A Constant-factor Approximation Algorithm for the Asymmetric Traveling Salesman Problem

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Claims a constant-factor algorithm for general Asymmetric TSP measured against the Held-Karp relaxation, replacing prior polylog bounds. Method: solve LP dual to get a laminar structure, reduce ATSP to a local-connectivity subproblem, contract reducible sets into irreducible instances (vertebrate pairs), use integer circulation rounding and flow-splitting to build Eulerian components, then stitch via Svensson’s black‑box reduction. Delivers an explicit albeit large constant (≈5,500) guarantee, and the algorithm is constructive and provably works.

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1. 0:00Start
2. 1:24Travelling salesman problem Given n cities and their pairwise distances, find a shortest tour visiting all n cities. . One of the best known NP-hard optimization problems Studied since the 19th century
3. 9:37ATSP - Graphic formulation Input: directed graph G =(V, E), edge weights w:E+R+ Find a minimum weight tour F. Tour = closed walk visiting every vertex at least once
4. 16:17Repeated cycle cover algorithm [Frieze, Galbiati & Maffioli '82] Relaxing connectivity: 1. Find minimum weight cycle cover 2. Contract and repeat
5. 47:28node-weighted case • The rerouted x is feasible to the circulation problem of weight OPT • Flow integrality: there exists integer solution of weight S OPT . After mapping back, every vertex with y 0 has in-degrees 2
6. 58:07Motivation: reducing by contraction • All sets in the family are singletons: node-weighted ATSP . Would like to reduce the problem by contracting nonsingleton sets in C