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Resource Leveling: Complexity of a UET two-processor scheduling variant and related problems

Pascale Bendotti, Luca Brunod Indrigo, Philippe Chrétienne, Bruno Escoffier

TL;DR

This work analyzes resource leveling variants of two-processor scheduling with a focus on minimizing the total overload cost under a makespan deadline. The central result provides a polynomial-time algorithm for the core case $L2 | prec, C_{max} \leq M, c_i = 1, p_i = 1 | F$ by leveraging maximum matchings on independence graphs and a two-machine schedule structure, characterizing the optimum $F^*(M)$ as a piecewise-linear function with three segments. It further broadens the scope by presenting polynomial or pseudo-polynomial algorithms for related special cases (in-tree precedences, $L1$, preemption-based problems) and establishing NP-hardness for several generalized leveling problems, linking resource leveling to classical scheduling via reductions. The paper also builds a comparative table of results to delineate tractable vs. hard instances and demonstrates the applicability of flow and matching techniques in the resource-leveling context. Overall, it advances theoretical understanding of scheduling under resource-overload costs and offers practical algorithmic strategies for exact solutions in key settings.

Abstract

This paper mainly focuses on a resource leveling variant of a two-processor scheduling problem. The latter problem is to schedule a set of dependent UET jobs on two identical processors with minimum makespan. It is known to be polynomial-time solvable. In the variant we consider, the resource constraint on processors is relaxed and the objective is no longer to minimize makespan. Instead, a deadline is imposed on the makespan and the objective is to minimize the total resource use exceeding a threshold resource level of two. This resource leveling criterion is known as the total overload cost. Sophisticated matching arguments allow us to provide a polynomial algorithm computing the optimal solution as a function of the makespan deadline. It extends a solving method from the literature for the two-processor scheduling problem. Moreover, the complexity of related resource leveling problems sharing the same objective is studied. These results lead to polynomial or pseudo-polynomial algorithms or NP-hardness proofs, allowing for an interesting comparison with classical machine scheduling problems.

Resource Leveling: Complexity of a UET two-processor scheduling variant and related problems

TL;DR

This work analyzes resource leveling variants of two-processor scheduling with a focus on minimizing the total overload cost under a makespan deadline. The central result provides a polynomial-time algorithm for the core case by leveraging maximum matchings on independence graphs and a two-machine schedule structure, characterizing the optimum as a piecewise-linear function with three segments. It further broadens the scope by presenting polynomial or pseudo-polynomial algorithms for related special cases (in-tree precedences, , preemption-based problems) and establishing NP-hardness for several generalized leveling problems, linking resource leveling to classical scheduling via reductions. The paper also builds a comparative table of results to delineate tractable vs. hard instances and demonstrates the applicability of flow and matching techniques in the resource-leveling context. Overall, it advances theoretical understanding of scheduling under resource-overload costs and offers practical algorithmic strategies for exact solutions in key settings.

Abstract

This paper mainly focuses on a resource leveling variant of a two-processor scheduling problem. The latter problem is to schedule a set of dependent UET jobs on two identical processors with minimum makespan. It is known to be polynomial-time solvable. In the variant we consider, the resource constraint on processors is relaxed and the objective is no longer to minimize makespan. Instead, a deadline is imposed on the makespan and the objective is to minimize the total resource use exceeding a threshold resource level of two. This resource leveling criterion is known as the total overload cost. Sophisticated matching arguments allow us to provide a polynomial algorithm computing the optimal solution as a function of the makespan deadline. It extends a solving method from the literature for the two-processor scheduling problem. Moreover, the complexity of related resource leveling problems sharing the same objective is studied. These results lead to polynomial or pseudo-polynomial algorithms or NP-hardness proofs, allowing for an interesting comparison with classical machine scheduling problems.
Paper Structure (18 sections, 22 theorems, 13 equations, 15 figures, 2 tables, 4 algorithms)

This paper contains 18 sections, 22 theorems, 13 equations, 15 figures, 2 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Related works
  3. Contributions
  4. Outline
  5. Problem description and notations
  6. A polynomial algorithm for $L2 | prec, C_{max} \leq M, c_i = 1, p_i = 1 | F$
  7. Solution for $M = |P|$
  8. Solution for $M > |P|$
  9. Algorithm
  10. Further tractable special cases
  11. A polynomial algorithm for $L | in \text{-} tree, C_{max} \leq M, p_i=1, c_i=1 | F$
  12. A polynomial algorithm for $L1 | prec, C_{max} \leq M, c_i > 0 | F$
  13. A polynomial algorithm for $L | r_i, d_i, c_i = 1, pmtn | F$
  14. A polynomial algorithm for $L2 | r_i, d_i, pmtn | F$
  15. A pseudo-polynomial algorithm for $L2 | C_{max} \leq M | F$
  16. ...and 3 more sections

Key Result

Proposition 1

Let $x$ be a feasible schedule for instance $I = (G, M)$ with $M = |P|$, where $P$ is a critical path of $G$. Let $a, b \in J$ be two independent jobs such that $x_a < x_b$ and such that $a$ is the only job scheduled at $x_a$. The schedule $x'$ resulting from the elementary operation on $a$ and $b$

Figures (15)

  • Figure 1: Illustration of function $F$ for an instance with five jobs and resource level $L = 2$.
  • Figure 2: Example of precedence graph and optimal schedule for $L2 | prec, C_{max} \leq M, c_i = 1, p_i = 1 | F$
  • Figure 3: Examples of graphs for $L2 | prec, C_{max} \leq M, c_i = 1, p_i = 1 | F$
  • Figure 4: Optimal objective value as a function of $M$
  • Figure 5: Example of augmenting sequence
  • ...and 10 more figures

Theorems & Definitions (47)

  • Definition 1: augmenting sequence
  • Definition 2: elementary operation
  • Proposition 1
  • proof
  • Remark 1
  • Lemma 1
  • proof
  • Proposition 2
  • proof
  • Lemma 2
  • ...and 37 more