combinatorial approach to guerra's interpolation method david gamarnik mit joint work with...
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![Page 1: Combinatorial approach to Guerra's interpolation method David Gamarnik MIT Joint work with Mohsen Bayati (Stanford) and Prasad Tetali (Georgia Tech) Physics](https://reader030.vdocuments.mx/reader030/viewer/2022032803/56649e215503460f94b0daea/html5/thumbnails/1.jpg)
Combinatorial approach to Guerra's interpolation method
David GamarnikMIT
Joint work with
Mohsen Bayati (Stanford) and Prasad Tetali (Georgia Tech)
Physics of Algorithms, Santa Fe
August, 2009
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Erdos-Renyi graph (diluted spin glass model) G(N,cN)
N nodes,
M=cN (K-hyper) edges chosen u.a.r. from NK possibilities
K=2
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K=3
Erdos-Renyi graph (diluted spin glass model) G(N,cN)
N nodes,
M=cN (K-hyper) edges chosen u.a.r. from NK possibilities
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Combinatorial models on G(N,cN)
• Independent set:
• Partial q-Coloring:
• Ising model, Max-Cut, K-SAT, NAE-K-SAT
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Optimization (ground state, zero temperature ¯=1 ):
Largest independent set, largest number of properly colored edges, Max-Cut, Max-K-SAT, etc.
Gibbs measure (positive temperature) 0<¯< 1 :
Combinatorial models on G(N,cN)
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Open problem. Groundstate limits
Does the following limit exist?..
Wormald [99], Aldous and Steele [03], Bollobas & Riordan [05], Janson & Tomason [08]
Yes … for K-SAT and Viana-Bray model.
Franz & Leone [03], Panchenko & Talagrand [04].
Use Guerra’s Interpolation Method leading to sub-additivity
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They show the existence of the limit
for finite ¯ and then take ¯!1
• What about other models, such as multi-spin (Coloring)?
• Direct proof for optimal solution (¯ =1)?
• Guerra’s interpolation method was used by F & L and T & P to prove that RS and RSB are valid bounds on the limit.
• Guerra’s interpolation method was used by Talagrand to prove validity of the Parisi formula for SK model.
Open problem. Groundstate limits
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Results. Groundstate limits
Theorem I. The following limit exists for all models (IS, Coloring, Max-Cut, K-SAT, NAE-K-SAT)
Remarks
• For the case of independent sets this resolves and open problemW [99], A & S [03], B & R [05], J & T [08]
• The proof is direct (¯=1), combinatorial and simple
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Results. Groundstate limits
Corollary (satisfiability threshold). For Coloring (K-SAT, NAE-K-SAT) models there exists c* such that, w.h.p.,
• The instance is nearly colorable (satisfiable) when c<c*
• Linearly in N many edges (clauses) have to be violated when c>c* .
Remarks
• For K-SAT already follows from F&L [03]
• Connections with the Satisfiability Conjecture.
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Results. Free energy limits at positive temperature
Theorem II. The following limit exists for all models (IS, Coloring, Max-Cut, K-SAT, NAE-K-SAT) for all 0<¯<1
Remarks
• For K-SAT already done by F&L [03]
• Open question for ¯< 0
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Results. Large deviations limits
Theorem III. The following limit exists for all models Coloring, K-SAT and NAE-K-SAT
Namely if the probability that the model is satisfiable (colorable) converges to zero exponentially fast, it does so at a constant rate.
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Proof sketch. Largest indepent set in G(N,cN)
IN – largest independent set in G(N,cN)
Claim: for every N1, N2 such that N1+N2=N
The existence of the limit
then follows by “near” sub-additivity .
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Interpolation between G(N,cN) and G(N1, cN1) + G(N2, cN2)
For t=1,2,…,cN generate cN-t blue edges and t red edges
Each blue edge u.r. connects any two of the N nodes.
Each red edge u.r. connects any two of the Nj nodes with prob Nj /N, j=1,2.
G(N,cN,t)
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• t=0 (no red edges) : G(N,cN)
Interpolation between G(N,cN) and G(N1, cN1) + G(N2, cN2)
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• t=cN (no blue edges) : G(N1, cN1) + G(N2, cN2)
Interpolation between G(N,cN) and G(N1, cN1) + G(N2, cN2)
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Claim: for every t=1,…,cN
Proof:
• G(N,cN,t+1) is obtained from G(N,cN,t) by deleting one blue edge and adding one red edge
• Let G0 be the graph obtained after deleting blue edge but before adding red edge. Then
G(N,cN,t+1)= G0+ red edge.
G(N,cN,t)= G0+ blue edge.
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Claim: for every graph G0 ,
Proof: Let I* be the set of nodes which belongs to every largest I.S.
I*
G0
Observation:
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Proof (continued):
I*
G0
>
I1*
I2*
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