Faster Planar Graph Algorithms for Connectivity Problems via Meanders

2026-10-08

AI Summary

This study addresses the subexponential algorithmic efficiency bottleneck for connectivity problems on planar graphs. By refining the sphere-cut decomposition-based dynamic programming framework and innovatively integrating meander system analysis with fast matrix multiplication techniques, this work reduces the upper bound on the number of meanders to O*(12.806^n). Consequently, it establishes a new record for the best deterministic complexity of planar graph problems under polynomial weights. Furthermore, this approach significantly improves the time complexity for classical NP-hard problems such as the Traveling Salesman Problem (TSP), achieving an O(2^{5.543√n}) algorithm for the Hamiltonian cycle problem.

Abstract

In this paper, we refine the dynamic programming framework based on the sphere cut decomposition designed by Dorn, Penninkx, Bodlaender, and Fomin (ESA 2005) to obtain faster subexponential algorithms for connectivity problems on planar graphs. We investigate the relationship between these problems and meanders, which are simple closed planar loops that intersect a fixed line in a given number of points. By combining dynamic programming with techniques from meandric system analysis and the use of fast matrix multiplication by Dorn (ESA 2006), we obtain improved algorithms for planar connectivity problems. We show that the number of meanders on $2n$ crossings $M_n$ is $\mathcal O^*(12.806^n)$, which improves the previous upper bound of $\mathcal O^*(12.901^n)$ by Albert and Paterson (FPSAC 2004). This then gives the best-known classical upper bounds on the deterministic time complexity of several planar graph problems with polynomially-bounded weights, namely $\mathcal O(2^{5.543\sqrt n})$ for the Planar Travelling Salesman problem, $\mathcal O(2^{5.796\sqrt n})$ for Planar Longest Cycle/Path, $\mathcal O(2^{8.251\sqrt n})$ for Planar Connected Dominating Set and $\mathcal O(2^{8.037\sqrt n})$ for Planar Steiner Tree. Notably, this leads to the best-known deterministic complexity $\mathcal O(2^{5.543\sqrt{n}})$ for the Planar Hamiltonian Cycle problem.

Problem

• Planar Graphs
• Connectivity Problems
• Subexponential Algorithms
• Meanders

Innovation

• Planar Graph Algorithms
• Sphere Cut Decomposition
• Meanders
• Dynamic Programming
• Subexponential Algorithms