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Worst-Case to Expander-Case Reductions: Derandomized and Generalized
Authors
Amir Abboud 
,
Nathan Wallheimer
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32nd Annual European Symposium on Algorithms (ESA 2024)
Series: Leibniz International Proceedings in Informatics (LIPIcs)
Conference: European Symposium on Algorithms (ESA) - License:
Creative Commons Attribution 4.0 International license
- Publication Date: 2024-09-23
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Acknowledgements
We would like to thank Thatchaphol Saranurak for clarifying a typo in the mentioned lower bound for the Densest Subgraph problem [Henzinger et al., 2015].
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Amir Abboud and Nathan Wallheimer. Worst-Case to Expander-Case Reductions: Derandomized and Generalized. In 32nd Annual European Symposium on Algorithms (ESA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 308, pp. 4:1-4:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.ESA.2024.4
https://doi.org/10.4230/LIPIcs.ESA.2024.4
Abstract
A recent paper by Abboud and Wallheimer [ITCS 2023] presents self-reductions for various fundamental graph problems, which transform worst-case instances to expanders, thus proving that the complexity remains unchanged if the input is assumed to be an expander. An interesting corollary of their self-reductions is that if some problem admits such reduction, then the popular algorithmic paradigm based on expander-decompositions is useless against it. In this paper, we improve their core gadget, which augments a graph to make it an expander while retaining its important structure. Our new core construction has the benefit of being simple to analyze and generalize while obtaining the following results:
- A derandomization of the self-reductions, showing that the equivalence between worst-case and expander-case holds even for deterministic algorithms, and ruling out the use of expander-decompositions as a derandomization tool.
- An extension of the results to other models of computation, such as the Fully Dynamic model and the Congested Clique model. In the former, we either improve or provide an alternative approach to some recent hardness results for dynamic expander graphs by Henzinger, Paz, and Sricharan [ESA 2022]. In addition, we continue this line of research by designing new self-reductions for more problems, such as Max-Cut and dynamic Densest Subgraph, and demonstrating that the core gadget can be utilized to lift lower bounds based on the OMv Conjecture to expanders.
Subject Classification
ACM Subject Classification
- Theory of computation → Design and analysis of algorithms
Keywords
- Fine-grained complexity
- expander graphs
- self-reductions
- worst-case to expander-case
- expander decomposition
- dynamic algorithms
- exact and parameterized complexity
- max-cut
- maximum matching
- k-clique detection
- densest subgraph
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References
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