Effective-Resistance-Reducing Flows,Spectrally Thin Trees, and ATSP

Nima AnariUC Berkeley

Shayan Oveis GharanUniv of Washington

Asymmetric TSP (ATSP)

Given a list of cities and their pairwise “distances”, satisfying the triangle inequality,

Find the shortest tour that

visits all cities exactly once.

Asymmetric:

2

Linear Programming Relaxation [Held-Karp’72]

3

𝑚𝑖𝑛 ∑𝑖 , 𝑗𝑐 (𝑖 , 𝑗 )𝑥 𝑖 , 𝑗 ¿ 𝑠 . 𝑡 . ∑

𝑗𝑥 𝑖 , 𝑗=∑

𝑗𝑥 𝑗 , 𝑖 ∀ 𝑖∈𝑉

∑𝑖∈𝑆 , 𝑗∉𝑆

𝑥 𝑖 , 𝑗≥1 ¿ 0≤𝑥 𝑖 , 𝑗 ¿ ¿

Integrality Gap:

Previous Works

Approximation Algorithms• log(n) [Frieze-Galbiati-Maffioli’82]

• .999 log(n) [Bläser’02]

• 0.842 log(n) [Kaplan-Lewenstein-Shafrir-Sviridenko’05]

• 0.666 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

ATSPThin Spanning

Tree

Spectrally Thin Spanning Tree

Max Effective Resistance

Our Contribution

Thin Spanning Trees

Def: Given a k-edge-connected graph .A spanning tree is -thin w.r.t. G if

7

Kn 2/n-thin tree

One-sided unweighted cut-sparsifierNo lower-bound on

∀𝑆⊆𝑉 ,|𝑇 (𝑆 ,𝑆 )|≤𝛼⋅∨𝐸 (𝑆 ,𝑆 )∨¿

Exercise: Show that (k-dim cube) has O(1/k) thin tree

𝑆 𝑆 𝑆 𝑆Ideally wBut even <0.99 is interesting

From Thin Trees to ATSP

[AGMOS’09]: If for any -connected graph , then the integrality gap of LP is .Furthermore, finding the tree algorithmically gives -approximation algorithm for ATSP.

Proof Idea: Max-flow/Min-cut thm.

8

Previous Works: Randomized Rounding

Thm: Any k-connected graph G has a thin tree

Pf. Sample each edge of G, indep, w.p. .By Karger’s cut counting argument, the sampled graph is -thin w.h.p.

[AGMOS’09]: Improved the above bound to .by sampling random spanning trees.

9

Main Result

10

Any -edge-connected graph has an -thin tree.

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

11

In Pursuit of Thin TreesBeyond Randomized Rounding

Graph Laplacian

Let For let

Laplacian Quadratic Form:

12

𝐿𝐺=[ 2 −1 −1 0−1 2 0 −1−1 0 2 −10 −1 −1 2 ]

E.g.,

Spectrally Thin Spanning Trees

Def: A spanning tree is -spectrally thin w.r.t. G if

Why?• Generalizes (combinatorial) thinness.– -spectral thinness implies -combinatorial thinness

• Testable in polynomial time.– Compute max eigenvalue of

13

Lem: The spectral thinness of any T is at least

Pf. If T is -spectrally thin, then any subgraph of T is -spectrally thin, so

is the spectral thinness of .

A Necessary Condition for Spectral Thinness

14

where

A k-con Graph with no Spectrally Thin Tree

For any spanning tree, T,

15

n/k vertices

Reff (𝑒 )≈1− 𝑘2

𝑛

k edges

k edges

1A

1A

A Sufficient Condition for Spectral Thinness

[Marcus-Spielman-Srivastava’13,Harvey-Olver’14]: Any G has an -spectrally thin tree.

So, any edge-transitive k-connected graph (e.g., ) has an O(1/k)-(spectrally) thin tree.

16

Spectrally Thin Trees (Summary)

17

k-edge connectivity

for all

O(1/k)-combinatorial thin tree

O(1/k)-spectrally thin tree

[MSS13]

?

Our Approach

18

Main Idea

19

Symmetrize L2 structure of G while preserving its L1 structure

An Example

20

n/k vertices

for all black edges

An Observation

An Application of [MSS’13]: If for any cut ,

Then G has a -spectrally thin tree.

21

Main Idea

22

Find a ``graph’’ D s.t. and

i.e.,

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

Thm: There is a k-connected graph G, s.t., for any

23

Proof Overview

24

k-connected graph G for

, F is -connected,

G has -comb thin tree

A General. of [MSS’13]

Main Tech Thm

has -spectrally thin tree

Note we may haveD is not a graph

Thm: Given a set of vectors s.t.

If then there is a basis s.t.…………………………......…,there are disjoint bases,

Thin Basis Problem

25

‖∑𝑒∈𝑇 𝑣𝑒𝑣𝑒𝑇‖≤𝑂 (𝜖) .

d Linearly independent set of vectors

Proof Overview

26

k-connected graph G for

, F is -connected,

G has -comb thin tree

A General. of [MSS’13]

Main Tech Thm

has -spectrally thin tree

A Weaker Goal: Satisfying Degree Cuts

Thm: Given a k-connected graph, , s.t., for all v,

for

Let then by Markov Ineq,

, for all v.

27

By [MSS13] implies existence of -thin edge covers.

A Convex Program for Optimum D

28

𝑚𝑖𝑛 max𝑣∈𝑉

𝔼𝑒∼𝛿 (𝑣 )Reff𝐷 (𝑒)

¿ 𝑠 . 𝑡 . 𝐷≼𝐶𝐿𝐺¿ ¿

Has exp. many constraints:

Recall convexity of matrix inv

If write ,then optimum is

Thm: For any k-connected graph, the optimum is.

Proof Idea: The dual is .

Main Result

29

Any -edge-connected graph has an -thin tree.

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

Conclusion

Main Idea: • Symmetrize L2 structure of G while preserving its L1

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