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The Computational Complexity of Factored Graphs

Authors Shreya Gupta , Boyang Huang , Russell Impagliazzo , Stanley Woo , Christopher Ye

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Author Details

Shreya Gupta
  • University of California San Diego, La Jolla, CA, USA
Boyang Huang
  • University of California San Diego, La Jolla, CA, USA
Russell Impagliazzo
  • University of California San Diego, La Jolla, CA, USA
Stanley Woo
  • University of California San Diego, La Jolla, CA, USA
Christopher Ye
  • University of California San Diego, La Jolla, CA, USA

Funding

  • Impagliazzo, Russell: Supported by NSF Award AF: Medium 2212136.
  • Ye, Christopher: Supported by NSF Award AF: Medium 2212136, NSF grants 1652303, 1909046, 2112533, and HDR TRIPODS Phase II grant 2217058.

Acknowledgements

We would like to thank Antonina Kolokolova, Anthony Ostuni, and anonymous reviewers for their many helpful comments and suggestions.

Cite As Get BibTex

Shreya Gupta, Boyang Huang, Russell Impagliazzo, Stanley Woo, and Christopher Ye. The Computational Complexity of Factored Graphs. In 16th Innovations in Theoretical Computer Science Conference (ITCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 325, pp. 58:1-58:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ITCS.2025.58

Abstract

While graphs and abstract data structures can be large and complex, practical instances are often regular or highly structured. If the instance has sufficient structure, we might hope to compress the object into a more succinct representation. An efficient algorithm (with respect to the compressed input size) could then lead to more efficient computations than algorithms taking the explicit, uncompressed object as input. This leads to a natural question: when does knowing the input instance has a more succinct representation make computation easier?
We initiate the study of the computational complexity of problems on factored graphs: graphs that are given as a formula of products and unions on smaller graphs. For any graph problem, we define a parameterized version that takes factored graphs as input, parameterized by the number of (smaller) ordinary graphs used to construct the factored graph. In this setting, we characterize the parameterized complexity of several natural graph problems, exhibiting a variety of complexities. We show that a decision version of lexicographically first maximal independent set is XP-complete, and therefore unconditionally not fixed-parameter tractable (FPT). On the other hand, we show that clique counting is FPT. Finally, we show that reachability is XNL-complete. Moreover, XNL is contained in FPT if and only if NL is contained in some fixed polynomial time.

Subject Classification

ACM Subject Classification
  • Theory of computation → Complexity theory and logic
  • Theory of computation → Complexity classes
  • Theory of computation → Problems, reductions and completeness
Keywords
  • Parameterized Complexity
  • Fine-grained complexity
  • Fixed-parameter tractability
  • Graph algorithms

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