アイテム詳細

要旨

We consider the single machine job sequencing problem with release dates. The main purpose of this paper is to investigate efficient and effective approximation algorithms with a bicriteria performance guarantee. That is, for some $(\rho_1, \rho_2)$, they find schedules simultaneously within a factor of $\rho_1$ of the minimum total weighted completion times and within a factor of $\rho_2$ of the minimum makespan. The main results of the paper are summarized as follows. First, we present a new $O(n\log n)$ algorithm with the performance guarantee $\left(1+\frac{1}{\beta}, 1+\beta\right)$ for any $\beta \in [0,1]$. For the problem with integer processing times and release dates, the algorithm has the bicriteria performance guarantee $\left(2-\frac{1}{p_{max}}, 2-\frac{1}{p_{max}}\right)$, where $p_{max}$ is the maximum processing time. Next, we study an elegant approximation algorithm introduced recently by Goemans. We show that its randomized version has expected bicriteria performance guarantee $(1.7735, 1.51)$ and the derandomized version has the guarantee $(1.7735, 2-\frac{1}{p_{max}})$. To establish the performance guarantee, we also use two LP relaxations and some randomization techniques as Goemans does, but take a different approach in the analysis, based on a decomposition theorem. Finally, we present a family of bad instances showing that it is impossible to achieve $\rho_1\leq 1.5$ with this LP lower bound.