An approximation algorithm for the longest cycle problem in solid grid graphs

TL;DR

This paper presents a linear-time approximation algorithm for the longest cycle problem in solid grid graphs, showing it is in APX by guaranteeing a cycle of at least two-thirds of the nodes.

Contribution

It introduces the first linear-time constant-factor approximation algorithm for the longest cycle problem in solid grid graphs, establishing its APX-completeness.

Findings

The algorithm finds a cycle of length at least (2n/3)+1 in 2-connected n-node solid grid graphs.

The problem is shown to be in APX, indicating it admits constant-factor approximation algorithms.

The approach advances understanding of cycle problems in grid graph structures.

Abstract

Although, the Hamiltonicity of solid grid graphs are polynomial-time decidable, the complexity of the longest cycle problem in these graphs is still open. In this paper, by presenting a linear-time constant-factor approximation algorithm, we show that the longest cycle problem in solid grid graphs is in APX. More precisely, our algorithm finds a cycle of length at least $\frac{2n}{3}+1$ in 2-connected $n$-node solid grid graphs. Keywords: Longest cycle, Hamiltonian cycle, Approximation algorithm, Solid grid graph.