Improving on Best-of-Many-Christofides for $T$-tours

TL;DR

This paper presents an improved approximation algorithm for the $T$-tour problem, achieving an $rac{11}{7}$-approximation ratio and establishing the integrality ratio of the LP relaxation, marking a significant advancement over previous analyses.

Contribution

The paper introduces the first improvement on Seb ext{"o}’s analysis of the Best-of-Many-Christofides algorithm for general $T$-tours, achieving an $rac{11}{7}$ approximation.

Findings

Achieved an $rac{11}{7}$-approximation ratio for the $T$-tour problem.

Proved the integrality ratio of the standard LP relaxation is at most $rac{11}{7}$.

First improvement over previous bounds for general $T$-tours.

Abstract

The $T$-tour problem is a natural generalization of TSP and Path TSP. Given a graph $G=(V,E)$, edge cost $c: E \to \mathbb{R}_{\ge 0}$, and an even cardinality set $T\subseteq V$, we want to compute a minimum-cost $T$-join connecting all vertices of $G$ (and possibly containing parallel edges). In this paper we give an $\frac{11}{7}$-approximation for the $T$-tour problem and show that the integrality ratio of the standard LP relaxation is at most $\frac{11}{7}$. Despite much progress for the special case Path TSP, for general $T$-tours this is the first improvement on Seb\H{o}'s analysis of the Best-of-Many-Christofides algorithm (Seb\H{o} [2013]).