Approximating the Held-Karp Bound for Metric TSP in Nearly Linear Work and Polylogarithmic Depth

TL;DR

This paper introduces a parallel algorithm that approximates the Held-Karp bound for Metric TSP efficiently in nearly linear work and polylogarithmic depth, advancing parallel optimization techniques for combinatorial problems.

Contribution

It presents a novel parallel algorithm with nearly linear work and polylogarithmic depth for approximating the Held-Karp bound and related problems, using core-sequences in MWU.

Findings

Achieves $(1+psilon)$-approximation in O(m/psilon^4) work and O(1/psilon^4) depth.

Introduces core-sequences to improve MWU iteration complexity and exploit cut structure.

Provides exponential depth improvement for certain LPs like kECSS.

Abstract

We present a nearly linear work parallel algorithm for approximating the Held-Karp bound for the Metric TSP problem. Given an edge-weighted undirected graph $G=(V,E)$ on $m$ edges and $\epsilon>0$, it returns a $(1+\epsilon)$-approximation to the Held-Karp bound with high probability, in $\tilde{O}(m/\epsilon^4)$ work and $\tilde{O}(1/\epsilon^4)$ depth. While a nearly linear time sequential algorithm was known for almost a decade (Chekuri and Quanrud'17), it was not known how to simultaneously achieve nearly linear work alongside polylogarithmic depth. Using a reduction by Chalermsook et al.'22, we also give a parallel algorithm for computing a $(1+\epsilon)$-approximate fractional solution to the $k$-edge-connected spanning subgraph (kECSS) problem, with similar complexity. To obtain these results, we introduce a notion of core-sequences for the parallel Multiplicative Weights Update (MWU) framework (Luby-Nisan'93, Young'01). For the Metric TSP and kECSS problems, core-sequences enable us to exploit the structure of approximate minimum cuts to reduce the cost per iteration and/or the number of iterations. The acceleration technique via core-sequences is generic and of independent interest. In particular, it improves the best-known iteration complexity of MWU algorithms for packing/covering LPs from $poly(\log nnz(A))$ to polylogarithmic in the product of cardinalities of the core-sequence sets, where $A$ is the constraint matrix of the LP. For certain implicitly defined LPs such as the kECSS LP, this yields an exponential improvement in depth.