Maximum-Size Properly Colored Forest

• Maximum-size properly colored forest is defined as the largest acyclic subgraph in an edge-colored graph that meets local color constraints by ensuring no two adjacent edges share the same color.
• It is analyzed using matroid theory and hypergraph frameworks, leading to approximation algorithms like the DBMIS technique with a 2/3 - ε guarantee.
• The problem bridges combinatorial optimization and classical spanning tree and matching issues, highlighting NP-hardness and sharp inapproximability boundaries in various graph classes.

A maximum-size properly colored forest is a largest (by edge count) acyclic subgraph of an edge-colored undirected graph such that no two adjacent edges share the same color. This fundamental combinatorial optimization problem, termed Max-PF in recent literature, generalizes several classical matching and spanning tree notions, intertwining constraints from graph acyclicity with local color exclusion properties. The study of Max-PF illuminates algorithmic interfaces between matroid theory, hypergraph degree constraints, and combinatorial local improvement frameworks (Bai et al., 23 Nov 2025, Bai et al., 1 Feb 2024).

1. Problem Definition and Mathematical Formulation

Let $G=(V,E)$ be an undirected loopless multigraph with edge-coloring $c\colon E\to[k]$, where $[k]=\{1,\ldots,k\}$. A subgraph $(V,F)$, with $F\subseteq E$, is a properly colored forest if:

• $(V,F)$ is acyclic (i.e., a forest),
• At every vertex $v\in V$ and for every color $i\in[k]$, at most one incident edge of color $i$ is present.

Equivalently, no pair of adjacent edges in $F$ have the same color. The Maximum-size Properly Colored Forest problem (Max-PF) seeks to maximize $|F|$ over all properly colored forests in $G$. The weighted version (Max-WPF) asks for $F$ maximizing $\sum_{e\in F}w(e)$ for edge weights $w\colon E\to\mathbb{R}_{\geq0}$ (Bai et al., 23 Nov 2025, Bai et al., 1 Feb 2024).

2. Structural Properties and Matroidal Viewpoint

Properly colored forests exhibit a deep connection to matroid theory. Any such forest $F$ can be decomposed so that for each color $i$, $F_i=\{e\in F:c(e)=i\}$ is a matching. This yields an intersection property: $F$ is simultaneously independent in the cycle matroid (acyclic) and respects partition-matroid-like local color constraints.

A key structural lemma asserts: if $U\subseteq V$ is a largest subset covered by matching in each $E_i$ (color class), then any maximum-size properly colored forest covers exactly $U$. Thus, finding a maximum matching-coverable set of vertices, via the sum of matching matroids and Edmonds–Fulkerson’s matroid-union theorem, forms a foundational preprocessing for Max-PF algorithms (Bai et al., 1 Feb 2024).

3. Approximation Algorithms

Max-PF is NP-hard; approximations are thus essential.

Algorithmic Approaches and Performance Guarantees

Algorithm/Framework	Approximation Ratio	Applicable Graph Class
Trivial union-of-matchings	$1/2$	Arbitrary $k$
Local-exchange (Algorithm A, Bai et al.)	$5/9$ [see below]	Multigraphs, all $k$
Degree Bounded Matroid Ind. Set (DBMIS)	$2/3-\epsilon$	Multigraphs, all $k$
Special cases (simple graphs, $k=2$)	$3/4$	Simple graphs, $k=2$
Special cases (multigraphs, $k=2$)	$3/5$	Multigraphs, $k=2$
Special cases (simple or $k=3$)	$4/7$	$k\leq 3$, no parallel edges
Complete multigraph, $k=2$	Exact (polytime)	Complete multigraph, $k=2$

Local-Exchange: The $5/9$-Approximation

Bai, Bérczi, Csáji, and Schwarcz [Eur. J. Comb. 132 (2026)] proved that, by repeated local "2-for-3" edge exchanges, one can improve on the naïve $1/2$-approximation and extract a properly colored forest with size at least $5/9$ of the optimum.

