Closing the Gap for Makespan Scheduling via Sparsification Techniques

Abstract

Makespan scheduling on identical machines is one of the most basic and fundamental packing problems studied in the discrete optimization literature. It asks for an assignment of n jobs to a set of m identical machines that minimizes the makespan. The problem is strongly NP-hard, and thus we do not expect a ([Formula: see text])-approximation algorithm with a running time that depends polynomially on [Formula: see text]. It has been recently shown that a subexponential running time on [Formula: see text] would imply that the Exponential Time Hypothesis (ETH) fails. A long sequence of algorithms have been developed that try to obtain low dependencies on [Formula: see text], the better of which achieves a quadratic running time on the exponent. In this paper we obtain an algorithm with an almost-linear dependency on [Formula: see text] in the exponent, which is tight under ETH up to logarithmic factors. Our main technical contribution is a new structural result on the configuration-IP integer linear program. More precisely, we show the existence of a highly symmetric and sparse optimal solution, in which all but a constant number of machines are assigned a configuration with small support. This structure can then be exploited by integer programming techniques and enumeration. We believe that our structural result is of independent interest and should find applications to other settings. We exemplify this by applying our structural results to the minimum makespan problem on related machines and to a larger class of objective functions on parallel machines. For all these cases, we obtain an efficient PTAS with running time with an almost-linear dependency on [Formula: see text] and polynomial in n.

Tightness of Sensitivity and Proximity Bounds for Integer Linear Programs

“…These ILPs often arise in the context of algorithmic applications. Probably the most famous one among such ILPs is the configuration ILP introduced by Gilmore and Gomory [10] and used for many packing and scheduling problems (e. g. [3,11,[17][18][19]).…”

A Log-Linear $$(2 +5/6)$$-Approximation Algorithm for Parallel Machine Scheduling with a Single Orthogonal Resource

“…A classic result on polynomial-time approximation scheme (PTAS) for P||C max was presented in [11]. In general, an efficient polynomial-time approximation scheme (EPTAS) for the Q||C max problem (with uniform machines) is known [12,13]. As the P||C max problem is strongly NP-Hard, there exists no fully polynomial-time approximation scheme unless P = NP.…”

Symmetry Exploitation for Online Machine Covering with Bounded Migration

“…Unlike other standard rounding techniques (e.g. [19,13]), the rounded sizes do not depend on OPT (or UB). This avoids possible migrations provoked by new rounded values, greatly simplifying our techniques.…”

“…Unlike known techniques used in previous work that yield similar results (see e.g. [13]), our rounding is well suited for online algorithms and helps simplifying the analysis as it does not depend on OPT (which varies through iterations).…”