Combinatorial approach to Guerra's interpolation method

1. Combinatorial approach to Guerra's interpolation method David Gamarnik MIT Joint work with Mohsen Bayati (Stanford) and Prasad Tetali (Georgia Tech) Physics of Algorithms, Santa Fe August, 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: • Partialq-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], 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

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 boundson 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, N2such that N1+N2=N The existence of the limit then follows by “near” sub-additivity .

13. Interpolation between G(N,cN) and G(N1, cN1) + G(N2, cN2) G(N,cN,t) • For t=1,2,…,cN generate cN-tblue edges and tred edges • Each blue edge u.r. connects any two of the N nodes. • Each rededge u.r. connects any two of the Njnodes with probNj/N, j=1,2.

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

15. Interpolation between G(N,cN) and G(N1, cN1) + G(N2, cN2) • t=cN (no blueedges) : 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+rededge. • 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. G0 I* Observation:

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

19. END