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A Spectral Approach to Approximately Counting Independent Sets in Dense Bipartite Graphs

Authors Charlie Carlson , Ewan Davies , Alexandra Kolla, Aditya Potukuchi

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Author Details

Charlie Carlson
  • Department of Computer Science, University of California Santa Barbara, CA, USA
Ewan Davies
  • Department of Computer Science, Colorado State University, Fort Collins, CO, USA
Alexandra Kolla
  • Computer Science and Engineering, University of California Santa Cruz, CA, USA
Aditya Potukuchi
  • Department of Electrical Engineering and Computer Science, York University, Toronto, Canada

Funding

  • Davies, Ewan: Supported in part by NSF grant CCF-2309707.
  • Potukuchi, Aditya: Supported in part by NSERC Discovery grants RGPIN-2023-05087 & DGECR-2023-00408.

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Charlie Carlson, Ewan Davies, Alexandra Kolla, and Aditya Potukuchi. A Spectral Approach to Approximately Counting Independent Sets in Dense Bipartite Graphs. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 35:1-35:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.ICALP.2024.35

Abstract

We give a randomized algorithm that approximates the number of independent sets in a dense, regular bipartite graph - in the language of approximate counting, we give an FPRAS for #BIS on the class of dense, regular bipartite graphs. Efficient counting algorithms typically apply to "high-temperature" problems on bounded-degree graphs, and our contribution is a notable exception as it applies to dense graphs in a low-temperature setting. Our methods give a counting-focused complement to the long line of work in combinatorial optimization showing that CSPs such as Max-Cut and Unique Games are easy on dense graphs via spectral arguments.
Our contributions include a novel extension of the method of graph containers that differs considerably from other recent low-temperature algorithms. The additional key insights come from spectral graph theory and have previously been successful in approximation algorithms. As a result, we can overcome some limitations that seem inherent to the aforementioned class of algorithms. In particular, we exploit the fact that dense, regular graphs exhibit a kind of small-set expansion (i.e., bounded threshold rank), which, via subspace enumeration, lets us enumerate small cuts efficiently.

Subject Classification

ACM Subject Classification
  • Theory of computation → Approximation algorithms analysis
  • Mathematics of computing → Approximation algorithms
  • Theory of computation → Algorithm design techniques
Keywords
  • approximate counting
  • independent sets
  • bipartite graphs
  • graph containers

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