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

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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

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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

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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)

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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

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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)

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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 )

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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:

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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

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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

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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 ?

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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

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“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 =

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“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

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“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

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“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

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“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

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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

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