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

Near-Linear Approximation Algorithms for Scheduling Problems with Batch Setup Times

“…Another remarkable point may be the investigation of unrelated/uniform machines; in fact, there is a known result of Correa et al [2] for the splittable case on unrelated machines. Also we only discussed sequence-independent setups here.…”

“…and this means that the jobs { j [2] | j ∈ C e } do expand the obligatory load L * e outside the large machines of e by exactly (x cks ) e w e , as mentioned before. Turning back to instance I new , we name the cheap load L…”

“…So we split them in the following way. For all jobs j ∈ C * i with i ∈ I * chp we create new job pieces j (1) and j (2) with processing times t…”

“…j ≤ T − 1 2 T = 1 2 T . Due to Lemma 4 we have to schedule a processing time of at least t (2) j of job j ∈ C * i outside the large machines. So we also have an obligatoy setup time s i outside the large machines.…”

“…Remark that i ∈I * chp (x ks ) i w i = Y −(x cks ) e w e so (x cks ) e w e is the time to fill with job load of class e. Therefore, we create new job pieces as follows. For all j ∈ C e let j [1] and j [2] be jobs with processing times t [1] j and t [2] j holding…”