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.
Highlights
The matroid partitioning problem is one of the most fundamental problems in combinatorial optimization
We study weighted versions of the matroid partitioning problem
For any feasible partition (I1, . . . , Ik) for (E, Ii)i∈[k], it is an optimal solution for the minimum (Op(1), Op(2))-value matroid partitioning problem instance (E, (Ii, wi)i∈[k]) if and only if it is optimal for the maximum (Op(1), Op(2))-value matroid partitioning problem instance (E, (Ii, wi)i∈[k]), where wmax = maxi∈[k] maxe∈E wi(e)
Summary
The matroid partitioning problem is one of the most fundamental problems in combinatorial optimization. Special cases of the minimum (max, )-value matroid partitioning problem have been extensively addressed in the scheduling literature under the name of the minimum makespan scheduling Since this problem is NP-hard, many papers have proposed polynomial-time approximation algorithms. Approximation algorithms for the maximum (min, )-value matroid partitioning problem are well-studied, see, e.g., [3, 13, 25, 20]. For the (max, min) and ( , min) cases, we give polynomial-time algorithms when the matroids and weights are respectively identical, and prove strong NP-hardness even to approximate for the general case. Due to the space limitation, we omit proofs of some results, which are found in [14]
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