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A New Perspective on Independent Sets in Bipartite Graphs

This paper presents a novel view on counting and sampling independent sets in bipartite graphs, analyzing polynomial-time sampling techniques. We explore the implications of maximum degree on the feasibility of efficient samplers, showing that while a polynomial-time sampler exists for graphs with maximum degree Δ ≤ 5, none exist for Δ ≥ 25 unless NP = RP. The study also examines Glauber dynamics in 6-regular bipartite graphs, illustrating its non-mixing behavior. This research emphasizes the complexity inherent in enumerating independent sets and their relevance to various combinatorial structures.

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A New Perspective on Independent Sets in Bipartite Graphs

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

  2. A different view of independent sets in bipartite graphs counting/sampling independent sets in general 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. A different view of independent sets in bipartite graphs counting/sampling 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). (max idependent set in bipartite graph  max matching) 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

  4. How hard is counting/sampling independent sets in bipartite graphs? Why do we care? * 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. A different view of independent sets in bipartite graphs 0 0 0 0 0 0 0 1 1 0 1 0 1 0 1 1 0 1 0 0 0 0 1 0 1 1 1 1 0 1 1 1 1 0 0 0 0 0 1 1 0 1 1 0 1 1 0 1 Ge, Štefankovič ’09 1 1 0 0 0 1 0 1 1 0 0 1 1 1 1 0 Independent sets in a bipartite graph. 0-1 matrices weighted by (1/2)rank (1 allowed at Auv if uv is an edge)

  6. A different view of independent sets in bipartite graphs 0 0 0 0 1 0 1 0 0 1 0 0 0 0 1 0 1 0 0 0 0 1 1 0 Ge, Štefankovič ’09 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 Auv if uv is an edge) #IS = 2|VU| - |E| 2-rk(A) A  B

  7. A different view of independent sets in bipartite graphs Question: Is there a polynomial-time sampler that produces matrices A  B with P(A)  2-rank(A) 0 0 0 0 1 0 1 0 0 1 0 0 0 0 1 0 1 0 0 0 0 1 1 0 Ge, Štefankovič ’09 1 1 0 0 1 1 1 0 Bij=0  Aij=0 Independent sets in a bipartite graph. 0-1 matrices weighted by (1/2)rank (1 allowed at Auv if uv is an edge) #IS = 2|V U| - |E| 2-rk(A) (everything over the F2) A  B

  8. Natural MC flip random entry + Metropolis filter. A = Xt with random (valid) entry flipped if rank(A)  rank(Xt) then Xt+1 = A if rank(A) > rank(Xt) then Xt+1 = A w.p. ½ Xt+1 = Xt w.p. ½ we conjectured it is mixing BAD NEWS: Goldberg,Jerrum’10: the chain is exponentially slow for some graphs.

  9. Our inspiration (Ising model): Fortuin-Kasteleyn 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 Newell Montroll ‘53 High temperature expansion: even subgraphs weighted by the number of edges

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

  11. Random cluster model R2 model R2(G,q,)=  qrk(S)|S| Z(G,q,)=  q(S)|S| 2 SE SE number of connected components of (G,S) rank (over F2) of the adjacency matrix of (G,S) Matchings Perfect matchings Independent sets (for bipartite only!) (Tutte polynomial) Ising model Potts model chromatic polynomial Flow polynomial More ?

  12. Complexity of exact evaluation ‘ R2(G,q,)=  qrk (S)|S| 2 SE Tutte polynomial ’ R2 model spanning trees BIS 2|E|-|V|+|isolated V| q Ge, Štefankovič ’09 Jaeger, Vertigan, Welsh ’90 easy if (x-1)(y-1)=1, or (1,1),(-1,-1),(0,-1),(-1,0) easy if q{0,1} or =0, or (1/2,-1) #P-hard elsewhere #P-hard elsewhere (GRH)

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

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

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

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

  18. Questions: Is there a polynomial-time sampler that produces matrices A  B with P(A)  2-rank(A) ? What other quantities does the R2 polynomial encode ? R2(G,q,)=  qrk(S)|S| 2 SE

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