shayan.pptx

Effective-Resistance-Reducing Flows,

Spectrally Thin Trees, and ATSP

Nima Anari UC Berkeley

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.

Linear Programming Relaxation

[Held-Karp’72]

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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

Main Result

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

Plan of the Talk

ATSP

ATSP Thin Spanning

Thin Spanning Trees

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

Kn 2/n-thin tree

One-sided unweighted cut-sparsifier No lower-bound on

One-sided unweighted cut-sparsifier No lower-bound on

∀ 𝑆 ⊆𝑉 ,

|

(

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|

(

)

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

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.

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.

Main Result

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

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

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

In Pursuit of Thin Trees

Graph Laplacian

Let

For let

Laplacian Quadratic Form:

𝐿_𝐺=

[

2 − 1 −1 0

−1 2 0 −1

−1 0 2 −1 0 − 1 −1 2

]

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.}

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

A k-con Graph with no Spectrally Thin Tree

For any spanning tree, T,

Reff (𝑒)≈ 1− 𝑘

2

𝑛

k edges k edges

A Sufficient Condition for Spectral Thinness

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

Main Idea

An Example

An Observation

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

Main Idea

Find a ``graph’’ D s.t. and

i.e.,

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. D+G has a spectrally thin tree and

An Impossibility Theorem

Proof Overview

k-connected graph G for

k-connected graph G for

,

F is -connected, ,

F is -connected,

G has -comb thin tree G has -comb

thin tree

A General. of [MSS’13] Main

Tech Thm

has -spectrally thin tree

has -spectrally thin tree

Note we may have Note we may have D is not

a graph D is not

Thm: Given a set of vectors s.t.

If then there is a basis s.t.

………...…,_{there are disjoint bases,}

Thin Basis Problem

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d Linearly independent set of vectors

Proof Overview

k-connected graph G for

k-connected graph G for

,

F is -connected, ,

F is -connected,

G has -comb thin tree 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.

A Convex Program for Optimum D

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Has exp. many constraints: Has exp. many constraints:

Recall convexity of matrix inv Recall convexity of matrix inv

If write ,

then optimum is If write ,

then optimum is

Main Result

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

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}