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Chvátal Gomory Rounding and Integrality Gaps. Mohit Singh Kunal Talwar MSR NE, McGillMSR SV. TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: A A A A A A A A A A A. Approximation Algorithm Design. Cleverly define Lower Bound on Optimum - PowerPoint PPT Presentation
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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