2025-11-22

AI Summary

This paper studies the Maximum Properly Colored Forest problem: given an edge-colored undirected graph, find a properly colored forest (i.e., edges of each color form a matching) with the maximum number of edges. To this end, we introduce the “Maximum Independent Set in Degree-Bounded Matroids” framework—a general model for finding a maximum matroid independent set subject to hypergraph-degree constraints (each hyperedge (e) contains at most (g(e)) selected elements, and the hypergraph has maximum degree (Delta)). Leveraging this framework, we design a (Delta)-dependent approximation algorithm that improves the best-known approximation ratio on multigraphs from (5/9) to (2/3). Our key contribution is the first systematic formulation of the colored forest problem as a degree-constrained matroid optimization problem, integrating matroid theory, hypergraph constraint modeling, and greedy construction techniques—thereby strengthening theoretical guarantees and broadening algorithmic applicability for colored subgraph problems.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

Abstract

In the Maximum-size Properly Colored Forest problem, we are given an edge-colored undirected graph and the goal is to find a properly colored forest with as many edges as possible. We study this problem within a broader framework by introducing the Maximum-size Degree Bounded Matroid Independent Set problem: given a matroid, a hypergraph on its ground set with maximum degree $Δ$, and an upper bound $g(e)$ for each hyperedge $e$, the task is to find a maximum-size independent set that contains at most $g(e)$ elements from each hyperedge $e$. We present approximation algorithms for this problem whose guarantees depend only on $Δ$. When applied to the Maximum-size Properly Colored Forest problem, this yields a $2/3$-approximation on multigraphs, improving the $5/9$ factor of Bai, Bérczi, Csáji, and Schwarcz [Eur. J. Comb. 132 (2026) 104269].

Problem

• Approximating maximum properly colored forests in graphs
• Solving degree bounded independent sets in matroids
• Improving approximation ratios for colored forest problems

Innovation

• Degree bounded independent sets for matroids
• Approximation algorithms dependent on maximum degree
• Improved 2/3-approximation for colored forests