A $(\frac32+\frac1{\mathrm{e}})$-Approximation Algorithm for Ordered TSP

TL;DR

This paper introduces a new approximation algorithm for the Ordered TSP that achieves a guarantee of approximately 1.868, improving significantly over previous methods and narrowing the gap with metric TSP.

Contribution

The paper presents the first approximation algorithm for Ordered TSP with a guarantee better than 2, using a novel LP relaxation and tree decomposition approach.

Findings

Achieves a $(rac32+rac1{ ext{e}})$-approximation ratio (~1.868).

Improves the previous best guarantee of 2.5 for Ordered TSP.

Reduces the gap between Ordered TSP and metric TSP approximability.

Abstract

We present a new $(\frac32+\frac1{\mathrm{e}})$-approximation algorithm for the Ordered Traveling Salesperson Problem (Ordered TSP). Ordered TSP is a variant of the classical metric Traveling Salesperson Problem (TSP) where a specified subset of vertices needs to appear on the output Hamiltonian cycle in a given order, and the task is to compute a cheapest such cycle. Our approximation guarantee of approximately $1.868$ holds with respect to the value of a natural new linear programming (LP) relaxation for Ordered TSP. Our result significantly improves upon the previously best known guarantee of $\frac52$ for this problem and thereby considerably reduces the gap between approximability of Ordered TSP and metric TSP. Our algorithm is based on a decomposition of the LP solution into weighted trees that serve as building blocks in our tour construction.