Approximating maximum properly colored forests via degree bounded independent sets

TL;DR

This paper introduces approximation algorithms for a broad class of problems involving degree-bounded matroid independent sets, achieving improved approximation ratios for the maximum properly colored forest problem in multigraphs.

Contribution

It presents a new framework and algorithms for degree-bounded matroid independent sets, leading to better approximation guarantees for properly colored forests.

Findings

Achieves a 2/3-approximation for maximum properly colored forests in multigraphs.

Improves previous approximation factor from 5/9 to 2/3.

Introduces a general approach applicable to degree-bounded matroid problems.

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 $\Delta$, 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 $\Delta$. 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, B\'erczi, Cs\'aji, and Schwarcz [Eur. J. Comb. 132 (2026) 104269].