Stochastic scheduling with Bernoulli-type jobs through policy stratification

TL;DR

This work studies scheduling $N$ independent Bernoulli-type jobs on $m$ identical machines to minimize the expected total completion time. It introduces a stratified scheduling framework that transforms an optimal non-anticipatory policy into a policy that starts jobs only at predefined time points, with controlled delays, enabling a polynomial-time dynamic-programming solution when the number of distinct job sizes is constant. The main theoretical contributions are a PTAS for the Bernoulli-job setting under a constant number of distinct sizes and a quasi-polynomial $O(\log N)$-approximation for the unrestricted case, built on a sequence of phase-based transformations and grouping techniques. The results significantly reduce the complexity gap in stochastic parallel-machine scheduling by showing near-optimality achievable with structured, tractable policies, and they provide concrete algorithmic pathways (DP, grouping, and time-grid stratification) to compute such policies. The findings have implications for practical stochastic scheduling where processing times are uncertain but Bernoulli-typed, offering near-optimal performance guarantees with feasible computation time.

Abstract

This paper addresses the problem of computing a scheduling policy that minimizes the total expected completion time of a set of $N$ jobs with stochastic processing times on $m$ parallel identical machines. When all processing times follow Bernoulli-type distributions, Gupta et al. (SODA '23) exhibited approximation algorithms with an approximation guarantee $\tilde{\text{O}}(\sqrt{m})$, where $m$ is the number of machines and $\tilde{\text{O}}(\cdot)$ suppresses polylogarithmic factors in $N$, improving upon an earlier ${\text{O}}(m)$ approximation by Eberle et al. (OR Letters '19) for a special case. The present paper shows that, quite unexpectedly, the problem with Bernoulli-type jobs admits a PTAS whenever the number of different job-size parameters is bounded by a constant. The result is based on a series of transformations of an optimal scheduling policy to a "stratified" policy that makes scheduling decisions at specific points in time only, while losing only a negligible factor in expected cost. An optimal stratified policy is computed using dynamic programming. Two technical issues are solved, namely (i) to ensure that, with at most a slight delay, the stratified policy has an information advantage over the optimal policy, allowing it to simulate its decisions, and (ii) to ensure that the delays do not accumulate, thus solving the trade-off between the complexity of the scheduling policy and its expected cost. Our results also imply a quasi-polynomial $\text{O}(\log N)$-approximation for the case with an arbitrary number of job sizes.

Figures (4)

• Figure 1: A specific machine under schedules $\hbox{${\mathcal{S}}$}$ and $\hbox{${\mathcal{S}}$}^1$ respectively. Blue jobs have length $p_1$, red jobs have length $p_2$, and the little circles denote zero length jobs. Note that the order of circles does not matter because we are depicting a schedule, not a policy. In our figures, $\varepsilon=1/8$, and $p_2=p_1/80$.
• Figure 2: A specific machine with respect to $\hbox{${\mathcal{S}}$}^1$ and $\hbox{${\mathcal{S}}$}^2$. Note the filling of type 2-spaces.
• Figure 3: Schedule $\hbox{${\mathcal{S}}$}^3$. Note how the interval size has increased and how each job starts at the same distance from the left endpoints in the corresponding intervals under $\hbox{${\mathcal{S}}$}^2$ and $\hbox{${\mathcal{S}}$}^3$.
• Figure 4: Schedule $\hbox{${\mathcal{S}}$}^4$. The first block of type-2 jobs corresponds to spaces created due to type-2 jobs scheduled earlier under $\hbox{${\mathcal{S}}$}^3$. The second block of type-2 jobs is moved as early as possible in the interval. (Note that job identities may change, because that happens simultaneously on all machines.) Also note that the relative order of type-1 jobs remains the same, and that such jobs only start on left-endpoints of intervals.

Theorems & Definitions (138)

• Theorem 1.1
• Theorem 1.2
• Lemma 2.0
• Lemma 2.1: Lemma 2.2 in GMZ2023
• Definition 2.2: Filling
• Theorem 2.3
• Definition 3.1: Stratified policy for two types
• Lemma 3.1
• Lemma 3.1
• Lemma 3.1