[Paper Review] The Soft Cumulative Constraint

Thierry Petit, Emmanuel Poder|ArXiv.org|Jul 6, 2009

Constraint Satisfaction and Optimization3 references3 citations

TL;DR

This paper introduces the SoftCumulative and RelSoftCumulative constraints to handle over-constrained cumulative scheduling problems by relaxing resource capacity limits with penalty variables. It extends the sweep algorithm and energy reasoning to filter start times and cost variables, enabling efficient lower-bound estimation and pruning while managing overloads across time intervals.

ABSTRACT

This research report presents an extension of Cumulative of Choco constraint solver, which is useful to encode over-constrained cumulative problems. This new global constraint uses sweep and task interval violation-based algorithms.

Motivation & Objective

• Address over-constrained cumulative scheduling problems where resource capacity is exceeded at certain time points.
• Develop a global constraint that relaxes capacity limits with associated cost variables to model soft violations.
• Extend traditional sweep and energy reasoning algorithms to handle overloads and cost variable domains.
• Provide tight lower bounds for the total cost objective using interval-based energetic reasoning and time-range aggregation.
• Enable efficient pruning of start time and cost variable domains while maintaining computational efficiency.

Proposed method

• Propose the SoftCumulativeSum constraint to model cumulative problems with soft capacity violations using a time-discretized cost array.
• Extend the sweep algorithm to track cumulative overloads and update costarray values for each time interval indexed by mult.
• Apply pruning rules that remove infeasible start time ranges when the sum of compulsory parts and minimum resource use exceeds ideal capacity.
• Introduce energy reasoning over task intervals using aggregated cost variables, computing lower bounds via min(costVar[j]) and over-estimate of overloads.
• Use Notation 4 to compute # (I,j) as the number of time points shared between interval I and time range j, enabling interval-based cost estimation.
• Maintain costarray to accumulate overloads per time range, updating costVar domains based on accumulated overloads exceeding their minimums.

Experimental results

Research questions

• RQ1 How can cumulative scheduling problems be effectively relaxed when resource capacity is exceeded at certain time points?
• RQ2 What filtering techniques can be applied to start time and cost variable domains in soft cumulative constraints?
• RQ3 How can the sweep algorithm be adapted to handle overloads and maintain tight lower bounds in relaxed cumulative problems?
• RQ4 What is the impact of time-range aggregation (via mult) on the accuracy of cost variable domain pruning and lower-bound estimation?
• RQ5 How can energy reasoning be extended to support soft constraints with penalty variables for overloads?

Key findings

• The SoftCumulativeSum constraint enables modeling of over-constrained cumulative problems by introducing a cost variable per time range to represent overloads.
• The sweep-based filtering algorithm achieves O(n log n) time complexity while pruning start time domains based on cumulative overloads.
• Pruning Rule 6 ensures that costVar[j] domains are reduced when accumulated overloads exceed their minimum values, improving bound tightening.
• The interval-based energetic reasoning provides a lower bound for the total cost by summing min(costVar[j]) and over-estimated overloads across intersecting time ranges.
• The method supports efficient cost variable domain reduction through incremental updates of costarray, maintaining computational efficiency.
• By using mult ≥ 2, the approach provides a tight over-estimate of overloads, ensuring correct lower bounds without under-estimation in interval reasoning.