1. Combinatorial approach to Guerra's interpolation method
David Gamarnik
MIT
Joint work with
Mohsen Bayati (Stanford) and Prasad Tetali (Georgia Tech)
Probabilistic Techniques and Applications, UCLA
October, 2009

2. 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

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
K=3

4. Combinatorial models on G(N,cN)
Independent set:
Partial q-Coloring:
Ising model, Max-Cut, K-SAT, NAE-K-SAT

5. Combinatorial models on G(N,cN)
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 :

6. Open problem. Groundstate limits
Does the following limit exist?..
Wormald [99], Aldous and Steele [03], Gamarnik, Nowicki & Swirszcz [05], Bollobas & Riordan [05], Janson & Thomason [08]
Yes … for K-SAT and Viana-Bray model.
Franz & Leone [03], Panchenko & Talagrand [04].
Use Guerra’s Interpolation Method leading to supper-additivity

7. Open problem. Groundstate limits
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.

8. 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 problem
W [99], A & S [03], B & R [05], J & T [08]
The proof is direct (¯=1), combinatorial and simple

9. 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.

10. 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

11. 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.

12. 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” supper-additivity .

13. Interpolation between G(N,cN) and G(N1, cN1) + G(N2, cN2)
G(N,t)
Fix 1· t· 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.

14. t=0 (no red edges) : G(N,cN)
Interpolation between G(N,cN) and G(N1, cN1) + G(N2, cN2)
t=0 (no red edges) : G(N,cN)

15. t=cN (no blue edges) : G(N1, cN1) + G(N2, cN2)
Interpolation between G(N,cN) and G(N1, cN1) + G(N2, cN2)
t=cN (no blue edges) : G(N1, cN1) + G(N2, cN2)

16. 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.

17. Claim: for every graph G0 ,
Proof: Let I* be the set of nodes which belongs to every largest I.S. in G0 G0 I*
Observation:

18. Proof (continued): G0
I1* I*
I2*
>

19. Large deviations limits for satisfiability (colorability)
exists

20. END