A New Approach to Online Scheduling: Approximating the Optimal Competitive Ratio

Abstract

We propose a new approach to competitive analysis in online scheduling by introducing the novel concept of competitiveratio approximation schemes. Such a scheme algorithmically constructs an online algorithm with a competitive ratio arbitrarily close to the best possible competitive ratio for any online algorithm. We study the problem of scheduling jobs online to minimize the weighted sum of completion times on parallel, related, and unrelated machines, and we derive both deterministic and randomized algorithms which are almost best possible among all online algorithms of the respective settings. We also generalize our techniques to arbitrary monomial cost functions and apply them to the makespan objective. Our method relies on an abstract characterization of online algorithms combined with various simplifications and transformations. We also contribute algorithmic means to compute the actual value of the best possible competitive ratio up to an arbitrary accuracy. This strongly contrasts (nearly) all previous manually obtained competitiveness results and, most importantly, it reduces the search for the optimal competitive ratio to a question that a computer can answer. We believe that our concept can also be applied to many other problems and yields a new perspective on online algorithms in general.

Online scheduling on parallel machines: A survey

“…2013 年, Gunther 等 [123] 在对实时在线排序问题的研究中, 提出了竞争比近似方案 (competitive ratio approximate scheme, CRAS) 的概念. 在线算法族 {A ε } 称为在线问题的竞争比近似方案, 若对任 意的 ε > 0, 算法 A ε 的竞争比至多为 (1 + ε)ρ * , 其中 ρ * 为该在线问题最好算法的竞争比.…”

Online scheduling with multi‐state machines

“…The goal in this line of research is to derive online algorithms with the smallest competitive ratios. Recently, Günther [20] develop an algorithmic approach that can approximate the bestpossible competitive ratios for many versions of these problems. We note that the solution techniques described in most of the papers cited above cannot be adapted to fit our problems because the effect of the machine states on the processing time is not taken into account.…”

“…. Due to Inequality (20), C N + T 0 , the objective of LP rand α,β (N), is an upper bound on the competitive ratio of RPAC α,β . Therefore, what is left is to show that the linear constraints are all valid for all problem instances I. Lemma 3.8 proves Constraints (2a) and (2b).…”

Competitive-Ratio Approximation Schemes for Minimizing the Makespan in the Online-List Model

“…Competitive-ratio approximation schemes were introduced by Günther et al in [18]. They focussed on scheduling problems in the online-time model and provide such schemes for various scheduling problems Pm| r j , (pmtn) | w j C j , Qm| r j , (pmtn) | w j C j (assuming a constant range of machine speeds without preemption), and Rm| r j , pmtn | w j C j .…”

“…Only recently, the concept of competitive-ratio approximation schemes was introduced in [18]. Such an approximation scheme is a procedure that computes a nearly optimal online algorithm and at the same time provides a nearly exact estimate of the optimal competitive ratio.…”