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.
Highlights
Minimum makespan scheduling is one of the foundational problems in the literature on approximation algorithms [6, 7]
We show how the structure can be applied to the minimum makespan problem on related machines and to a larger class of objective functions on parallel machines
72:2 Closing the Gap for Makespan Scheduling via Sparsification Techniques characterized by a non-negative processing time pj ∈ Z>0
Summary
Minimum makespan scheduling is one of the foundational problems in the literature on approximation algorithms [6, 7]. The load of a machine is the total processing time of jobs assigned to it, and our objective is to minimize the makespan, that is, the maximum machine load This problem is usually denoted P ||Cmax. If we aim for a (1 + ε)-approximate solution, for some ε > 0, we can assume that all processing times are integral and T is a constant number, namely T ∈ O(1/ε2) This can be achieved with well known rounding and scaling techniques [1, 2, 8] which will be specified later. In this article we derive new insights on this ILP that help us to design faster algorithms for P ||Cmax and other more general problems These including makespan scheduling on related machines Q||Cmax, and a more general class of objective functions on parallel machines.
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