Tight Approximation Results for Matroid Optimization with a Linear Constraint

Abstract

We study the following class of matroid optimization problems with a linear constraint (𝒫-MOL). Given a matroid ℳ = (E,ℐ), two nonnegative weight functions v,w:E → ℝ_{≥ 0}, and a threshold L ∈ ℝ_{≥ 0}, find opt v(S) where S is either an independent set or a base of ℳ satisfying a budget-type constraint: w(S) ≤ L or w(S) ≥ L, and opt ∈ {min,max}. 𝒫-MOL provides a unified representation for a broad family of NP-hard optimization problems, including budgeted matroid independent set, constrained minimum-basis, and knapsack-cover variants with a matroid constraint. Also, it naturally extends to multiple matroid constraints. In particular, we consider the matroid intersection cover (MIC) problem, where feasibility is defined by the common independent sets of two matroids and one seeks minimum v(S) subject to w(S) ≥ L.
Our main result is a unified efficient polynomial-time approximation scheme (EPTAS) for all nontrivial 𝒫-MOL variants, obtained by generalizing a technique of Hassin and Levin (SIAM J. Comput., 2004) for solving the constrained minimum spanning tree problem. Specifically, for any fixed ε > 0, we present an algorithm running in time |E|^O(1) ⋅ (1/ε²)^O(1/ε) that outputs a feasible solution S whose value is at most (1+ε)OPT for minimization variants and at least (1-ε)OPT for maximization variants. This resolves the complexity status of all members of 𝒫-MOL, as none of these problems admits a fully polynomial-time approximation scheme (Doron-Arad, Kulik and Shachnai, ICALP'24). Finally, we separate the 𝒫-MOL family from its extension to matroid intersection. We show that an EPTAS is unlikely to exist for the matroid intersection variant of 𝒫-MOL under a covering constraint, whereas an EPTAS is known to exist under a budget constraint. This highlights a qualitative difference between these two types of linear constraints that does not arise in the single-matroid setting.