Spectrally Thin Trees and ATSP Nima Anari UC Berkeley Shayan Oveis Gharan Univ of Washington Asymmetric TSP ATSP Given a list of cities and their pairwise distances satisfying the triangle inequality ID: 422185
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Slide1
Effective-Resistance-Reducing Flows,Spectrally Thin Trees, and ATSP
Nima
AnariUC Berkeley
Shayan
Oveis
Gharan
Univ
of WashingtonSlide2
Asymmetric TSP (ATSP)
Given a list of cities and their pairwise “distances”, satisfying the triangle inequality,
Find
the shortest tour
t
hat visits all cities exactly once.Asymmetric:
2Slide3
Linear Programming Relaxation [Held-Karp’72]
3
Integrality
Gap:Slide4
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].4Slide5
Main Result
5
For any cost function, the integrality gap of the LP relaxation is
polyloglog
(n).Slide6
Plan of the Talk
6
ATSP
Thin Spanning Tree
Spectrally Thin Spanning Tree
Max Effective ResistanceOur ContributionSlide7
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-
sparsifier
No lower-bound on
Exercise
: Show that
(k-dim cube) has O(1/k) thin
tree
Ideally w
But even
<0.99 is interesting
Slide8
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. 8Slide9
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.
9Slide10
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).Slide11
11
In Pursuit of Thin Trees
Beyond Randomized RoundingSlide12
Graph Laplacian
Let
For
let
Laplacian Quadratic Form:
12
E.g.,Slide13
Spectrally Thin Spanning Trees
Def: A spanning tree
is -spectrally
thin
w.r.t. G
ifWhy?Generalizes (combinatorial) thinness.-spectral thinness implies -combinatorial thinness
Testable in polynomial time.
Compute max eigenvalue of
13Slide14
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
w
hereSlide15
A k-con Graph with no Spectrally Thin Tree
For any spanning tree, T,
15
n/k vertices
k
edges
k
edges
0
k/n
2k/n
1/2
1/2
1-k/n
1-2k/n
0
0
0
0
1
is small if there are many
short
parallel paths from
i
to j
1A
1ASlide16
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. 16Slide17
Spectrally Thin Trees (Summary)
17
k-edge
connectivity
for
all O(1/k)-combinatorial thin treeO(1/k)-spectrally thin tree
[MSS13]
?Slide18
Our Approach18Slide19
Main Idea
19
Symmetrize L
2
structure of G
while preserving its L1 structureSlide20
An Example
20
n/k vertices
for all black edges
Slide21
An Observation
An Application of [MSS’13]:
If for any cut
,
Then G has a
-spectrally thin tree. 21Slide22
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.Slide23
An Impossibility Theorem
Thm: There is a k-connected graph G,
s.t., for any
23Slide24
Proof Overview
24
k-connected graph G for
,
F is -connected,
G has
-comb thin tree
A General.
o
f [MSS’13]
Main
Tech
Thm
has
-spectrally thin tree
Note we may have
D is not a graphSlide25
Thm: Given a set of vectors
s.t.
If then there is a basis
s.t.
…………………………......…,
there are
disjoint
bases,
Thin Basis Problem
25
d
Linearly
independent set of vectorsSlide26
Proof Overview
26
k-connected graph G for
,
F is -connected,
G has
-comb thin tree
A General.
o
f [MSS’13]
Main
Tech
Thm
has
-spectrally thin tree
Slide27
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.
Slide28
A Convex Program for Optimum D
28
Has exp. many constraints:
Recall convexity of matrix
inv
If write
,
t
hen optimum
is
Thm
: For any k-connected graph, the optimum is
.
Proof Idea
: The dual is
.
Slide29
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).Slide30
ConclusionMain Idea:
Symmetrize L2 structure of
G while preserving its L1 structureTools:
Interlacing polynomials/Real Stable polynomials
Convex
optimizationGraph partitioningHigh dimensional geometry30Slide31
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