A (Slightly) Improved Deterministic Approximation Algorithm for Metric TSP

TL;DR

This paper presents a derandomized, deterministic approximation algorithm for metric TSP that improves upon the classic 3/2 ratio by a tiny epsilon, using the method of conditional expectation.

Contribution

It introduces a derandomization of the max entropy algorithm for metric TSP, achieving a deterministic 3/2 - epsilon approximation with epsilon > 10^{-36}.

Findings

Deterministic 3/2 - epsilon approximation for metric TSP.

Polynomial-time computation of the objective function's expected value.

Application of the method of conditional expectation to TSP approximation.

Abstract

We show that the max entropy algorithm can be derandomized (with respect to a particular objective function) to give a deterministic $3/2-\epsilon$ approximation algorithm for metric TSP for some $\epsilon > 10^{-36}$. To obtain our result, we apply the method of conditional expectation to an objective function constructed in prior work which was used to certify that the expected cost of the algorithm is at most $3/2-\epsilon$ times the cost of an optimal solution to the subtour elimination LP. The proof in this work involves showing that the expected value of this objective function can be computed in polynomial time (at all stages of the algorithm's execution).