A Submodular Optimization Approach to Bicriteria Scheduling Problems with Controllable Processing Times on Parallel Machines

Abstract

In this paper, we present a general methodology for designing polynomial-time algorithms for bicriteria scheduling problems on parallel machines with controllable processing times. For each considered problem, the two criteria are the makespan and the total compression cost and the solution is delivered in the form of the breakpoints of the e¢cient frontier. We reformulate the scheduling problems in terms of optimization over submodular polyhedra and give e¢cient procedures for computing the corresponding rank functions. As a result, for two of the considered problems we obtain the …rst polynomial-time algorithms, while for the third problem we considerably improve the known running time.

Handling Scheduling Problems with Controllable Parameters by Methods of Submodular Optimization

“…Theorem 2 provides the foundation to an approach that finds the efficiency frontier of the bicriteria scheduling problems Q|p(j) = p(j) − x(j), pmtn| (C max , W ) and αm|r(j), p(j) = p(j) − x(j), pmtn| (C max , W ) with α ∈ {P, Q} in a closed form [32].…”

“…(18) Thus, in order to be able to compute the (piecewise-linear) function W (d), we first have to compute the functions ψ t (d), 1 ≤ t ≤ n, for all relevant values of d. It is shown in [32], that after the functions ψ t (d), 1 ≤ t ≤ n, for all relevant values of d are found, their weighted sum by (18) can be computed in O (nm log n) time. This (piecewise-linear) function W (d) fully defines the efficiency frontier for the corresponding bicriteria scheduling problem.…”

“…For problem Q|r(j), pmtn|C max , an algorithm for that requires O(mn + n log n) time is due to [24]. Prior to work of our team on the links between the SCPT problems and SO [32], no purpose-built algorithms had been known for problems Q|p(j) = p(j) − x(j), pmtn, C(j) ≤ d|W and α|r(j), p(j) = p(j) − x(j), pmtn, C(j) ≤ d|W with α ∈ {P, Q}, as well as for their bicriteria counterparts. We consider these problems in Section 4.3 and Section 4.2, respectively.…”

“…As a result, a powerful toolkit of the SO techniques [5,26] can be used for design and justification of algorithms for solving a wide range of the SCPT problems. In this paper, we present convincing examples of a positive mutual influence of scheduling and submodular optimization, mainly based on our recent work [32]- [35].…”

Machine Speed Scaling by Adapting Methods for Convex Optimization with Submodular Constraints

“…This "step change" research allows us to develop a common toolkit for solving scheduling problems of a similar nature. The success of this new methodology for the CPT models has been demonstrated in a series of papers Strusevich 2005, 2008;Shakhlevich et al 2009;Shioura et al 2013Shioura et al , 2015Shioura et al , 2016. As a result, powerful methods of submodular optimization have been used to develop and justify the fastest available algorithms for both single criterion and bicriteria problems with CPT.…”