Optimal Matroid Partitioning Problems

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$forall$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.