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

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.