# On Solving Travelling Salesman Problem with Vertex Requisitions

## TL;DR

This paper introduces a more efficient algorithm for the NP-hard Travelling Salesman Problem with Vertex Requisitions, enabling faster solutions for most instances and improving local search and integer programming approaches.

## Contribution

It presents a new algorithm with reduced time complexity for the problem, solving nearly all feasible instances in linear time and aiding in neighborhood enumeration and integer programming modeling.

## Key findings

- Almost all feasible instances solved in O(n) time
- Enhanced local search neighborhood enumeration
- Integer programming model with O(n) binary variables

## Abstract

We consider the Travelling Salesman Problem with Vertex Requisitions, where for each position of the tour at most two possible vertices are given. It is known that the problem is strongly NP-hard. The proposed algorithm for this problem has less time complexity compared to the previously known one. In particular, almost all feasible instances of the problem are solvable in O(n) time using the new algorithm, where n is the number of vertices. The developed approach also helps in fast enumeration of a neighborhood in the local search and yields an integer programming model with O(n) binary variables for the problem.