Degree Bounded Matroid Independent Set (DBMIS) and the $2/3$-Approximation

Embedding Max-PF as a Max-DBMIS instance, where:

• The ground set is edges $E$,
• The matroid is the graphic matroid (cycle-free sets),
• For each vertex-color pair $(v,i)$, a hyperedge $e_{v,i}$ consists of all edges of color $i$ incident to $v$, and $g(e_{v,i})=1$.

Each edge lies in at most $\Delta=2$ hyperedges; thus, the DBMIS instance has $\Delta=2$. The reduction to a matroid $(\Delta+1)$-parity instance and application of Lee–Sviridenko–Vondrák’s $2/k-\epsilon$ approximation for matroid $k$-parity yields a polynomial-time $2/3-\epsilon$ approximation for Max-PF (Bai et al., 23 Nov 2025). This strictly improves the previous $5/9$ guarantee.

Special and Exact Cases

For complete multigraphs with $k=2$, an exact polynomial-time solution is achievable via contraction and application of the 2-color Hamiltonian-path result of Bang–Jensen & Gutin. For simple graphs or small $k$, refined decomposition and coloring arguments provide stronger ratios (Bai et al., 1 Feb 2024).

4. Hardness and Inapproximability

Hardness of approximation results for Max-PF, via $L$-reductions from MAX-SNP-hard problems (Longest Path, $(1,2)$-TSP, Maximum Linear Forest), are as follows:

• For $k=2$ in simple graphs, Max-PF is MAX-SNP-hard; NP-hard to approximate within any factor $<1601/1602$ even when a properly colored spanning tree exists.
• For $k=3$ in complete simple graphs, NP-hard to approximate within $1-1/3204$.
• For $k=2$ in (noncomplete) multigraphs, NP-hard to approximate within $533/534$.
• For Max-PT (maximum-size properly colored tree), inapproximability is significantly stronger: in simple or multigraphs, hard to approximate within $n^{1-\varepsilon}$ for any $\varepsilon>0$ (Bai et al., 1 Feb 2024).

5. Extensions and Generalizations

The DBMIS framework generalizes Max-PF: Given a matroid $M=(U,\mathcal{I})$, a hypergraph $H$ of maximum degree $\Delta$, bounds $g(e)$ on hyperedges, the goal is to maximize $|I|$ for $I\in\mathcal{I}$ with $|I\cap e| \leq g(e)$ for all $e\in E(H)$. The central result (Bai et al., 23 Nov 2025) provides a $(2/(\Delta+1)-\epsilon)$-approximation for this problem.

For Max-PF, embedding with $\Delta=2$ yields the $2/3$-approximation. The reduction maps feasible solutions between Max-DBMIS and matroid $(\Delta+1)$-parity bijectively, preserving size and weight.

Weighted versions (Max-WPF, Max-WDBMIS) can be approximated via the guarantee $(\ln 4)/(\Delta+2)\approx 0.346$ for $\Delta=2$.

6. Algorithmic Complexity and Empirical Status

All methods described operate in polynomial time. For fixed $\epsilon$ and $\Delta=2$, both the reduction to matroid parity and the parity-approximation subroutine require time polynomial in $|V|$, $|E|$, and $k$. Space usage is also polynomial. No empirical or experimental evaluations of the algorithms are reported; all results are worst-case theoretical guarantees (Bai et al., 23 Nov 2025, Bai et al., 1 Feb 2024).

7. Comparison with Properly Colored Tree and Related Problems

The corresponding tree version (Max-PT), seeking the largest properly colored tree (not necessarily spanning), is much harder, exhibiting polylogarithmic or worse inapproximability barriers even in highly structured graphs. By contrast, Max-PF admits constant-factor approximations strictly exceeding $1/2$. The connection to Degree Bounded Spanning Tree, $(1,2)$-TSP, and matroid union/matching covers further situates Max-PF within the landscape of combinatorial optimization and coloring (Bai et al., 1 Feb 2024).