1. A different view of independent sets in bipartite graphs Qi Ge Daniel Štefankovič University of Rochester

2. counting/sampling independent sets in general graphs: A different view of independent sets in bipartite graphs polynomial time sampler for  5 (Dyer,Greenhill ’00, Luby,Vigoda’99, Weitz’06). no polynomial time sampler (unless NP=RP) for  25 (Dyer, Frieze, Jerrum ’02). Glauber dynamics does not mix in polynomial time for 6-regular bipartite graphs (example: union of 6 random matchings) (Dyer, Frieze, Jerrum ’02).  = maximum degree of G

3. counting/sampling independent sets in bipartite graphs: A different view of independent sets in bipartite graphs polynomial time sampler for  5 (Dyer,Greenhill ’00, Luby,Vigoda’99, Weitz’06). no polynomial time sampler (unless NP=RP) for  25 (Dyer, Frieze, Jerrum ’02). Glauber dynamics does not mix in polynomial time for 6-regular bipartite graphs (example: union of 6 random matchings) (Dyer, Frieze, Jerrum ’02). (max idependent set in bipartite graph  max matching)  = maximum degree of G

4. Why do we care? How hard is counting/sampling independent sets in bipartite graphs? * bipartite independent sets equivalent to * enumerating solutions of a linear Datalog program * downsets in a poset (Dyer, Goldberg, Greenhill, Jerrum’03) * ferromagnetic Ising with mixed external field (Goldberg,Jerrum’07) * stable matchings (Chebolu, Goldberg, Martin’10)

5. 0 0 1 1 0 1 0 1 0 1 0 1 0 1 1 1 0 0 1 0 1 1 0 1 0 1 1 0 0 1 1 0 0 1 1 1 0 Independent sets in a bipartite graph. 0-1 matrices weighted by (1/2) rank (1 allowed at A uv if uv is an edge) Ge, Štefankovič ’09 A different view of independent sets in bipartite graphs

6. 0 1 0 0 1 0 1 0 0 0 1 1 0 1 0 1 1 0 A different view of independent sets in bipartite graphs Ge, Štefankovič ’09 #IS = 2 |V  U| - |E|  2 -rk(A) A  B Independent sets in a bipartite graph. 0-1 matrices weighted by (1/2) rank (1 allowed at A uv if uv is an edge)

7. 0 1 0 0 1 0 1 0 0 0 1 1 0 1 0 1 1 0 A different view of independent sets in bipartite graphs Ge, Štefankovič ’09 #IS = 2 |V  U| - |E|  2 -rk(A) A  B Independent sets in a bipartite graph. 0-1 matrices weighted by (1/2) rank (1 allowed at A uv if uv is an edge) Question: Is there a polynomial-time sampler that produces matrices A  B with P(A)  2 -rank(A) B ij =0  A ij =0 (everything over the F 2 )

8. Natural MC flip random entry + Metropolis filter. A = X t with random (valid) entry flipped if rank(A)  rank(X t ) then X t+1 = A if rank(A) > rank(X t ) then X t+1 = A w.p. ½ X t+1 = X t w.p. ½ we conjectured it is mixing Goldberg,Jerrum’10: the chain is exponentially slow for some graphs. BAD NEWS:

9. Ising model: assignment of spins to sites weighted by the number of neighbors that agree Random cluster model: subgraphs weighted by the number of components and the number of edges High temperature expansion: even subgraphs weighted by the number of edges Our inspiration (Ising model): Fortuin-Kasteleyn Newell Montroll ‘53

10. Random cluster model Z(G,q,  )=  q  (S)  |S| SESE number of connected components of (G,S) (Tutte polynomial) Ising model Potts model chromatic polynomial Flow polynomial

11. Random cluster model Z(G,q,  )=  q  (S)  |S| SESE R 2 model R 2 (G,q,  )=  q rk(S)  |S| SESE 2 number of connected components of (G,S) rank (over F 2 ) of the adjacency matrix of (G,S) (Tutte polynomial) Ising model Potts model chromatic polynomial Flow polynomial Matchings Perfect matchings Independent sets (for bipartite only!) More ?

12. R 2 model ’ easy if (x-1)(y-1)=1, or (1,1),(-1,-1),(0,-1),(-1,0) #P-hard elsewhere Tutte polynomial easy if q  {0,1} or  =0, or (1/2,-1) #P-hard elsewhere (GRH) Complexity of exact evaluation Ge, Štefankovič ’09 Jaeger, Vertigan, Welsh ’90 2 |E|-|V|+|isolated V| spanning trees R 2 (G,q,  )=  q rk (S)  |S| SESE 2 ‘ BIS q

13. “high-temperature expansion”   (1-  (  (u),  (v)) U  {0,1} V  {0,1} {u,v}  E  1,1) = 1  0,1) =  (1,0) =  (0,0) = -1 where 2 |E| #BIS =

14. “high-temperature expansion”   (1-  (  (u),  (v)) U  {0,1} V  {0,1} {u,v}  E  1,1) = 1  0,1) =  (1,0) =  (0,0) = -1 where 2 |E| #BIS = =  (-1) |S|     (  (u),  (v)) SESE U  {0,1} V  {0,1} {u,v}  S

15. “high-temperature expansion”   (1-  (  (u),  (v)) U  {0,1} V  {0,1} {u,v}  E  1,1) = 1  0,1) =  (1,0) =  (0,0) = -1 where 2 |E| #BIS = =  (-1) |S|     (  (u),  (v)) SESE U  {0,1} V  {0,1} {u,v}  S = { 0 if some v  V has an odd number of neighbors in (U  V,S) labeled by 1 (-2) |V| otherwise

16. “high-temperature expansion” 2 |E| #BIS = =  (-1) |S|     (  (u),  (v)) SESE U  {0,1} V  {0,1} {u,v}  S bipartite adjacency matrix of (U  V,S) = 2 |V|  SESE number of u such that u T A = 0 (mod 2) = 2 |V|+|U|  SESE 2 - rank (A)) 2

17. “high-temperature expansion” – curious f(A, ) =  |v| ( ) |Av| 1- 1+ f(A,1) = 2 rank (A) 1 1 2 f(A,1) = f(A,1) T But in fact: f(A, ) = f(A, ) T

18. Questions: Is there a polynomial-time sampler that produces matrices A  B with P(A)  2 -rank(A) ? What other quantities does the R 2 polynomial encode ? R 2 (G,q,  )=  q rk(S)  |S| SESE 2