Max-DBMIS: Degree Bounded Matroid Independent Set

• Max-DBMIS is a framework that extends classical matroid optimization by introducing hyperedge degree constraints, applicable to problems like colored forests and matching.
• It utilizes LP relaxation and combinatorial reduction (e.g., matroid (Δ+1)-parity and p-exchange local search) to develop approximation algorithms whose guarantees depend on the hypergraph's maximum degree.
• Special cases, such as the Maximum-size Properly Colored Forest, demonstrate improved ratios (e.g., 2/3 approximation for Δ=2) and inspire further research into matroidal properties and hypergraph structures.

The Maximum-size Degree Bounded Matroid Independent Set (Max-DBMIS) problem extends classical matroid optimization by imposing upper bounds on the participation of elements across hyperedges of a hypergraph with bounded maximum degree. This framework generalizes a variety of combinatorial optimization problems, including the Maximum-size Properly Colored Forest problem, and admits a suite of linear and combinatorial approximation algorithms whose guarantees depend solely on the hypergraph’s maximum degree.

1. Formal Problem Definition

Let $\M=(E, \I)$ be a matroid on finite ground set $E$, where $\I$ denotes the family of independent subsets. Let $H=(E, \E)$ be a hypergraph on the same ground set, and each element $v \in E$ appears in at most $\Delta$ hyperedges. To each hyperedge $e \in \E$, associate a nonnegative integer upper bound $g(e)$. The goal is to find $I \subseteq E$ of maximal cardinality such that $I \in \I$ (matroid independence) and $|I \cap e| \leq g(e)$ for all $e \in \E$.

The formal optimization problems are: $\max\left\{|I| : I \in \I,\, |I \cap e| \leq g(e)\ \forall e \in \E\right\}$ For the weighted variant, with nonnegative weights $w : E \to \mathbb{R}_{\ge 0}$,

$\max\left\{\sum_{v \in I} w(v) : I \in \I,\, |I \cap e| \leq g(e)\ \forall e \in \E\right\}$

This definition subsumes classical matching and $k$-dimensional matching; specific instances such as the Maximum-size Properly Colored Forest problem are specializations where the matroid and hypergraph structure are induced by the underlying colored graph (Bai et al., 23 Nov 2025).

2. Linear Programming Relaxation and Separation

A natural LP relaxation introduces a variable $x_v \in [0,1]$ for each $v \in E$:

• Objective: maximize $\sum_{v \in E} x_v$
• Subject to:
  • Matroid constraints: $\sum_{v \in A} x_v \leq r_\M(A)$ for all $A \subseteq E$, with $r_\M$ the matroid rank function
  • Hyperedge constraints: $\sum_{v \in e} x_v \leq g(e)$ for all $e \in \E$
  • Box constraints: $0 \leq x_v \leq 1$ for all $v \in E$

The LP can be separated efficiently with calls to a matroid-rank oracle and simple counting for the degree constraints. However, this relaxation is not known to admit tight rounding directly for general $(\Delta, g)$ parameters; instead, reduction techniques are applied (Bai et al., 23 Nov 2025).

3. Approximation Techniques

3.1. Reduction to Matroid $(\Delta+1)$-Parity

For general $g$, the canonical approach is a combinatorial reduction to matroid $(\Delta+1)$-parity:

• Each $v \in E$ is represented by a gadget $e'_v$ of size $\Delta+1$: $\{v^M\} \cup \{v^e : e \in \E, v \in e\} \cup D_v$ where $|D_v| = \Delta - \deg_H(v)$.
• The augmented ground set $E'$ consists of all such gadgets.
• Copy the original matroid on the $v^M$ layer, use uniform matroids of rank $g(e)$ on $v^e$-copies per hyperedge, make $D_v$ free, and take the direct sum.
• Assign weight $w(v)$ to gadget $e'_v$. Selection of $\M'$-independent gadgets is equivalent to an independent $I \subseteq E$ respecting degree bounds.

Applying the best-known approximations for this reduction yields:

• Unweighted: $(2/(\Delta+1) - \varepsilon)$-approximation (Lee–Sviridenko–Vondrák 2010 for $k$-parity with $k = \Delta+1$)
• Weighted: $(\ln 4)/(\Delta+2)$-approximation (Singer–Thiéry 2025)

3.2. Local Search for Unit Hyperedge-Bounds

For $g(e) \leq 1$:

• The problem reduces to the intersection of $\Delta+1$ matroids: the original plus $\Delta$ matchings from König’s theorem applied to the conflict graph.
• A $p$-exchange local search, with $p=\lceil 1/\varepsilon \rceil$, yields a $1/(\Delta+\varepsilon)$-approximation.

These algorithms run in polynomial time, with complexity determined chiefly by the matroid-parity or local search routines (Bai et al., 23 Nov 2025).

4. Approximation Guarantees and Algorithmic Properties

The following table collates the known approximation factors for Max-DBMIS based on the algorithmic strategy and input structure:

Setting	Approximation Ratio	Method
General $g$ (Unweighted)	$2/(\Delta+1) - \varepsilon$	Matroid $(\Delta+1)$-parity
General $g$ (Weighted)	$(\ln 4)/(\Delta+2)$	Matroid $(\Delta+1)$-parity
All $g(e) \leq 1$	$1/(\Delta+\varepsilon)$	$p$-exchange local search

For the unweighted case, the ratio improves to $2/3$ for $\Delta = 2$ (as in properly colored forests), surpassing the previous $5/9$ bound of Bai, Bérczi, Csáji, and Schwarcz. The reduction and analysis preserve feasibility and objective, and the matroid-parity-based approach is polynomial with respect to oracle complexity (Bai et al., 23 Nov 2025).

5. Special Cases: Properly Colored Forests

In the Maximum-size Properly Colored Forest problem, the ground set is the set of edges of an edge-colored graph, color classes yield hyperedges, and the graphic matroid encodes acyclicity:

• Hyperedge degree: Each edge is in two hyperedges, so $\Delta=2$.
• Hyperedge bounds: $g(e) = 1$.
• Algorithmic result: Both reduction and local search yield a $2/3$-approximation, strictly improving on previous $5/9$-approximations.

This demonstrates the tight applicability of Max-DBMIS analysis in colored forest optimization, underscoring the relevance of the matroidal framework and the combinatorial reduction for achieving improved approximation bounds (Bai et al., 23 Nov 2025).

6. Extensions, Limitations, and Related Work

If all $g(e) \leq 1$, Max-DBMIS is a $(\Delta+1)$-extendible system, but no better than $1/\Delta$-approximation is generally achievable. In the special case where $\M$ is free and $g(e)=1$ with $\Delta=k$, the problem reduces to $k$-Dimensional Matching, for which approximability is shown to be hard beyond $\Theta(1/k)$ (Lee–Svensson–Thiéry, STOC 2025).

For weighted and general matroids, the $(\ln 4)/(\Delta+2)$ factor is tight for large $\Delta$ by reduction from $k$-Dimensional Matching. If the matroid is strongly base-orderable or the hypergraph admits special structure (such as totally unimodular incidence constraints), improved guarantees may be possible, but these remain open research directions.

A plausible implication is that further gains in approximation ratio likely require either stronger matroidal properties or more restrictive hypergraph structure; for arbitrary instances, the currently established ratios are provably best possible given existing complexity-theoretic assumptions (Bai et al., 23 Nov 2025).