Approximating maximum-size properly colored forests

TL;DR

This work studies the Maximum-size Properly Colored Forest (Max-PF) problem on edge-colored graphs, relating it to fundamental constrained-spanning-tree problems and matroid theory. It establishes strong hardness results (MAX-SNP-hardness and tight inapproximability) for Max-PF and Max-PT across various graph classes, and delivers a sequence of approximation algorithms anchored in the sum of color-class matchings and multi-matroid intersections. A central contribution is a general 5/9-approximation for Max-PF in $k$-edge-colored multigraphs, with improved ratios (up to $3/4$, $5/8$, or $4/7$) for simple graphs and small color counts, plus a specialized $1/\\sqrt{(2+\varepsilon)(n-1)}$-approximation for Max-PT on complete multigraphs. These results connect matroid-theoretic methods with classical optimization problems and offer concrete algorithmic benchmarks for properly colored structures in colored graphs.

Abstract

In the Properly Colored Spanning Tree problem, we are given an edge-colored undirected graph and the goal is to find a properly colored spanning tree, i.e., a spanning tree in which any two adjacent edges have distinct colors. The problem is interesting not only from a graph coloring point of view, but is also closely related to the Degree Bounded Spanning Tree and (1,2)-Traveling Salesman problems, two classical questions that have attracted considerable interest in combinatorial optimization and approximation theory. Previous work on properly colored spanning trees has mainly focused on determining the existence of such a tree and hence has not considered the question from an algorithmic perspective. We propose an optimization version called Maximum-size Properly Colored Forest problem, which aims to find a properly colored forest with as many edges as possible. We consider the problem in different graph classes and for different numbers of colors, and present polynomial-time approximation algorithms as well as inapproximability results for these settings. Our proof technique relies on the sum of matching matroids defined by the color classes, a connection that might be of independent combinatorial interest. We also consider the Maximum-size Properly Colored Tree problem, which asks for the maximum size of a properly colored tree not necessarily spanning all the vertices. We show that the optimum is significantly more difficult to approximate than in the forest case, and provide an approximation algorithm for complete multigraphs.

Figures (5)

• Figure 1: Illustration of the proof of Theorem \ref{['thm:2snp']}.
• Figure 2: Illustration of the proof of Theorem \ref{['thm:3snp']}.
• Figure 3: Illustration of the proof of Theorem \ref{['thm:maxpt-2']}.
• Figure 4: Tight examples for Algorithm \ref{['algo:maxpf']} in different settings. Thick edges denote the properly colored forest found by the algorithm, while edges with a grey outline form an optimal solution. The graphs are obtained by repeating the blocks enclosed by the dashed boxes $\ell$ times.
• Figure 5: Tight examples for Algorithm \ref{['alg-2-colors']} for $k=2$ and $3$. Thick edges denote the properly colored forest found by the algorithm, while edges with a grey outline form an optimal solution. The graphs are obtained by repeating the blocks enclosed by the dashed boxes $\ell$ times.

Theorems & Definitions (60)

• Theorem 2.1: Bang-Jensen and Gutin bang1997alternating
• Theorem 2.2: Borozan et al. borozan2019maximum
• Theorem 2.3: Karpinski and Schmied karpinski2012approximation
• Theorem 2.4: Csaba, Karpinski and Krystacsaba2002approximability
• Theorem 2.5: De la Vega and Karpinski de1999approximation
• Theorem 2.6: Björklund, Husfeldt and Khanna bjorklund2004approximating
• Theorem 2.7: De la Vega and Karpinski de1999approximation
• Corollary 3.1
• proof
• Corollary 3.2