Generalizing many well-known and natural scheduling problems, scheduling with
job-specific cost functions has gained a lot of attention recently. In this
setting, each job incurs a cost depending on its completion time, given by a
private cost function, and one seeks to schedule the jobs to minimize the total
sum of these costs. The framework captures many important scheduling objectives
such as weighted flow time or weighted tardiness. Still, the general case as
well as the mentioned special cases are far from being very well understood
yet, even for only one machine. Aiming for better general understanding of this
problem, in this paper we focus on the case of uniform job release dates on one
machine for which the state of the art is a 4-approximation algorithm. This is
true even for a special case that is equivalent to the covering version of the
well-studied and prominent unsplittable flow on a path problem, which is
interesting in its own right. For that covering problem, we present a
quasi-polynomial time (1+ϵ)-approximation algorithm that yields an
(e+ϵ)-approximation for the above scheduling problem. Moreover, for
the latter we devise the best possible resource augmentation result regarding
speed: a polynomial time algorithm which computes a solution with \emph{optimal
}cost at 1+ϵ speedup. Finally, we present an elegant QPTAS for the
special case where the cost functions of the jobs fall into at most logn
many classes. This algorithm allows the jobs even to have up to logn many
distinct release dates.Comment: 2 pages, 1 figur