Optimal Matroid Partitioning Problems

Abstract

This paper studies optimal matroid partitioning problems for various objective functions. In the problem, we are given k weighted-matroids on the same ground set. Our goal is to find a feasible partition that minimizes (maximizes) the value of an objective function. A typical objective is the maximum over all subsets of the total weights of the elements in a subset, which is extensively studied in the scheduling literature. Likewise, as an objective function, we handle the maximum/minimum/sum over all subsets of the maximum/minimum/total weight(s) of the elements in a subset. In this paper, we determine the computational complexity of the optimal partitioning problem with the above-described objective functions. Namely, for each objective function, we either provide a polynomial time algorithm or prove NP-hardness. We also discuss the approximability for the NP-hard cases.

Omitted Proof in Theorem 2

Lemma 2

([2]) Let $(E, \mathcal {I})$ be a partition matroid defined by $\mathcal {I} = \{ I \,:\,|I \cap S_i| \le \eta _i \ (i\in [p]) \}$ with weight w, where $(S_1,\dots ,S_p)$ is a partition of E, $\eta _1,\dots ,\eta _p$ are positive integers, and $|S_i|=\eta _i\cdot k$ for each $i\in [p]$. Let $I_{i,j}$ consist of $\eta _i$ elements with the $\eta _i$ largest weights in $S_i\setminus (\bigcup _{h=1}^{j-1}I_{i,h})$. Then $(\bigcup _{i\in [p]} I_{i,1},\dots ,\bigcup _{i\in [p]} I_{i,k})$ is an optimal solution to the minimum $(\sum ,\max )$-value matroid partitioning problem $(E, (\mathcal {I}, w), k)$.

Proof

Let $(I_1^*,\dots ,I_k^*)$ be an optimal partition. Without loss of generality, we may assume that $\max _{e\in I_1^*}w(e)\ge \dots \ge \max _{e\in I_k^*}w(e)$. Let j be any index in [k]. In addition, let $(i', e_j)$ be the pair of an index and an element attaining $\max _{i \in [p]} \max _{e \in I_{i,j}}w(e)$. We claim that $\max _{e \in I^*_j} w(e) \ge w(e_j)$. To show this, we suppose the contrary. We denote $S = \bigcup _{h < j} I_{i', h} \cup \{e_j\}$. Note that $|S| = (j-1)\eta _{i'}+1$ and $w(e) \ge w(e_j)$ for all $e \in S$. Since $(E, \mathcal {I})$ is a partition matroid, at most $(j-1)\eta _{i'}$ elements in S are contained in $I^*_1, \ldots , I^*_{j-1}$. By assumption $\max _{e \in I^*_j} w(e) < w(e_j)$, there is an index $\ell > j$ such that $I^*_{\ell }$ has some element $e' \in I^*_{\ell } \cap S$. Then we have $\max _{e \in I^*_{\ell }}w(e) \ge w(e') \ge w(e_j) > \max _{e\in I^*_j} w(e)$, which contradicts the assumption $\max _{e\in I^*_j} w(e) \ge \max _{e \in I^*_{\ell }}w(e)$.

Thus, we have

$$\begin{aligned} \max _{e\in I_j^*}w(e) \ge w(e_j)=\max _{i\in [p]}\max _{e\in I_{i,j}}w(e)=\max _{e\in \bigcup _{i\in [p]}I_{i,j}}w(e). \end{aligned}$$

Therefore, $(\bigcup _{i\in [p]} I_{i,1},\dots ,\bigcup _{i\in [p]} I_{i,k})$ is also an optimal solution. $\square$