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

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

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

Combinatorial models on G(N,cN)

• Independent set:

• Partial q-Coloring:

• Ising model, Max-Cut, K-SAT, NAE-K-SAT

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)

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

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

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

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.

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

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.

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 .

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)

• t=0 (no red edges) : G(N,cN)

Interpolation between G(N,cN) and G(N1, cN1) + G(N2, cN2)

• t=cN (no blue edges) : G(N1, cN1) + G(N2, cN2)

Interpolation between G(N,cN) and G(N1, cN1) + G(N2, cN2)

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.

Claim: for every graph G0 ,

Proof: Let I* be the set of nodes which belongs to every largest I.S.

I*

G0

Observation:

Proof (continued):

I*

G0

>

I1*

I2*

END