Backtracking algorithms for constructing the Hamiltonian decomposition of a 4-regular multigraph

TL;DR

This paper introduces two backtracking algorithms for constructing Hamiltonian decompositions of 4-regular multigraphs, addressing vertex non-adjacency verification in complex polytope problems, with varying success on different graph types.

Contribution

It presents novel backtracking algorithms for Hamiltonian decomposition, including a chain fixing procedure, and evaluates their performance on directed and undirected multigraphs.

Findings

Algorithms underperform compared to heuristics on undirected graphs.

Chain fixing algorithm performs well on directed graphs with existing solutions.

Algorithms show promise on infeasible directed instances.

Abstract

We consider a Hamiltonian decomposition problem of partitioning a regular graph into edge-disjoint Hamiltonian cycles. It is known that verifying vertex non-adjacency in the 1-skeleton of the symmetric and asymmetric traveling salesperson polytopes is NP-complete. On the other hand, a sufficient condition for two vertices to be non-adjacent can be formulated as a combinatorial problem of finding a second Hamiltonian decomposition of a 4-regular multigraph. We present two backtracking algorithms for constructing a second Hamiltonian decomposition and verifying vertex non-adjacency: an algorithm based on a simple path extension and an algorithm based on the chain edge fixing procedure. Based on the results of computational experiments for undirected multigraphs, both backtracking algorithms lost to the known general variable neighborhood search heuristics. However, for directed multigraphs, the algorithm based on chain fixing of edges showed results comparable to heuristics on instances with an existing solution and better results on infeasible instances where the Hamiltonian decomposition does not exist.