Download presentation

Presentation is loading. Please wait.

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| SESE 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| SESE R 2 model R 2 (G,q, )= q rk(S) |S| SESE 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| SESE 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)) SESE 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)) SESE 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)) SESE U {0,1} V {0,1} {u,v} S bipartite adjacency matrix of (U V,S) = 2 |V| SESE number of u such that u T A = 0 (mod 2) = 2 |V|+|U| SESE 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| SESE 2

Similar presentations

OK

CS774. Markov Random Field : Theory and Application Lecture 17 Kyomin Jung KAIST Nov 05 2009.

CS774. Markov Random Field : Theory and Application Lecture 17 Kyomin Jung KAIST Nov 05 2009.

© 2018 SlidePlayer.com Inc.

All rights reserved.

To make this website work, we log user data and share it with processors. To use this website, you must agree to our Privacy Policy, including cookie policy.

Ads by Google

Pancreas anatomy and physiology ppt on cells Ppt on id ego superego image Ppt on c programming tutorial Ppt on layer 3 switching hub Ppt on acid-base indicators animations free Ppt on 60 years of indian parliament today Ppt on edge detection image Ppt on artificial intelligence download Ppt on personal grooming and etiquette Ppt on natural heritage of india