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

TL;DR

This work addresses the General Scheduling Problem (GSP), unifying several preemptive single-machine objectives by allowing job-dependent nondecreasing costs of completion time. The authors deliver a quasi-polynomial-time $(2+\varepsilon)$-approximation for GSP (assuming quasi-polynomially bounded processing times) and improve to a $1+\varepsilon$-approximation for weighted tardiness, via a reduction to a geometric rectangle covering problem (RCP). The reduction discretizes cost growth with milestone sequences and translates scheduling decisions into selecting prefixes of non-overlapping rectangles, which are solved recursively by partitioning the plane into left and right subproblems and handling crossing rows with carefully designed row types and artificial rays. Central to the approach is proving that an optimal GSP solution can be transformed into a near-optimal, structured RCP solution, enabling quasi-polynomial-time branching and solving subproblems independently. The results advance constant-factor approximation goals for GSP and provide near-optimal, scalable methods for tardiness under heterogeneous costs, with implications for both theory and practice in scheduling under complex cost structures.

Abstract

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.

Figures (2)

• Figure 1: Rectangles and rays in an instance of the rectangle covering problem to which we reduce GSP.
• Figure 2: The shaded region indicates a given subproblem. It is split into two parts by the blue dotted line. We lose a factor of $2$ for the centered rows crossing this line in this step, but not for the other rows.

Theorems & Definitions (66)

• Theorem 1
• Theorem 2
• Lemma 3
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• Lemma 5
• Lemma 6
• proof
• Lemma 7
• proof
• Lemma 8