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On inferring cumulative constraints

Konstantin Sidorov

TL;DR

This work targets the inefficiency of propagating Cumulative constraints in isolation when scheduling involves multiple resources. It reframes Cumulative constraints as linear inequalities over occupancy vectors, enabling cutting-plane style inference via lifting to produce new Cumulative constraints that capture cross-resource contention. The approach enumerates cover sets, lifts their inequalities, and injects a limited number of strengthened constraints at the root, resulting in tighter bounds and faster search on RCPSP and RCPSP/max benchmarks, including 25 new lower bounds and five new best solutions (eight directly from lifted constraints). The findings demonstrate that global, occupancy-vector based inference can provide meaningful practical benefits with limited preprocessing overhead, offering a viable path to integrating multi-resource reasoning within CP solvers.

Abstract

Cumulative constraints are central in scheduling with constraint programming, yet propagation is typically performed per constraint, missing multi-resource interactions and causing severe slowdowns on some benchmarks. I present a preprocessing method for inferring additional cumulative constraints that capture such interactions without search-time probing. This approach interprets cumulative constraints as linear inequalities over occupancy vectors and generates valid inequalities by (i) discovering covers, the sets of tasks that cannot run in parallel, (ii) strengthening the cover inequalities for the discovered sets with lifting, and (iii) injecting the resulting constraints back into the scheduling problem instance. Experiments on standard RCPSP and RCPSP/max test suites show that these inferred constraints improve search performance and tighten objective bounds on favorable instances, while incurring little degradation on unfavorable ones. Additionally, these experiments discover 25 new lower bounds and five new best solutions; eight of the lower bounds are obtained directly from the inferred constraints.

On inferring cumulative constraints

TL;DR

This work targets the inefficiency of propagating Cumulative constraints in isolation when scheduling involves multiple resources. It reframes Cumulative constraints as linear inequalities over occupancy vectors, enabling cutting-plane style inference via lifting to produce new Cumulative constraints that capture cross-resource contention. The approach enumerates cover sets, lifts their inequalities, and injects a limited number of strengthened constraints at the root, resulting in tighter bounds and faster search on RCPSP and RCPSP/max benchmarks, including 25 new lower bounds and five new best solutions (eight directly from lifted constraints). The findings demonstrate that global, occupancy-vector based inference can provide meaningful practical benefits with limited preprocessing overhead, offering a viable path to integrating multi-resource reasoning within CP solvers.

Abstract

Cumulative constraints are central in scheduling with constraint programming, yet propagation is typically performed per constraint, missing multi-resource interactions and causing severe slowdowns on some benchmarks. I present a preprocessing method for inferring additional cumulative constraints that capture such interactions without search-time probing. This approach interprets cumulative constraints as linear inequalities over occupancy vectors and generates valid inequalities by (i) discovering covers, the sets of tasks that cannot run in parallel, (ii) strengthening the cover inequalities for the discovered sets with lifting, and (iii) injecting the resulting constraints back into the scheduling problem instance. Experiments on standard RCPSP and RCPSP/max test suites show that these inferred constraints improve search performance and tighten objective bounds on favorable instances, while incurring little degradation on unfavorable ones. Additionally, these experiments discover 25 new lower bounds and five new best solutions; eight of the lower bounds are obtained directly from the inferred constraints.
Paper Structure (13 sections, 3 theorems, 9 equations, 6 figures, 4 tables, 2 algorithms)

This paper contains 13 sections, 3 theorems, 9 equations, 6 figures, 4 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Constraint programming
  4. Polyhedral geometry
  5. From Cumulative to linear inequalities
  6. Inferring Cumulative constraints by lifting
  7. Experimental evaluation
  8. Evaluation against direct CP solver application
  9. Evaluation against UɴL
  10. Related work
  11. Conclusions
  12. Novel bounds
  13. Disjointness structure of instance #40 from UBO50

Key Result

Proposition 4

A cover inequality$\boldsymbol{x}(C) \le \# C - 1$ is a valid inequality for $\mathcal{P}(\boldsymbol{a}^T,b)$ if $C$ is its cover.

Figures (6)

  • Figure 1: Impact of using lifting on time to optimality. Every point corresponds to a pair of runs on the same instances, using the same underlying solver with the same search direction. Diagonal lines are evenly spaced and correspond to a tenfold relative change between the durations.
  • Figure 2: Comparison of integrals between the baseline and lifting approach.
  • Figure 3: Ratios of dual integrals between baseline runs and the corresponding runs with lifting across the right-hand side values of the best inferred constraint.
  • Figure 4: Distribution of time required to run the inference of Cumulative constraints.
  • Figure 5: Comparison between the lifting approach and UnL.
  • ...and 1 more figures

Theorems & Definitions (12)

  • Definition 1
  • Definition 2
  • Definition 3
  • Proposition 4
  • Example 5
  • Definition 6
  • Lemma 7
  • proof
  • Example 9
  • Definition 10
  • ...and 2 more