Effective-Resistance-Reducing Flows, Spectrally Thin Trees, and ATSP Nima Anari UC Berkeley Shayan Oveis Gharan Univ of Washington

Asymmetric TSP (ATSP) 2

Linear Programming Relaxation [Held-Karp’72] 3 Integrality Gap:

Previous Works Approximation Algorithms log(n) [Frieze-Galbiati-Maffioli’82].999 log(n) [Bläser’02] log(n) [Kaplan-Lewenstein-Shafrir-Sviridenko’05] log(n) [Feige-Singh’07] O(logn/loglogn) [Asadpour-Goemans-Madry-O-Saberi’09] O(1) (planar/bd genus) [O-Saberi’10,Erickson-Sidiropoulos’13] Integrality Gap ≥ 2 [Charikar-Goemans-Karloff’06] ≤ O(logn/loglogn) [AGMOS’09]. 4

Main Result 5 For any cost function, the integrality gap of the LP relaxation is polyloglog(n).

Plan of the Talk 6 ATSP Thin Spanning Tree Spectrally Thin Spanning Tree Max Effective Resistance Our Contribution

Thin Spanning Trees 7 Kn2/n-thin tree

From Thin Trees to ATSP 8

Previous Works: Randomized Rounding 9

Main Result 10 For any cost function, the integrality gap of the LP Relaxation is polyloglog(n).

11 In Pursuit of Thin Trees Beyond Randomized Rounding

Graph Laplacian 12 E.g.,

Spectrally Thin Spanning Trees 13

A Necessary Condition for Spectral Thinness 14 where

A k-con Graph with no Spectrally Thin Tree 15 n/k vertices k edges 0 k/n 2k/n1/2 1-k/n 1-2k/n A

A Sufficient Condition for Spectral Thinness 16

Spectrally Thin Trees (Summary) 17 k-edge connectivity k-edge connectivity O(1/k)-combinatorial thin tree O(1/k)-spectrally thin tree [MSS13] ? ?

Our Approach 18

Main Idea 19 Symmetrize L 2 structure of G while preserving its L 1 structure

An Example 20 n/k vertices

An Observation 21

Main Idea 22 D+G has a spectrally thin tree and any spectrally thin tree of G+D is (comb) thin in G. Bypasses Spectral Thinness Barrier.

An Impossibility Theorem 23

Proof Overview 24 A General. of [MSS’13] Main Tech Thm D is not a graph

…………………………......…, Thin Basis Problem 25 d Linearly independent set of vectors

Proof Overview 26 A General. of [MSS’13] Main Tech Thm

A Weaker Goal: Satisfying Degree Cuts 27

A Convex Program for Optimum D 28

Main Result 29 For any cost function, the integrality gap of the LP Relaxation is polyloglog(n).

Conclusion Main Idea: Symmetrize L 2 structure of G while preserving its L 1 structure Tools: Interlacing polynomials/Real Stable polynomials Convex optimization Graph partitioning High dimensional geometry 30

Future Works/Open Problems Algorithmic proof of [MSS’13] and our extension. Existence of C/k thin trees and constant factor approximation algorithms for ATSP. Subsequent work: Svensson designed a 27-app algorithm for ATSP when c(.,.) is a graph metric. 31