Mohit Singh Kunal TalwarMSR NE, McGill MSR SV

Chvátal Gomory Rounding and

Integrality Gaps

LPLB

Approximation Algorithm DesignCleverly define Lower Bound on OptimumThink hardShow that

Write natural linear program Think hardShow that

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

OPT

Alg

Would like to establish limits on what we can hope to do

APX-hardness: usually good evidence. Often unavailable

LP Gaps show limits of specific LPShow gap between and

When no good algorithms known

OPT

Alg

LP

Write natural linear program Think hardShow that

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

Ways to automate getting tighter relaxationsLovász-SchrijverSherali-AdamsLassereChvátal-Gomory

Often (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

[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,CL

10LS/SA/Lassere/LS+ gaps for several problems

MaxCutUnique Label CoverSparsest CutCSPLINMatching…

Gap for LS, SA etc.

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

Gaps remain large for many rounds of CG

Vertex Cover: Gap ) after roundsMaxCut: Gap after roundsUnique Label Cover: Gap after rounds-: Gap after rounds

Same as SA gaps.

This talk: What about Chvátal-Gomory

[Gomory 1958]For a polyhedron Let where

Let is polyhedron obtained after j rounds of CG

Defining Chvátal-Gomory Cuts

-uniform hypergraph: Each edge with Goal: find largest subset of disjoint edges

s.t.

Hypergraph Matching

Graph maximum matchingSA takes rounds to get withinCG gets to integer hull in 1 round

APX hard-inapproximable)approximation[Chan Lau 10] Gap after rounds of SA

There is a poly size LP with gap

Hypergraph matching

is an intersecting family if for all

[Chan Lau 10] LP + intersecting

has gap at most

Intersecting family

is an intersecting family if for all

Fix valid for valid for valid for valid for

So valid for

Intersecting family via CG

Extremal combinatorics resultFor any intersecting family in a k-regular

hypergraph, there is one of size

Implies thatIntegrality gap of is bounded by

I.e. for hypergraph matching, round CG is nearly a factor of two better than round SA.

Small families suffice

Max Cut LP

[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

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

is feasible for

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

Case 1: Case 2:

Proof by induction

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.

:

By definition, -1 valid for

Therefore done.

:

Similar proofs for unique games, CSPs, VC

CG 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