Sampling Binary Contingency Tables with a Greedy Start

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Sampling Binary Contingency Tables with a Greedy
Start

• Ivona Bezáková
• (U. Chicago)
• Nayantara Bhatnagar
• Eric Vigoda
• (Georgia Tech)

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Darwins Finches
A Field Guide to the Birds of Galapagos Taken
from http//www.rit.edu/rhrsbi/
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Sampling Binary Contingency Tables

• Problem 1 Given row sums r1 rn and column sums
  c1 cm, generate a random 0-1 matrix with these
  marginals.
• Equivalently

r1 r2 r3
c1 c2 c3
cn
rn
Generate a random bipartite graph with a given
degree sequence.
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Previous Work

• Jerrum, Sinclair 90,
• Kannan, Tetali, Vempala 97
• Regular degree sequences.
• Cooper, Dyer, Greenhill 05
• Regular degree sequences.
• General graphs (non-bipartite).
• Jerrum, Sinclair, Vigoda 01
• All degree sequences.
• By reduction to permanent.
• Quadratic blowup in instance size.
• O(n14) algorithm using
• Bezáková, tefankovic, Vazirani, Vigoda 06.
• Our Results
• All degree sequences.
• Direct algorithm.
• O(n11) worst case.

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Binary Contingency Tables with Cell Bounds

• Problem 2 Given a bipartite graph G, and a
  degree sequence r1 rn, c1 cm generate a
  random subgraph with these degrees.

1 2 1 1 1
2 1
1 1
3
2

• Special cases
• G is Kn,m Binary Contingency Tables.
• G is arbitrary, ri cj 1 Permanent.

Main Result An FPAUS to generate a random
subgraph of G with the desired degree sequence.
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A Simple Markov Chain Jerrum, Sinclair
Perfect tables P
Near-Perfect tables N (u,v)
u
v
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A Simple Markov Chain
Perfect table
Pick e ? E u.a.r from current graph, delete it.
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A Simple Markov Chain
Near-perfect table
Pick (x,y) ? E U (u,v). If (x,y) (u,v), add it.
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A Simple Markov Chain
Near-Perfect table
Pick (x,y) ? E U (u,v).
u
y
u
v
x
v
u
y
v
x
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Reweighting the Chain

• P set of perfect tables.
• N set of near-perfect tables.
• Simple Markov Chain
• Stationary distribution uniform on P ?N
• JS Mixes rapidly if N poly(n).P .
• Problem What if N gtgt P ?

• Reweighted Markov Chain
• For G?P, w(G) 1.
• For G?N (u,v), w(G)
• Define transitions s.t. p(G) a w(G).
• In the stationary dist, p( P ) p(N (u,v)).
• So we end at a perfect table w.p 1/(n2 1).

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Annealing for MatchingsJerrum, Sinclair, Vigoda

• Given graph H, compute

For matching M, let ?(M) ? of non-edges of
H. Let ?(P ) ? ?(M) for M ? P Let ?(N
(u,v)) ? ?(M) for M ? N (u,v)
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Annealing for Matchings

• ?(P ) is a polynomial in ?.
• Changing ? by tiny amount changes ?(P ) by at
  most constant factor.
• Boost approx. factor by sampling.
• Faster annealing by BSVV 06.

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Annealing for Matchings
Computing ?(P ) for ? 1 - 1/n Let Pi
matchings in Kn.n with i edges from Hc. ?(P )
P0 ?P1 ?2P2 ?nPn Hence ?(P ) changes by
at most e. The factor e can be improved by
sampling.
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Annealing for Binary Contingency Tables
? 1
?
? 0
At ? 1, ?(P ) P and ?(N (u,v)) N (u,v)
At ? 0, ?(P ) 1 and ?(N (u,v )) 1 or 0.

Need non-zero approx. for ?(N (u,v )) at ? gt 0.
There is a starting graph, GREEDY, for which low
degree coefficients of ?(N (u,v )) are gt 0.
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The Greedy Graph

• Constructing GREEDY
• Sort both sides by increasing degree.
• Repeat
• Match the highest deg. vertex v from left to
  d(v) highest vertices from right.
• Sort the right by residual degree, break ties
  by previous round.

1 1 3 3
2 2 2 2
2 1 1 1
1 1 0 0
1 1 3 0
1 1 0 0
0 0 0 0
0 0 0 0

• Characterization Given degree sequence is
  feasible iff the algorithm succeeds.
• Tie-break rule is defined inductively.

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The Greedy Start

• Lemma If there is a near-perfect graph with
  holes at u and v, there is an alternating path of
  length at most 5 from u to v in GREEDY.

? Non-zero approx. for ?( N (u,v)) at ? 0.
u
v
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The Greedy Start

• Lemma If there is a near-perfect graph with
  holes at u and v, there is an alternating path of
  length at most 5 from u to v in GREEDY.

? Non-zero approx. for ?( N (u,v)) at ? 0.
u
v
Greedy algorithm with arbitrary tie-breaking.
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The Greedy Start

• Lemma If there is a near-perfect graph with
  holes at u and v, there is an alternating path
  from u to v in GREEDY of length at most 5.

u
u
v
v
Recall that
?(P ) ? ?edges of H not in G
H?P
1 ?H H ? Gc 1 ?2 H H ?
Gc 2 ?3()
?(N(u,v) ) H H?Gc 0
?H H?Gc 1 ?2() ?3()
By lemma, one of first 3 terms is non-zero.
Computing these terms gives an approx.
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Tables with Cell Bounds
To generate subgraphs of a given graph G
Input graph G
Degrees r1 r2 r3 1 c1 c2 c3 1
First Phase
GREEDY(r,c)
GREEDY G
Second Phase
G
GREEDY G