How Unsplittable-Flow-Covering helps Scheduling with Job-Dependent Cost Functions

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 $(1+\epsilon)$-approximation algorithm that yields an $(e+\epsilon)$-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 $1+\epsilon$ speedup. Finally, we present an elegant QPTAS for the special case where the cost functions of the jobs fall into at most $\log n$ many classes. This algorithm allows the jobs even to have up to $\log n$ many distinct release dates.Comment: 2 pages, 1 figur

Competing Provers Yield Improved Karp-Lipton Collapse Results

Via competing provers, we show that if a language A is self-reducible and has polynomial-size circuits then S2(A)=S2. Building on this, we strengthen the Kaemper-AFK Theorem, namely, we prove that if NP subseteq (NP intersect coNP)/poly then the polynomial hierarchy collapses to S2(NP intersect coNP). We also strengthen Yap's Theorem, namely, we prove that if NP subseteq coNP/poly then the polynomial hierarchy collapses to S2(NP). Under the same assumptions, the best previously known collapses were to ZPP(NP) and ZPP(NP(NP)) respectively ([KW98,BCK+94], building on [KL80,AFK89,Kaem91,Yap83]). It is known that S2 subseteq ZPP(NP) [Cai01]. That result and its relativized version show that our new collapses indeed improve the previously known results. Since the Kaemper-AFK Theorem and Yap's Theorem are used in the literature as bridges in a variety of results---ranging from the study of unique solutions to issues of approximation---our results implicitly strengthen all those results