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 $\text{polyloglog}(n)$. In other words, there is a polynomial time algorithm that approximates the value of the optimum tour within a factor of $\text{polyloglog}(n)$, where $\text{polyloglog}(n)$ is a bounded degree polynomial of $\log\log(n)$. We prove this by showing that any $k$-edge-connected unweighted graph has a $\text{polyloglog}(n)/k$-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 $k\geq 7\log(n)$, we show that there is a matrix $D$ that "preserves" the structure of all cuts of $G$ such that for a set $F\subseteq E$ that induces an $\Omega(k)$-edge-connected graph, the effective resistance of every edge in $F$ w.r.t. $D$ is at most $\text{polylog}(k)/k$. Then, we use a recent extension of the seminal work of Marcus, Spielman, and Srivastava [MSS13] by the authors [AO14] to prove the existence of a $\text{polylog}(k)/k$-spectrally thin tree with respect to $D$. Such a tree is $\text{polylog}(k)/k$-combinatorially thin with respect to $G$ as $D$ preserves the structure of cuts of $G$.