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Optimal Competitive Ratio for Optimization Problems with Congestion Effects

Authors Miriam Fischer, Dario Paccagnan , Cosimo Vinci

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ACM Subject Classification
  • Theory of computation → Online algorithms
  • Mathematics of computing → Mathematical optimization
  • Theory of computation → Algorithmic game theory
Keywords
  • Online Algorithms
  • Competitive Ratio
  • Algorithmic Game Theory
  • Greedy Algorithms
  • Congestion Games

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Abstract

In this work we study online optimization problems with congestion effects. These are problems where tasks arrive online and a decision maker is required to allocate them on the fly to available resources in order to minimize the cost suffered, which grows with the amount of resources used. This class of problems corresponds to the online counterpart of well-known studied problems, including optimization problems with diseconomies of scale [Konstantin Makarychev and Maxim Sviridenko, 2018], minimum cost in congestion games [Gairing and Paccagnan, 2023], and load balancing problems [Baruch Awerbuch et al., 1995]. 
Within this setting, our work settles the problem of designing online algorithms with optimal competitive ratio, i.e., algorithms whose incurred cost is as close as possible to that of an oracle with complete knowledge of the future instance ahead of time. We provide three contributions underpinning this result. First, we show that no online algorithm can achieve a competitive ratio below a given factor depending solely on the resource costs. Second, we show that, when guided by carefully modified cost functions, the greedy algorithm achieves a competitive ratio matching this lower bound and thus is optimal. Finally, we show how to compute such modified cost functions in polynomial time.

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Miriam Fischer, Dario Paccagnan, and Cosimo Vinci. Optimal Competitive Ratio for Optimization Problems with Congestion Effects. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 9:1-9:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2025.9

Author Details

Miriam Fischer
  • Imperial College, London, UK
Dario Paccagnan
  • Imperial College, London, UK
Cosimo Vinci
  • University of Salento, Lecce, Italy

Funding

This work is partially supported by: a Google DeepMind Scholarship awarded to Miriam Fischer; the PNRR MIUR project FAIR - Future AI Research (PE00000013), Spoke 9 - Green-aware AI; MUR - PNRR IF Agro@intesa; the Project SERICS (PE00000014) under the NRRP MUR program funded by the EU - NGEU; GNCS-INdAM.

Acknowledgements

We thank the reviewers for valuable comments and suggestions.

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