The local–global conjecture for scheduling with non-linear cost

Abstract

We consider the classical scheduling problem on a single machine, on which we need to schedule sequentially n given jobs. Every job j has a processing time p j and a priority weight w j , and for a given schedule a completion time C j . In this paper we consider the problem of minimizing the objective value j w j C β j for some fixed constant β > 0. This non-linearity is motivated for example by the learning effect of a machine improving its efficiency over time, or by the speed scaling model. For β = 1, the well-known Smith's rule that orders job in the non-increasing order of w j /p j give the optimum schedule. However, for β = 1, the complexity status of this problem is open. Among other things, a key issue here is that the ordering between a pair of jobs is not well-defined, and might depend on where the jobs lie in the schedule and also on the jobs between them. We investigate this question systematically and substantially generalize the previously known results in this direction. These results lead to interesting new dominance properties among schedules which lead to huge speed up in exact algorithms for the problem. An experimental study evaluates the impact of these properties on the exact algorithm

On Submodular Search and Machine Scheduling

“…Indeed, without any restrictions on h, it is a difficult to believe anything can be said in general. If there are no precedence constraints and h(C j ) = C β j , β ≥ 0, this is the problem 1|| w j C β j , as studied in [10], in which it is shown that the problem of minimizing total weighted completion time plus total energy requirement (see [17,37]) can be reduced to 1|| w j C β j , β ∈ (0, 1). We discuss the problem 1|| w j h(C j ) further in Subsection 2.4, in which we bound the approximation ratio of our algorithm by a simple expression in terms of h.…”

“…We may also consider the more specific problem 1|| w j h(C j ) for some non-decreasing function h. In [37], a polynomial time approximation scheme is given for the problem. If h(C j ) = C β j for β = 1, then it is the problem considered in [10]. It is unknown whether there exists a polynomial time algorithm to compute an optimal schedule for this problem, or if it is NP hard (see [10] and the references therein).…”

“…If h(C j ) = C β j for β = 1, then it is the problem considered in [10]. It is unknown whether there exists a polynomial time algorithm to compute an optimal schedule for this problem, or if it is NP hard (see [10] and the references therein). For concave or convex h, it is shown in [52] that Smith's rule (for the original problem 1|| j w j C j ) yields a ( √ 3 + 1)/2 ≈ 1.37approximation to the problem 1|| j w j h(C j ), while [28] gives explicit formulas for the exact approximation ratio of Smith's rule for this problem in terms of a maximization over two continuous variables.…”

A Fast Exact Algorithm for Airplane Refueling Problem

“…Following a long series of improvements [2,6,12,15,16], Drr and Vsquez [7] conjectured that for all cost functions of the form f j (t) = w j t β , β > 0 and all jobs i, j, i ≺ l j implies i ≺ g j. Latter, Bansal et al [4] confirmed this conjecture, and they also gave a counter example of the generalized conjecture that i ≺ l[a,b] j implies i ≺ g[a,b] j. For airplane refueling problem, Vsquez [17]…”