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Compact formulations and valid inequalities for parallel machine scheduling with conflicts

Phablo F. S. Moura, Roel Leus, Hande Yaman

TL;DR

This work tackles Parallel Machine Scheduling with Conflicts (pmc) by linking it to vertex coloring and developing a compact Representatives Formulation (RF) that reduces symmetry compared with the natural Assignment Formulation (AF). A polyhedral analysis shows that the pmc polytope is full-dimensional and that many facets of the stable-set polytope lift to pmc under RF, enabling strong external and subgraph-induced inequalities. The authors implement a branch-and-cut framework using RF with clique and odd-cycle cuts and demonstrate, on benchmark and hard instances, that the RF-based bc-r+ approach often outperforms state-of-the-art methods, particularly as the conflict graph density increases or the optimal value is farther from the trivial lower bound. The results suggest the approach is competitive and scalable, with potential extensions to branch-and-price via a set-covering decomposition of the representatives model.

Abstract

The problem of scheduling conflicting jobs on parallel machines consists in assigning a set of jobs to a set of machines so that no two conflicting jobs are allocated to the same machine, and the maximum processing time among all machines is minimized. We propose a new compact mixed integer linear formulation based on the representatives model for the vertex coloring problem, which overcomes a number of issues inherent in the natural assignment model. We present a polyhedral study of the associated polytope, and describe classes of valid inequalities inherited from the stable set polytope. We describe branch-and-cut algorithms for the problem, and report on computational experiments with benchmark instances. Our computational results on the hardest instances of the benchmark set show that the proposed algorithms are superior (either in running time or quality of the solutions) to the current state-of-the-art methods. We find that our new method performs better than the existing ones especially when the gap between the optimal value and the trivial lower bound (i.e., the sum of all processing times divided by the number of machines) increases.

Compact formulations and valid inequalities for parallel machine scheduling with conflicts

TL;DR

This work tackles Parallel Machine Scheduling with Conflicts (pmc) by linking it to vertex coloring and developing a compact Representatives Formulation (RF) that reduces symmetry compared with the natural Assignment Formulation (AF). A polyhedral analysis shows that the pmc polytope is full-dimensional and that many facets of the stable-set polytope lift to pmc under RF, enabling strong external and subgraph-induced inequalities. The authors implement a branch-and-cut framework using RF with clique and odd-cycle cuts and demonstrate, on benchmark and hard instances, that the RF-based bc-r+ approach often outperforms state-of-the-art methods, particularly as the conflict graph density increases or the optimal value is farther from the trivial lower bound. The results suggest the approach is competitive and scalable, with potential extensions to branch-and-price via a set-covering decomposition of the representatives model.

Abstract

The problem of scheduling conflicting jobs on parallel machines consists in assigning a set of jobs to a set of machines so that no two conflicting jobs are allocated to the same machine, and the maximum processing time among all machines is minimized. We propose a new compact mixed integer linear formulation based on the representatives model for the vertex coloring problem, which overcomes a number of issues inherent in the natural assignment model. We present a polyhedral study of the associated polytope, and describe classes of valid inequalities inherited from the stable set polytope. We describe branch-and-cut algorithms for the problem, and report on computational experiments with benchmark instances. Our computational results on the hardest instances of the benchmark set show that the proposed algorithms are superior (either in running time or quality of the solutions) to the current state-of-the-art methods. We find that our new method performs better than the existing ones especially when the gap between the optimal value and the trivial lower bound (i.e., the sum of all processing times divided by the number of machines) increases.
Paper Structure (14 sections, 11 theorems, 20 equations, 5 figures, 3 tables)

This paper contains 14 sections, 11 theorems, 20 equations, 5 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Assignment formulation
  3. Representatives formulation
  4. Representatives polytope
  5. Stable set inequalities
  6. Subgraph-induced inequalities
  7. Computational experiments
  8. Dataset
  9. Comparison of the compact models
  10. Comparison with the state of the art
  11. Hard instances with sparse conflict graphs ($d\le 0.2$)
  12. Hard instances with dense conflict graphs ($d\ge 0.3$)
  13. Computational results on structured instances
  14. Concluding remarks and further research

Key Result

Proposition 2

Let $\underline{\tilde{y}}$ and $\underline{y}$ denote the optimal values of the linear relaxations of (AF) and (RF), respectively, on instance $(G,p,m)$ of pmc. It holds that $\underline{\tilde{y}} - \underline{y}$ is not bounded.

Figures (5)

  • Figure 1: A web graph and its complement.
  • Figure 2: Gaps produced by bc-r+, kl, and bt on dense instances ($d\ge 0.3$) where the corresponding algorithm computes a primal solution. $\bigstar$ Recall that bt is the only method that does not find a feasible solution to all these instances (precisely, 23 instances with $d=0.3$, 10 with $d=0.4$, and 21 with $d=0.5$.)
  • Figure 3: Running times (with logarithmic scale) of bc-r+, and kl on random graphs with $100$ vertices, $d=0.1$, and $m=5$.
  • Figure 4: Running times (with logarithmic scale) of bc-r+, kl, and bt on random graphs with $50$ vertices, $d=0.2$, and $m=5$.
  • Figure 5: Average gaps (in %) for each processing time interval ($a=[1,10]$, $b=[1,50]$, and $c=[1,100]$) obtained by bc-r+, kl, and bt on dense conflict graphs.

Theorems & Definitions (19)

  • Proposition 2
  • proof
  • Lemma 3
  • proof
  • Theorem 4
  • proof
  • Theorem 5
  • proof
  • Theorem 6: CamMouSan16
  • Theorem 7
  • ...and 9 more