[Paper Review] Collective Schedules: Scheduling Meets Computational Social Choice

Fanny Pascual, Krzysztof Rzaḑca+1 more|arXiv (Cornell University)|Mar 20, 2018

Game Theory and Voting Systems21 references3 citations

TL;DR

This paper introduces the concept of collective schedules, integrating scheduling theory with computational social choice to address public planning problems where job durations and societal preferences must be balanced. It extends core social choice tools—positional scoring rules, Kemeny rule, and Condorcet principle—to account for job lengths, demonstrating that these extensions yield computationally feasible and fair scheduling solutions, with PTA-Condorcet-based methods showing strong scalability and near-optimal performance on realistic instances.

ABSTRACT

When scheduling public works or events in a shared facility one needs to accommodate preferences of a population. We formalize this problem by introducing the notion of a collective schedule. We show how to extend fundamental tools from social choice theory---positional scoring rules, the Kemeny rule and the Condorcet principle---to collective scheduling. We study the computational complexity of finding collective schedules. We also experimentally demonstrate that optimal collective schedules can be found for instances with realistic sizes.

Motivation & Objective

• To formalize the problem of public scheduling as a collective decision-making process that balances job durations and societal preferences.
• To extend fundamental tools from social choice theory—positional scoring rules, Kemeny rule, and Condorcet principle—to incorporate job lengths in scheduling.
• To analyze the computational complexity of collective scheduling under these extended models.
• To evaluate the fairness and optimality of proposed scheduling mechanisms through empirical experiments on realistic-sized instances.
• To identify trade-offs between efficiency, fairness, and compliance with democratic principles in collective scheduling.

Proposed method

• Proposes a formal model of collective scheduling where jobs are sequenced to minimize societal cost, defined as the sum or maximum of agents' tardiness relative to their preferred schedules.
• Extends positional scoring rules by weighting scores based on job duration, ensuring longer jobs have greater impact on the final ranking.
• Adapts the Kemeny rule to use swap distances between rankings while incorporating job duration in the cost function for pairwise comparisons.
• Generalizes the Condorcet principle to account for job lengths by defining a PTA-Condorcet condition: if a majority prefers job A before B, and A is shorter, then A should be scheduled before B unless the duration difference justifies otherwise.
• Develops a PTA-Condorcet-based scheduling mechanism that checks for majority-preferred order violations and prioritizes shorter, highly supported jobs to reduce overall societal impact.
• Employs experimental evaluation on synthetic and realistic instances to compare performance of cost-based and PTA-Condorcet-based methods in terms of tardiness, fairness (Gini index), and adherence to democratic principles.

Experimental results

Research questions

• RQ1 How can classical social choice mechanisms be meaningfully extended to incorporate job durations in collective scheduling?
• RQ2 What is the computational complexity of finding optimal collective schedules when job lengths are considered?
• RQ3 How do different scheduling mechanisms—based on total tardiness, maximum tardiness, or PTA-Condorcet consistency—compare in terms of fairness and optimality?
• RQ4 To what extent do PTA-Condorcet-based methods maintain fairness and efficiency while remaining computationally scalable?
• RQ5 What trade-offs exist between adherence to the Condorcet principle and optimization of collective cost functions like sum or max tardiness?

Key findings

• The PTA-Condorcet-based scheduling mechanism achieves an average of only 3% degradation in total tardiness (Σ-T) and 24% in maximum tardiness (max-T) compared to optimal schedules, indicating strong performance despite not explicitly optimizing these criteria.
• The Σ-T mechanism produces fairer schedules than max-T, as evidenced by a smaller average Gini index, even though max-T focuses on minimizing the worst-case agent delay.
• The max-T objective increases societal inequality by disproportionately affecting agents not at the worst-case, leading to higher Gini indices than Σ-T.
• PTA-Condorcet paradoxes—where majority preferences are violated in the final schedule—occur frequently in Σ-T and max-T schedules, with ratios of 1.0 and 1.1 respectively, indicating poor compliance with majoritarian fairness.
• Cost-based scheduling methods (e.g., Σ-T, max-T) are feasible only for moderate-sized instances, while PTA-Condorcet-based methods scale effectively to large instances, suggesting superior practical applicability.
• The study identifies a clear computational boundary: problems become NP-hard when jobs have non-unit durations, highlighting additional complexity from scheduling beyond standard voting models.