Interruptible algorithms for multiproblem solving

Abstract

In this paper we address the problem of designing an interruptible system in a setting in which n problem instances, all equally important, must be solved concurrently. The system involves scheduling executions of contract algorithms (which offer a trade-off between allowable computation time and quality of the solution) in m identical parallel processors. When an interruption occurs, the system must report a solution to each of the n problem instances. The quality of this output is then compared to the best-possible algorithm that has foreknowledge of the interruption time and must, likewise, produce solutions to all n problem instances. This extends the well-studied setting in which only one problem instance is queried at interruption time.In this work we first introduce new measures for evaluating the performance of interruptible systems in this setting. In particular, we propose the deficiency of a schedule as a performance measure that meets the requirements of the problem at hand. We then present a schedule whose performance we prove that is within a small factor from optimal in the general, multiprocessor setting. We also show several lower bounds on the deficiency of schedules on a single processor. More precisely, we prove a general lower bound of (n + 1)/n, an improved lower bound for the two-problem setting (n = 2), and a tight lower bound for the class of round-robin schedules. Our techniques can also yield a simpler, alternative proof of the main result of Bernstein et al. [4] concerning the performance of cyclic schedules in multiprocessor environments.

Competitive Sequencing with Noisy Advice

“…for all 1 ≤ k < n. This follows from an induction argument on k. Indeed, for a given n ≥ 2, the case k = 1 is a mere reformulation of (6). Then, in view of (7), assuming that (8) holds for some 1 ≤ k ≤ n − 2, we obtain…”

“…Beyond these simple settings, some of the above problems, and most notably contract scheduling, have been studied under more complex settings, for which optimizing the corresponding performance measure becomes much more challenging. This includes settings in which one must schedule executions of the contract algorithm for several problem instances, as well as the setting in which the executions can be scheduled in several parallel processors, see, e.g., [12,35,11,24,7,6,4,21]. All these works show theoretical upper and lower bounds on the performance of the corresponding schedules.…”

“…To achieve this result, we introduce a new application of Gal's functional theorem [19]. This is the main tool for showing lower bounds in search theory but also in contract scheduling: see, e.g., Chapters 7 and 9 in [1] and [25,32], for its applications in search theory, as well as [24,7,6,4] for its applications in contract scheduling. Note that all previous work has applied this theorem in a "black-box" manner: specifically, a typical application of this theorem gives a lower bound to the performance of the algorithms, as a function of a parameter α X , that depends on the sequence X (see Theorem 11 for details); say that f (α X ) is this lower bound expression.…”

Further connections between contract-scheduling and ray-searching problems

“…However, it is possible to extend the setting, and the definition of the acceleration ratio, so as to model schedules of contract algorithms in multiple processors. Bernstein et al (2002) were the first to introduce and study such an extension in multiple, identical processors, followed by López-Ortiz et al (2014) and Angelopoulos and López-Ortiz (2009).…”

Contract Scheduling With Predictions

“…[6] considered the problem in which the interruption is not a fixed deadline, but there is a "grace period" within which the system is allowed to complete the execution of the contract. Measures alternative to the acceleration ratio were proposed and studied in [5]. More recently, [4] studied contract scheduling in the setting in which the schedule is deemed "complete" once a contract reaches some prespecified end guarantees.…”