Online Makespan Scheduling with Sublinear Advice

Abstract

Online algorithms are of importance for many practical applications. Typical examples involve scheduling and routing algorithms employed in operating systems and computer networks. Beside their practical significance, online algorithms are extensively studied from a theoretical point of view. Due to the nature of online problems, these algorithms fail to be optimal in general. The quality of their output is usually measured by the so called competitive ratio. A few years ago, another measure, the advice complexity, has been introduced to give a deeper insight into the nature of an online problem and to capture the essential information contained in the instances of the problem.The makespan scheduling problem is a generic scheduling problem where the task is to schedule jobs with different processing times on identical machines. The objective is to distribute the processing times as evenly as possible among the machines. This is why this problem is sometimes referred to as the load balancing problem. When considering the online version of the problem, the jobs arrive gradually and have to be assigned to a machine immediately at their arrival. In the work at hand, we study the advice complexity of the online makespan scheduling problem restricted to two machines.For this problem it proves to be that already little advice helps a great deal to improve the competitive ratio in comparison to purely deterministic online algorithms. Therefore, this work mainly focuses on upper bounds. First, we give linear upper and lower bounds on the advice complexity for optimal algorithms with advice, with the bounds being tight up to one bit. Afterwards, we introduce a very simple online algorithm with logarithmic advice complexity and a competitive ratio of 4 3 . We also show that with a more involved strategy an online algorithm can get arbitrarily close to this competitive ratio using only a constant number of advice bits. As the main result we prove that even a constant number of advice bits is sufficient to achieve an almost optimal solution.

Reordering buffer management with advice

“…In the semi-online advice model, the advice consisting of O(log(1/ε)/ε) log(m/ε) advice bits in total would indicate the index of the best schedule. This result was rediscovered by Dohrau [14] explicitly for the semi-online advice model. Dorrigiv et al…”

“…Specifically, for any ε > 0, the algorithm has a competitive ratio of 1 + ε, using O(log(1/ε)) advice bits per request, i.e., it is a LAAS. Unlike for other problems where a LAAS is known [26,3,14], there is no offline PTAS known for the reordering buffer management problem. In fact, our result can be interpreted as an offline (1 + ε)-approximation algorithm with a running time of 2 O(n log 1/ε) (see Conclusions).…”

Online algorithms with advice for bin packing and scheduling problems

“…To the best of our knowledge, the only scheduling problems studied to date in the framework of online computation with advice is a special case of the job shop scheduling problem [BKK + 09, KK11] , and, makespan scheduling on identical machines in [Doh15]. In both cases, the semionline advice model is used.…”

“…In both cases, the semionline advice model is used. In [Doh15], an algorithm that is (1 + ε)-competitive and uses advice of constant size in total is presented. Boyar et al [BKLL14] studied the bin packing problem with advice, using the semi-online advice model of [BKK + 09] and presented a 3/2-competitive algorithm, using log n + o(log n) bits of advice in total, and a (4/3 + ε)-competitive algorithm, using 2n + o(n) bits of advice in total, where n is the length of the request sequence.…”

A Technique to Obtain Hardness Results for Randomized Online Algorithms – A Survey

“…Since then, new results on job shop scheduling [28], the k-server problem [8,23,32], and disjoint path allocation [2] were obtained. Additionally, many other online problems were studied including buffer management [14], online set cover [29], string guessing [7], graph exploration [17], online independent set [15], online knapsack [10], online makespan scheduling [19,33], online bin packing [12,33], online Steiner tree [1], list update [13] and online graph coloring [3,4,22,34]. Online algorithms using both advice and randomization were investigated by Böckenhauer et al [6].…”