A $(2+\varepsilon)$-approximation algorithm for the general scheduling problem in quasipolynomial time

We study the general scheduling problem (GSP) which generalizes and unifies several well-studied preemptive single-machine scheduling problems, such as weighted flow time, weighted sum of completion time, and minimizing the total weight of tardy jobs. We are given a set of jobs with their processing times and release times and seek to compute a (possibly preemptive) schedule for them on one machine. Each job incurs a cost that depends on its completion time in the computed schedule, as given by a separate job-dependent cost function for each job, and our objective is to minimize the total resulting cost of all jobs. The best known result for GSP is a polynomial time $O(\log\log P)$-approximation algorithm [Bansal and Pruhs, FOCS 2010, SICOMP 2014]. We give a quasi-polynomial time $(2+ε)$-approximation algorithm for GSP, assuming that the jobs' processing times are quasi-polynomially bounded integers. For the special case of the weighted tardiness objective, we even obtain an improved approximation ratio of $1+ε$. For this case, no better result had been known than the mentioned $O(\log\log P)$-approximation for the general case of GSP. Our algorithms use a reduction to an auxiliary geometric covering problem. In contrast to a related reduction for the special case of weighted flow time [Rohwedder, Wiese, STOC 2021][Armbruster, Rohwedder, Wiese, STOC 2023] for GSP it seems no longer possible to establish a tree-like structure for the rectangles to guide an algorithm that solves this geometric problem. Despite the lack of structure due to the problem itself, we show that an optimal solution can be transformed into a near-optimal solution that has certain structural properties. Due to those we can guess a substantial part of the solution quickly and partition the remaining problem in an intricate way, such that we can independently solve each part recursively.