TY - JOUR

T1 - Strong LP formulation for scheduling splittable jobs on unrelated machines

AU - Correa, J.

AU - Marchetti-Spaccamela, A.

AU - Matuschke, J.

AU - Stougie, L.

AU - Svensson, O.

AU - Verdugo, V.

AU - Verschae, J.

PY - 2015

Y1 - 2015

N2 - 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 (Formula presented.), where (Formula presented.) 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 (Formula presented.), which is larger than the inapproximability bound of 1.5 for the original problem.

AB - 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 (Formula presented.), where (Formula presented.) 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 (Formula presented.), which is larger than the inapproximability bound of 1.5 for the original problem.

U2 - 10.1007/s10107-014-0831-8

DO - 10.1007/s10107-014-0831-8

M3 - Article

VL - 154

SP - 305

EP - 328

JO - Mathematical Programming

JF - Mathematical Programming

SN - 0025-5610

IS - 1

ER -