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Butterflies are an important motif found in bipartite graphs that provide a structural way for finding dense regions within the graph. Beyond counting butterflies and enumerating them, other metrics and peeling for bipartite graphs are designed around counting butterfly motifs. The importance of counting butterflies has led to many works on efficient implementations for butterfly counting, given certain situational or hardware constraints. However, most algorithms are based on first counting the building block of the butterfly motif, and from that calculating the total possible number of butterflies in the graph. In this paper, using a linear algebra approach, we show that many provably correct algorithms for counting butterflies can be systematically derived. Moreover, we show how this formulation facilitates butterfly peeling algorithms that find the k-tip and k-wing subgraphs within a bipartite graph.
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A Spectral Approach to Approximately Counting Independent Sets in Dense Bipartite Graphs
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.
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- Award ID(s):
- 2309707
- PAR ID:
- 10588155
- Editor(s):
- Bringmann, Karl; Grohe, Martin; Puppis, Gabriele; Svensson, Ola
- Publisher / Repository:
- Schloss Dagstuhl – Leibniz-Zentrum für Informatik
- Date Published:
- Volume:
- 297
- ISSN:
- 1868-8969
- ISBN:
- 978-3-95977-322-5
- Page Range / eLocation ID:
- 35:1-35:18
- Subject(s) / Keyword(s):
- approximate counting independent sets bipartite graphs graph containers Theory of computation → Approximation algorithms analysis Mathematics of computing → Approximation algorithms Theory of computation → Algorithm design techniques
- Format(s):
- Medium: X Size: 18 pages; 813823 bytes Other: application/pdf
- Size(s):
- 18 pages 813823 bytes
- Right(s):
- Creative Commons Attribution 4.0 International license; info:eu-repo/semantics/openAccess
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Conference Paper:
- https://doi.org/10.4230/LIPIcs.ICALP.2024.35
