Mohit - PowerPoint Presentation

Slide1

Mohit Singh Kunal TalwarMSR NE, McGill MSR SV

Chvátal Gomory Rounding and Integrality Gaps

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ASlide2

Approximation Algorithm Design

Cleverly define

Lower Bound

on

Optimum

Think hard

Show that

Write natural linear program Think hardShow that

Write natural linear program Add cleverly designed constraints to get Think hardShow that

 Slide3

Would like to establish limits on what we can hope to doAPX-hardness: usually good evidence. Often unavailableLP Gaps show limits of specific LP

Show gap between and

When no good algorithms known

Write natural linear program

Think hard

Show

that

Write natural linear program

Add cleverly designed constraints to get Think hardShow that

 Slide4

Ways to automate getting tighter relaxationsLovász-SchrijverSherali-AdamsLassere

Chvátal-GomoryOften (at least retrospectively), improve LP/SDP gapsMatching, MaxCut, Sparsest Cut, Unique Label Cover[

Arora-Bollobás-Lovász

2002

]

Can we establish limits for these procedures?Cut Generating Procedures

Write natural linear program

Add cleverly designed constraints to get

Think hardShow that  Slide5

[Arora-Bollobás-Lovász

2002] Vertex Cover: Large Class of LPs has integrality gap

.Implies gaps for LS, SA.

GMTT07,DK07,S08,CMM09,MS09,T09,RS09,KS09,CL10

LS/SA/

Lassere/LS+ gaps for several problemsMaxCut

Unique Label CoverSparsest CutCSPLINMatching…

Gap for LS, SA etc.Slide6

Hypergraph matching in k-uniform hypergraphs

rounds of CG bring gap down to

[

Chan Lau 10

] SA gap is at least

even after

rounds.

This talk: What about Chvátal-Gomory Slide7

Gaps remain large for many rounds of CGVertex Cover

: Gap ) after

rounds

MaxCut

:

Gap

after

roundsUnique Label Cover:

Gap after

rounds-: Gap after rounds

Same as SA gaps. 

This talk: What about Chvátal-GomorySlide8

[Gomory 1958]

For a polyhedron Let

where

Let

is polyhedron obtained after j rounds of CG

Defining

Chvátal-Gomory CutsSlide9

-uniform hypergraph:

Each edge

with

Goal: find largest subset of disjoint edges

s.t.

Hypergraph MatchingSlide10

Graph maximum matching

SA takes

rounds to get within

CG gets to integer hull in 1 round

APX hard

-

inapproximable

)approximation

[

Chan Lau 10] Gap after rounds of SAThere is a poly size LP with gap

 Hypergraph

matchingSlide11

is an

intersecting family if

for all

[Chan Lau 10]

LP +

intersecting

has gap at most

Intersecting

familySlide12

is an

intersecting family if

for all

Fix

valid for

valid for

valid for

valid for

So

valid for

Intersecting family via CG

 Slide13

Extremal combinatorics resultFor any intersecting family in a k-regular hypergraph

, there is one of size

Implies that

Integrality gap of

is bounded by

I.e. for

hypergraph

matching,

round CG is nearly a factor of two better than

round SA. Small families sufficeSlide14

Max Cut LPSlide15

[Charikar

Makarychev Makarychev 09]

for any subset

s.t.

a distribution

overs solutions

such that

is integral for any

Max Cut SA gap

Survives

rounds of SA

 Slide16

Observation: for any constraint

in

,

are integers and

(can add arbitrary positive multiple of

to remove negative coefficients and get stronger constraint)

Main idea: show that

Max Cut CG

i

s feasible for

 Slide17

Base case: k=0. InspectionInduction Step. Need to show

in

holds for

Case 1:

Case 2:

Proof by inductionSlide18

Let

for some

Recall

a distribution

overs solutions

s.t.

is integral for any

For each

,

For

, set For

, set

, for arbitrary fixed For , set New integral. Agrees with

on

.

. Therefore done.

:

 Slide19

By definition,

-1 valid for

Therefore done.

:

 Slide20

Similar proofs for unique games, CSPs, VCCG hierarchy often not much better than SANoticeably better for Hypergraph matching

What other problems show large gap between clever LP and LS/SA? Does CG capture them?Conclusions