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Lower Bounds on Pure Dynamic Programming for Connectivity Problems on Graphs of Bounded Path-Width

Authors Kacper Kluk , Jesper Nederlof

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LIPIcs.ICALP.2026.130.pdf
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ACM Subject Classification
  • Theory of computation → Parameterized complexity and exact algorithms
Keywords
  • Parameterized Complexity
  • treewidth
  • tropical circuits
  • connectivity problems

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Abstract

We give unconditional parameterized complexity lower bounds on pure dynamic programming algorithms - as modeled by tropical circuits - for connectivity problems such as the Traveling Salesperson Problem. Our lower bounds are higher than the currently fastest algorithms that rely on algebra and give evidence that these algebraic aspects are unavoidable for competitive worst case running times. 
Specifically, we study input graphs with a small width parameter such as treewidth and pathwidth and show that for any k there exists a graph G of pathwidth at most k and k^𝒪(1) vertices such that any tropical circuit calculating the optimal value of a Traveling Salesperson round tour uses at least 2^Ω(k log log k) gates. We establish this result by linking tropical circuit complexity to the nondeterministic communication complexity of specific compatibility matrices. These matrices encode whether two partial solutions combine into a full solution, and Raz and Spieker [Combinatorica 1995] previously proved a lower bound for this complexity measure.

Cite As Get BibTex

Kacper Kluk and Jesper Nederlof. Lower Bounds on Pure Dynamic Programming for Connectivity Problems on Graphs of Bounded Path-Width. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 130:1-130:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.ICALP.2026.130

Author Details

Kacper Kluk
  • Institute of Informatics, University of Warsaw, Poland
Jesper Nederlof
  • Department of Information and Computing Sciences, Utrecht University, The Netherlands

Funding

  • Kluk, Kacper: Supported by Polish National Science Centre SONATA BIS-12 grant number 2022/46/E/ST6/00143.
  • Nederlof, Jesper: supported by the project COALESCE that has received funding from the European Research Council (ERC), grant agreement No 853234.

Acknowledgements

We thank Wim van den Broeck for making us aware of relations between our work and [Mateus de Oliveira Oliveira and Wim Van den Broeck, 2024], and Daniel Dadush for making us aware of direction connections of tropical circuits and extension complexity as outlined in [Hertrich and Loho, 2024].

References

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