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Minimizing Completion Times of Stochastic Jobs on Parallel Machines is Hard

Benjamin Moseley, Kirk Pruhs, Marc Uetz, Rudy Zhou

TL;DR

The paper addresses the hardness of stochastic scheduling on parallel identical machines with the objective $\mathbb{E}[\sum_j w_j C_j]$. It develops reductions from knapsack counting to show that evaluating and optimizing under common stochastic policies (WSEPT and SEPT) are #P-hard, even when processing times are restricted to independent two-point distributions and weights are equal. The results provide the first hardness proofs for scheduling independent stochastic jobs with a min-sum objective outside reliance on deterministic hardness, and imply that computing the exact minimum or even the exact cost of fixed policies is intractable in general. This establishes a clear computational barrier and suggests that efficient polynomial-time solutions are unlikely unless #P = P, with significant implications for theory and practice in stochastic scheduling and approximation approaches.

Abstract

This paper considers the scheduling of stochastic jobs on parallel identical machines to minimize the expected total weighted completion time. While this is a classical problem with a significant body of research on approximation algorithms over the past two decades, constant-factor performance guarantees are currently known only under very restrictive assumptions on the input distributions, even when all job weights are identical. This algorithmic difficulty is striking given the lack of corresponding complexity results: to date, it is conceivable that the problem could be solved optimally in polynomial time. We address this gap with hardness results that demonstrate the problem's inherent intractability. For the special case of discrete two-point processing time distributions and unit weights, we prove that deciding whether there exists a scheduling policy with expected cost at most a given threshold is #P-hard. Furthermore, we show that evaluating the expected objective value of the standard (W)SEPT greedy policy is itself #P-hard. These represent the first hardness results for scheduling independent stochastic jobs and min-sum objective that do not merely rely on the intractability of the underlying deterministic counterparts.

Minimizing Completion Times of Stochastic Jobs on Parallel Machines is Hard

TL;DR

The paper addresses the hardness of stochastic scheduling on parallel identical machines with the objective . It develops reductions from knapsack counting to show that evaluating and optimizing under common stochastic policies (WSEPT and SEPT) are #P-hard, even when processing times are restricted to independent two-point distributions and weights are equal. The results provide the first hardness proofs for scheduling independent stochastic jobs with a min-sum objective outside reliance on deterministic hardness, and imply that computing the exact minimum or even the exact cost of fixed policies is intractable in general. This establishes a clear computational barrier and suggests that efficient polynomial-time solutions are unlikely unless #P = P, with significant implications for theory and practice in stochastic scheduling and approximation approaches.

Abstract

This paper considers the scheduling of stochastic jobs on parallel identical machines to minimize the expected total weighted completion time. While this is a classical problem with a significant body of research on approximation algorithms over the past two decades, constant-factor performance guarantees are currently known only under very restrictive assumptions on the input distributions, even when all job weights are identical. This algorithmic difficulty is striking given the lack of corresponding complexity results: to date, it is conceivable that the problem could be solved optimally in polynomial time. We address this gap with hardness results that demonstrate the problem's inherent intractability. For the special case of discrete two-point processing time distributions and unit weights, we prove that deciding whether there exists a scheduling policy with expected cost at most a given threshold is #P-hard. Furthermore, we show that evaluating the expected objective value of the standard (W)SEPT greedy policy is itself #P-hard. These represent the first hardness results for scheduling independent stochastic jobs and min-sum objective that do not merely rely on the intractability of the underlying deterministic counterparts.
Paper Structure (6 sections, 11 theorems, 8 equations, 1 figure)

This paper contains 6 sections, 11 theorems, 8 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Results
  3. Hardness of evaluating WSEPT and SEPT
  4. Hardness of WSEPT
  5. Hardness of SEPT
  6. Stochastic scheduling is presumably hard(er)

Key Result

Lemma 1

The problem to count the number of feasible solutions for any instance $(B,s_1,\dots,s_n)$ of problem Knapsack is a #P-hard problem, and this is true also if we assume that $B+1 <\sum_{j\in[n]} s_j\le 3B/2$.

Figures (1)

  • Figure 1: Illustration for an instance with $n=m=9$: Schedules for the two scheduling instances with blocker jobs (blue), knapsack jobs (striped), and dummy job (tiled) for some NO realization of the processing times. Note that except for the knapsack job on machine no. 2, all start times are identical.

Theorems & Definitions (21)

  • Definition 1: Knapsack
  • Lemma 1
  • Theorem 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Theorem 2
  • proof
  • ...and 11 more