Strong LP formulations for scheduling splittable jobs on unrelated machines

Abstract

A natural extension of the makespan minimization problem on unrelated machines is to allow jobs to be partially processed by different machines while incurring an arbitrary setup time. In this paper we present increasingly stronger LP-relaxations for this problem and their implications on the approximability of the problem. First we show that the straightforward LP, extending the approach for the original problem, has an integrality gap of 3 and yields an approximation algorithm of the same factor. By applying a lift-and-project procedure, we are able to improve both the integrality gap and the implied approximation factor to 1 + φ, where φ is the golden ratio. Since this bound remains tight for the seemingly stronger machine configuration LP, we propose a new job configuration LP that is based on an infinite continuum of fractional assignments of each job to the machines. We prove that this LP has a finite representation and can be solved in polynomial time up to any accuracy. Interestingly, we show that our problem cannot be approximated within a factor better than e e−1 ≈ 1.582 (unless P = N P), which is larger than the inapproximability bound of 1.5 for the original problem.Mathematics Subject Classification Primary 90B35 · 68W25; Secondary 68Q25 · 90C10

Malleable scheduling beyond identical machines

“…All our approximation results imply corresponding upper bounds on the integrality gap of the linear programming relaxation [LP(C)]. Based on an adaptation of a construction in [4], we show a lower bound of 1 + ϕ ≈ 2.618 on the integrality gap of [LP(C)] for malleable job scheduling on unrelated machines, where ϕ is the golden ratio. For the cases of restricted assignment and uniformly related machines, respectively, we obtain an integrality gap of 2.…”

“…Again, we assume oracle access to the processing time functions. 4 The notion of speed-implementable processing times allows us to quantify the fundamental assumptions of monotonicity and diminishing utility in a clean and natural way. More specifically, we make the following two assumptions on speed-implementable functions: 1.…”

“…If a fraction x ij ∈ (0, 1] of job j is to be scheduled on machine i, the load that is incurred on the machine is s ij + p ij x ij . Correa et al [4] provide an (1 + ϕ)-approximation algorithm for this setting (where ϕ is the golden ratio), which is based on an adaptation of the classic LP rounding result by Lenstra, Shmoys, and Tardos [16] for the traditional unrelated machine scheduling problem. We remark that the generalized malleable setting considered in this paper also induces a natural generalization of the splittable setting beyond setup times, when dropping the requirement that jobs need to be executed in unison.…”

“…We remark that the generalized malleable setting considered in this paper also induces a natural generalization of the splittable setting beyond setup times, when dropping the requirement that jobs need to be executed in unison. As D. Fotakis, J. Matuschke, and O. Papadigenopoulos 17:5 in [4], we provide a rounding framework based on a variant of the assignment LP from [16]. However, the fact that processing times are only given implicitly as functions in our setting makes it necessary to very carefully choose the coefficients of the assignment LP in order to ensure a constant integrality gap.…”

“…Specifically, we introduce the notion of critical speed and use the critical speed to define the load coefficients incurred by large jobs on machines in the LP relaxation by proportionally distributing the work volume according to machine speeds. For the rounding, we exploit the sparsity of our relaxation's extreme points (as in [16]) and generalize the approach of [4], in order to carefully distinguish between jobs assigned to a single machine and jobs shared by multiple machines.…”