A Linear-size Conversion of HCP to 3HCP

TL;DR

This paper presents a new algorithm that converts any Hamiltonian cycle problem (HCP) instance into a 3HCP instance with only linear growth in size, improving efficiency over previous methods.

Contribution

It introduces a linear-size conversion method from HCP to 3HCP, utilizing new subgraph structures called s-gates for general s, and extends to bounded degree cases.

Findings

Conversion from HCP to 3HCP is linear in size.

New subgraph structures (s-gates) enable size-efficient conversions.

Quadratic growth reduces to linear when maximum degree is bounded.

Abstract

We provide an algorithm that converts any instance of the Hamiltonian cycle problem (HCP) into a cubic instance of HCP (3HCP), and prove that the input size of the new instance is only a linear function of that of the original instance. This is achieved by first considering various restrictions of HCP. Known conversions from directed HCP to undirected HCP, and sub-cubic HCP to cubic HCP are given. We introduce a subgraph called a 4-gate and show that it may be used to convert sub-quartic HCP into sub-cubic HCP. We further generalise this idea by first introducing the 5-gate, and then the s-gate for any s >= 4. We prove that these subgraphs may be used to convert general instances of HCP into cubic HCP instances, where the input size of the converted instance is a quadratic function of that of the original instance. This result improves upon the previously best known approach which results in cubic growth in the size of the instance. We further prove that the quadratic function is reduced to a linear function if the maximum initial degree is bounded above by a constant. Motivated by this result, we describe an algorithm to convert general HCP to HCP of bounded degree and prove that it results in only linear growth. All of the above results are then used in the proof that any instance of HCP may be converted to an equivalent instance 3HCP with only linear growth in the input size.