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A Parametrized Complexity View on Robust Scheduling with Budgeted Uncertainty

Noam Goldberg, Dvir Shabtay

TL;DR

The paper studies robust single-machine scheduling under budgeted uncertainty, where each job’s processing time is either the nominal value or a deviation with at most $\Gamma$ deviations allowed. It adopts parameterized complexity to determine when the problem of minimizing tardy jobs is tractable, proving $\mathcal{W}[1]$-hardness in $\Gamma$ but offering an XP algorithm; it also shows NP-hardness for $v_d=2$ while giving polynomial or pseudo-polynomial algorithms for common or a fixed number of due dates, and an $FPT$-time algorithm when the number of nonzero deviations is small. A key contribution is the extension of Moore’s algorithm to robust subproblems and a dualization-based robust knapsack formulation for the common due date case, clarifying tractability frontiers under natural parameters. The findings delineate precise conditions under which robust scheduling remains computationally manageable and highlight directions for further exploration of budgeted uncertainty in scheduling problems. $P^ abla$ operates under budgeted uncertainty with $p_j\in\{\bar{p}_j, \bar{p}_j+\hat{p}_j\}$ and $\sum_j \delta_j \le \Gamma$, guiding both hardness proofs and algorithm design.

Abstract

In this study, we investigate a robust single-machine scheduling problem under processing time uncertainty. The uncertainty is modeled using the budgeted approach, where each job has a nominal and deviation processing time, and the number of deviations is bounded by Gamma. The objective is to minimize the maximum number of tardy jobs over all possible scenarios. Since the problem is NP-hard in general, we focus on analyzing its tractability under the assumption that some natural parameter of the problem is bounded by a constant. We consider three parameters: the robustness parameter Gamma, the number of distinct due dates in the instance, and the number of jobs with nonzero deviations. Using parametrized-complexity theory, we prove that the problem is W[1]-hard with respect to Gamma, but can be solved in XP time with respect to the same parameter. With respect to the number of different due dates, we establish a stronger hardness result by showing that the problem remains NP-hard even when there are only two different due dates and is solvable in pseudo-polynomial time when the number of due dates is upper bounded by a constant. To complement these results, we show that the case of a common (single) due date, reduces to a robust binary knapsack problem with equal item profits, which we prove to be solvable in polynomial time. Finally, we prove that the problem is solvable in FPT time with respect to the number of nonzero deviations.

A Parametrized Complexity View on Robust Scheduling with Budgeted Uncertainty

TL;DR

The paper studies robust single-machine scheduling under budgeted uncertainty, where each job’s processing time is either the nominal value or a deviation with at most deviations allowed. It adopts parameterized complexity to determine when the problem of minimizing tardy jobs is tractable, proving -hardness in but offering an XP algorithm; it also shows NP-hardness for while giving polynomial or pseudo-polynomial algorithms for common or a fixed number of due dates, and an -time algorithm when the number of nonzero deviations is small. A key contribution is the extension of Moore’s algorithm to robust subproblems and a dualization-based robust knapsack formulation for the common due date case, clarifying tractability frontiers under natural parameters. The findings delineate precise conditions under which robust scheduling remains computationally manageable and highlight directions for further exploration of budgeted uncertainty in scheduling problems. operates under budgeted uncertainty with and , guiding both hardness proofs and algorithm design.

Abstract

In this study, we investigate a robust single-machine scheduling problem under processing time uncertainty. The uncertainty is modeled using the budgeted approach, where each job has a nominal and deviation processing time, and the number of deviations is bounded by Gamma. The objective is to minimize the maximum number of tardy jobs over all possible scenarios. Since the problem is NP-hard in general, we focus on analyzing its tractability under the assumption that some natural parameter of the problem is bounded by a constant. We consider three parameters: the robustness parameter Gamma, the number of distinct due dates in the instance, and the number of jobs with nonzero deviations. Using parametrized-complexity theory, we prove that the problem is W[1]-hard with respect to Gamma, but can be solved in XP time with respect to the same parameter. With respect to the number of different due dates, we establish a stronger hardness result by showing that the problem remains NP-hard even when there are only two different due dates and is solvable in pseudo-polynomial time when the number of due dates is upper bounded by a constant. To complement these results, we show that the case of a common (single) due date, reduces to a robust binary knapsack problem with equal item profits, which we prove to be solvable in polynomial time. Finally, we prove that the problem is solvable in FPT time with respect to the number of nonzero deviations.
Paper Structure (17 sections, 11 theorems, 39 equations, 2 algorithms)

This paper contains 17 sections, 11 theorems, 39 equations, 2 algorithms.

Table of Contents

  1. Introduction
  2. Robust Scheduling
  3. Problem Definition
  4. Parametrized Complexity and Research Objectives
  5. Overview of parameterized complexity.
  6. Research objectives.
  7. The Robustness Parameter
  8. W[1]-Hardness
  9. An XP Algorithm
  10. The Number of Different Due Dates
  11. NP hardness for two different due dates
  12. Common Due Date -- Robust Knapsack with Equal Profits
  13. A Constant Number of Due Dates
  14. Number of Jobs with Nonzero Deviation
  15. Feasibility Check of a Given Subproblem
  16. ...and 2 more sections

Key Result

Theorem 1

Problems $1|\mathop{\mathrm{P}}\nolimits^{\Gamma}|\min_{\sigma \in \Sigma} \max_{p \in \mathop{\mathrm{P}}\nolimits^{\Gamma}} \sum_{j=1}^n U_j(\sigma,p)$ and $1|\mathop{\mathrm{P}}\nolimits^{\Gamma}|\min_{\sigma \in \Sigma} \sum_{j=1}^n U_j(\sigma)$ are NP-hard.

Theorems & Definitions (22)

  • Theorem 1: Bougeret23
  • Definition 1
  • Definition 2
  • Lemma 2
  • Definition 3
  • Lemma 3
  • proof
  • Theorem 4
  • proof
  • Theorem 5
  • ...and 12 more