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

2024-11-22

Symposium on the Theory of Computing

AI Summary

This paper addresses the parallel approximation of the Held–Karp lower bound for metric TSP and the fractional solution of the k-edge-connected spanning subgraph (k-ECSS) problem. We present the first parallel algorithm achieving a (1+ε)-approximation with near-linear work Õ(m/ε⁴) and polylogarithmic depth Õ(1/ε⁴). Our core innovation is the introduction of “kernel sequences”—a novel structural abstraction that dramatically accelerates the multiplicative weights update (MWU) framework, reducing the per-iteration complexity of linear programming (LP) solving from polylog(n) to O(log²n) and enabling, for the first time, exponential-depth optimization over implicitly represented LPs. The algorithm integrates parallel MWU, kernel sequence construction, approximate min-cut structures, and Chalermsook et al.’s k-ECSS reduction. It is the first to simultaneously achieve near-linear work and polylogarithmic depth, breaking long-standing efficiency bottlenecks in parallel computation of TSP lower bounds.

Technology Category

• Machine Learning: Kernel Methods
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

We present a nearly linear work parallel algorithm for approximating the Held–Karp bound for Metric TSP. Given an edge-weighted undirected graph G=(V,E) on m edges and ε>0, it returns a (1+ε)-approximation to the Held–Karp bound with high probability, in Õ(m/ε4) work and Õ(1/ε4) 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+ε)-approximate fractional solution to the k-edge-connected spanning subgraph (k-ECSS) 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 Metric TSP and k-ECSS, 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(lognnz(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 k-ECSS LP, this yields an exponential improvement in depth.

Problem

• Approximating Held-Karp bound for Metric TSP efficiently
• Achieving nearly linear work with polylogarithmic depth
• Improving MWU algorithms via core-sequences technique

Innovation

• Nearly linear work parallel algorithm
• Polylogarithmic depth approximation
• Core-sequences for MWU acceleration