Abstract
We study the problem of counting all cycles or self-avoiding walks (SAWs) on triangulated planar graphs. We present a subexponential \(2^{O(\sqrt{n})}\) time algorithm for this counting problem. Among the technical ingredients used in this algorithm are the planar separator theorem and a delicate analysis using pairs of Motzkin paths and Motzkin numbers. We can then adapt this algorithm to uniformly sample SAWs, in subexponential time. Our work is motivated by the problem of gerrymandered districting maps.
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Notes
- 1.
The algorithm in [4] is applicable for grid graphs only, and it was explicitly calculated that the number of self avoiding walks connecting two diagonal corners in a \(19 \times 19\) grid graph is \( > 10^{88}\). Our algorithm for planar graphs is based on a recursive, thus different, approach.
- 2.
It is known that asymptotically we have \( \left|\mathcal {L}_{A} \right|\le O(3^{|E_{A}|})\).
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Acknowledgements
We sincerely thank the three anonymous referees for their careful reading of the paper and helpful suggestions.
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Cai, JY., Maran, A. (2022). Counting Cycles on Planar Graphs in Subexponential Time. In: Zhang, Y., Miao, D., Möhring, R. (eds) Computing and Combinatorics. COCOON 2022. Lecture Notes in Computer Science, vol 13595. Springer, Cham. https://doi.org/10.1007/978-3-031-22105-7_24
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