Non-Approximability Results for Scheduling Problems with Minsum Criteria

Abstract

We provide several non-approximability results for deterministic scheduling problems whose objective is to minimize the total job completion time. Unless P = N P, none of the problems under consideration can be approximated in polynomial time within arbitrarily good precision. Most of our results are derived by APX-hardness proofs. We show that, whereas scheduling on unrelated machines with unit weights is polynomially solvable, the problem becomes APX-hard if release dates or weights are added. We further show APX-hardness for scheduling in flow shops, job shops, and open shops. We also investigate the problems of scheduling on parallel machines with precedence constraints and unit processing times, and two variants of the latter problem with unit communication delays; for these problems we provide lower bounds on the worst-case behavior of any polynomial-time approximation algorithm through the gap reduction technique.

Fifty years of scheduling: a survey of milestones

“…They show that for any of these problems with integer processing times, there is a polynomialtime algorithm that determines whether a schedule S with C max (S) = 3 exists; by contrast, unless P = N P, it is impossible to verify in polynomial time whether there exists a schedule S with C max (S) = 4. Thus, the existence of a polynomial-time approximation algorithm with a worst-case ratio strictly less than 5/4 would imply that P = N P. Hoogeveen et al (2001) demonstrate that many scheduling problems with a variable number of machines to minimize the sum of the completion times (including the flow shop and the open shop) are Max SNP-hard, and this implies that for any of these problems the existence of a PTAS would imply that P = N P.…”

Combinatorial Algorithms for Data Migration to Minimize Average Completion Time

“…This problem is a special case of the data migration problem [7]. Open shop scheduling problem has been studied in [3,12,20,21].…”

Improved Bounds for Flow Shop Scheduling

“…Queyranne & Sviridenko [15] showed that an approximation algorithm for the above mentioned problems that produces a schedule with makespan a factor O(ρ) away from the lower bound lb can be used to obtain a O(ρ)-approximation algorithms for other objectives, including the sum of weighted completion times. The only known inapproximability result is by Hoogeveen, Schuurman & Woeginger [7], who showed that F || C j is NP-hard to approximate within a ratio better than 1 + for some small > 0.…”