Optimal Matroid Partitioning Problems

TL;DR

This paper investigates optimal matroid partitioning problems with various objectives, providing polynomial algorithms or NP-hardness proofs, and demonstrates a PTAS for a specific NP-hard case.

Contribution

It introduces new complexity results and algorithms for different objective functions in matroid partitioning problems, including a PTAS for a strongly NP-hard case.

Findings

Polynomial-time algorithms for some objectives.

NP-hardness proofs for others.

Existence of a PTAS for a specific NP-hard case.

Abstract

This paper studies optimal matroid partitioning problems for various objective functions. In the problem, we are given a finite set $E$ and $k$ weighted matroids $(E, \mathcal{I}_i, w_i)$, $i = 1, \dots, k$, and our task is to find a minimum partition $(I_1,\dots,I_k)$ of $E$ such that $I_i \in \mathcal{I}_i$ for all $i$. For each objective function, we give a polynomial-time algorithm or prove NP-hardness. In particular, for the case when the given weighted matroids are identical and the objective function is the sum of the maximum weight in each set (i.e., $\sum_{i=1}^k\max_{e\in I_i}w_i(e)$), we show that the problem is strongly NP-hard but admits a PTAS.