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Abstract:
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.
