Effective-Resistance-Reducing Flows, Spectrally Thin Trees, and Asymmetric TSP

We show that the integrality gap of the natural LP relaxation of the Asymmetric Traveling Salesman Problem is polyloglog(n). In other words, there is a polynomial time algorithm that approximates the value of the optimum tour within a factor of polyloglog(n), where polyloglog(n) is a bounded degree polynomial of loglog(n). We prove this by showing that any k-edge-connected unweighted graph has a polyloglog(n)/k-thin spanning tree. Our main new ingredient is a procedure, albeit an exponentially sized convex program, that “transforms” graphs that do not admit any spectrally thin trees into those that provably have spectrally thin trees. More precisely, given a k-edge-connected graph G = (V, E) where k ≥ 7 log(n), we show that there is a matrix D that “preserves” the structure of all cuts of G such that for a set F ⊆ E that induces an Ω(k)-edge-connected graph, the effective resistance of every edge in F w.r.t. D is at most polylog(k)/k. Then, we use our extension of the seminal work of Marcus, Spielman, and Srivastava [1], fully explained in [2], to prove the existence of a polylog(k)/k-spectrally thin tree with respect to D. Such a tree is polylog(k)/k-combinatorially thin with respect to G as D preserves the structure of cuts of G.