Message Passing for the Coloring Problem: Gallager Meets Alon and Kahale

1. Message Passing for the Coloring Problem:Gallager Meets Alon and Kahale Sonny Ben-Shimon and Dan Vilenchik Tel Aviv University AofA June, 2007 TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAA

2. “noisy channel” LDPC codes • Low Density Parity Check Codes • Cis a binary linear code (i.e. a linear subspace of GF[2]n) • Ais a m£n binary matrix s.t. Ac=0for every codeword c in C • Every row of A has a Hamming-weight of at most k (constant) c c’ Message Passing Algorithm for the Coloring Problem

3. Its preferred value from the previous round What is should be to satisfy the constraint Gallager’s Decoding Algorithm • From the matrix Abuild an auxiliary bipartite graph B[A] • Left side consists of the variables • x1,x2,…,xn • Right side consists of the constraints • C1,C2,…,Cr Message Passing Algorithm for the Coloring Problem

4. Gallager’s Decoding Algorithm x1 C1 x2 C2 x3 C3 x4 C4 x5 Message Passing Algorithm for the Coloring Problem

5. Decoding via Message Passing • [SiSp]: Assume B[A] is an expander graph and dist(c,c’)·n/const then Gallager’s decoding algorithm converges Message Passing Algorithm for the Coloring Problem

6. Random 3-colorable graphs • The planted modelGn,p,3[Kučera] • Partition the vertex set, V, into 3 color classes of size n/3each. • Denote by *:V! {1,2,3} the partition • Include each random edge between two distinct color classed with probability p=p(n) • [AK] if p(n)¸const/n then * can be recovered w.h.p. in polynomial time Message Passing Algorithm for the Coloring Problem

7. Alon and Kahale’s Coloring Algorithm Obtain an initial 3-coloring of the graph (using spectral methods) Not necessarily proper • For i = 1 to logn do: • for all v2V greedily color v • While 9v2V with <const neighbors colored in some other color uncolor v U½V - the set of uncolored vertices If there exists a connected component in G[U]of size at least logn – FAIL Else, exhaustively extend the coloring of VnU to G[U] Message Passing Algorithm for the Coloring Problem

8. Similarities of Decoding and Coloring • Two 3-colorings , are at distancet if • They disagree on the color of at least t vertices in every permutation of the color classes • There exists one permutation obtaining equality • Given a sampled graph G from Gn,p,3,:V! {1,2,3} s.t. dist(, * )·n/const we try to recover * •  in [AK] is given by the spectral method Not necessarily proper So, maybe we can use Gallager’s algorithm?! Message Passing Algorithm for the Coloring Problem

9. Its most likely color from the previous round What color xi shouldn’t be Gallager for Coloring • From the graph Gbuild an auxiliary bipartite graph B[G] • Left side consists of the variables (vertices) • x1,x2,…,xn • Right side consists of the constraints (edges) • C1,C2,…,Cr This is basically what [AK] does! Message Passing Algorithm for the Coloring Problem

10. Proof Outline – d-regular case • Given a sampled graph G from Gn,p,3 • Assume every vertex has np/3 neighbors in the other color classes Claim: W.h.p. there is no U½Vs.t. |U|<n/60 and e(U)>np|U|/10 • Let Ui ={ v | i(v)  *(v) } • |U0|<n/90 • Assume 2|Uj|>|Uj-1| for the first time Message Passing Algorithm for the Coloring Problem

11. Proof Outline – d-regular case Every u2Uj has ¸ 2np/9 neighbors inUj-1 Green Yellow g2 g1 y1 y2 b1 b2 Blue u Message Passing Algorithm for the Coloring Problem

12. Proof Outline – d-regular case • Let U=Uj[Uj-1 • |U|<1.5|Uj|< n/60 • e(U) ¸2np|Uj|/9 > np|U|/10 Contradiction! • This algorithm converges after logn iterations • But if only life was perfect… Message Passing Algorithm for the Coloring Problem

13. Proof Outline – Remarks • Every vertex is expected to have np/3 neighbors in every other color class • if p¸logn/ n then the degrees of vertices will be relatively “close” to the expectation • for p=const/n some vertices can have very low degree (less than 3). One cannot expect to recover their color. Message Passing Algorithm for the Coloring Problem

14. Proof Outline – The Core • Definition of H=Core(G): maximal subset of vertices s.t. • 8 v2H has at least (1-²)np/3H-neighbors in the other color classes • v has at most ²‘np neighbors outside of H • It follows easily that on the vertices of H, the algorithm converges • But, is there such an H? and is it big? Message Passing Algorithm for the Coloring Problem

15. V1 V2 V3 Proof Outline – Outside The Core Corollary: • (1-exp{-(np)})n vertices are frozen in every proper 3-coloring • Only one cluster of exponential size V1 V2 V3 Message Passing Algorithm for the Coloring Problem

16. Concluding Remarks • If the factor graph is even “close” to an expander graph, one can recover most of the coloring (codeword) • May be interesting to apply this scheme to general constraint systems Message Passing Algorithm for the Coloring Problem

17. Merci… Message Passing Algorithm for the Coloring Problem