Approximating maximum properly colored forests via degree bounded independent sets
Yuhang Bai, Kristóf Bérczi, Johanna K. Siemelink
TL;DR
The paper introduces the Maximum-size Degree Bounded Matroid Independent Set (Max-DBMIS) framework to approximate maximum independent sets under per-hyperedge bounds in an edge-colored graph, with guarantees that depend solely on the hypergraph degree Δ. By reducing Max-DBMIS to matroid parity with k=Δ+1, the authors obtain a polynomial-time (2/(Δ+1) − ε)-approximation for the unweighted case and a (ln 4)/(Δ+2)-approximation for the weighted case; with unit bounds this yields a 1/(Δ+ε) guarantee. Applying the framework to Maximum-size Properly Colored Forests (Max-GPF) yields a 2/3-approximation, improving the previous 5/9 bound, with analogous ratios in weighted and unit-bound variants. The paper further extends the approach to properly colored forests with bundles, arborescences, and b-matchings, via local search, matroid-intersection refinements, and reductions to hierarchical b-matching, highlighting the broad applicability and potential for improved Δ-dependent guarantees in color-constrained combinatorial structures.
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].