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A k-swap Local Search for Makespan Scheduling

Lars Rohwedder, Ashkan Safari, Tjark Vredeveld

TL;DR

This work investigates makespan minimization on $m$ identical parallel machines using a generalized $k$-swap neighborhood that exchanges up to $k$ jobs between two machines. It develops a fast randomized method to find improving $k$-swaps in $O(n^{\lceil k/2 \rceil + O(1)})$ time with high probability and provides derandomization, a lower bound based on the $k$-sum conjecture, and both a polynomial upper bound for $k=2$ with $m=2$ and an exponential lower bound for $k\ge 3$. Theoretical results are complemented by computational experiments showing practical efficiency of the randomized approach, especially for larger $k$ or higher job density. The paper also establishes a $O(n^4)$ bound on iterations to reach a 2-swap optimum when $m=2$, while proving exponential lower bounds for $k\ge 3$, offering nuanced insights into local search dynamics for scheduling.

Abstract

Local search is a widely used technique for tackling challenging optimization problems, offering significant advantages in terms of computational efficiency and exhibiting strong empirical behavior across a wide range of problem domains. In this paper, we address the problem of scheduling a set of jobs on identical parallel machines with the objective of makespan minimization. For this problem, we consider a local search neighborhood, called $k$-swap, which is a generalized version of the widely-used swap and jump neighborhoods. The $k$-swap neighborhood is obtained by swapping at most $k$ jobs between two machines. First, we propose an algorithm for finding an improving neighbor in the $k$-swap neighborhood which is faster than the naive approach, and prove an almost matching lower bound on any such an algorithm. Then, we analyze the number of local search steps required to converge to a local optimum with respect to the $k$-swap neighborhood. For $k \geq 3$, we provide an exponential lower bound regardless of the number of machines, and for $k = 2$ (similar to the swap neighborhood), we provide a polynomial upper bound for the case of having two machines. Finally, we conduct computational experiments on various families of instances.

A k-swap Local Search for Makespan Scheduling

TL;DR

This work investigates makespan minimization on identical parallel machines using a generalized -swap neighborhood that exchanges up to jobs between two machines. It develops a fast randomized method to find improving -swaps in time with high probability and provides derandomization, a lower bound based on the -sum conjecture, and both a polynomial upper bound for with and an exponential lower bound for . Theoretical results are complemented by computational experiments showing practical efficiency of the randomized approach, especially for larger or higher job density. The paper also establishes a bound on iterations to reach a 2-swap optimum when , while proving exponential lower bounds for , offering nuanced insights into local search dynamics for scheduling.

Abstract

Local search is a widely used technique for tackling challenging optimization problems, offering significant advantages in terms of computational efficiency and exhibiting strong empirical behavior across a wide range of problem domains. In this paper, we address the problem of scheduling a set of jobs on identical parallel machines with the objective of makespan minimization. For this problem, we consider a local search neighborhood, called -swap, which is a generalized version of the widely-used swap and jump neighborhoods. The -swap neighborhood is obtained by swapping at most jobs between two machines. First, we propose an algorithm for finding an improving neighbor in the -swap neighborhood which is faster than the naive approach, and prove an almost matching lower bound on any such an algorithm. Then, we analyze the number of local search steps required to converge to a local optimum with respect to the -swap neighborhood. For , we provide an exponential lower bound regardless of the number of machines, and for (similar to the swap neighborhood), we provide a polynomial upper bound for the case of having two machines. Finally, we conduct computational experiments on various families of instances.
Paper Structure (14 sections, 19 theorems, 5 equations, 7 figures, 4 tables, 1 algorithm)

This paper contains 14 sections, 19 theorems, 5 equations, 7 figures, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related work
  3. Formal model
  4. Our contribution
  5. Searching the Neighborhood
  6. Probability of Success
  7. Running Time
  8. Derandomization
  9. Lower Bound on the Running Time
  10. Running Time of Finding a Local Optimum
  11. Obtaining a 2-swap Optimal Solution
  12. An Exponential Lower Bound for $k \geq 3$
  13. Computational Results
  14. Concluding Remarks

Key Result

Theorem 1.1

We are able to find a better solution in the k-swap neighborhood with the probability of success of at least $1 - \frac{1}{e}$ in $O(m^2 \cdot k^{3/2} \cdot n^{\lceil\frac{k}{2}\rceil} \cdot \log n)$ by running Algorithm Alg:k-Swap, for every possibility of swapping exactly $\mathcal{K} = 1, \dotsc

Figures (7)

  • Figure 1: $k$-swap operator.
  • Figure 2: Mappings of the numbers $\{ c_1, c_2, ... , c_{k^2} \}$ to 0 and 1.
  • Figure 3: Improving $\sigma$ by making $\omega_j = 1$ and $\omega_i = 0$ for any $j \in \{3, ... , n\}$ and all $i \in \{1, ... , j-1\}$.
  • Figure 4: Average running time (in milliseconds) of the randomized and naive operators for different values of $k$ in class $C_7$.
  • Figure 5: Average running time (in milliseconds) of the randomized and naive operators for different values of $k$ in classes $C_5$ (left diagram) and $C_8$ (right diagram).
  • ...and 2 more figures

Theorems & Definitions (27)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Lemma 2.1
  • Lemma 2.2
  • Theorem 2.3
  • Lemma 2.4
  • proof
  • ...and 17 more