When Perm(A) = |M(G)| where M(G) is the - - PowerPoint PPT Presentation

APPROXIMATING THE PERMANENT

Mark Jerrum and Alistair Sinclair Presented by Dzung Nguyen

PERMANENT PROBLEM



When Perm(A) = |M(G)| where M(G) is the set of perfect matchings of the bipartite graph G=(U, V, E) with (

, ) iff 1

i j ij

u v E a  

A =



rows columns

1 1 1 1 1 1 1 1            

OUTLINE

1.

Reduction from counting problem to generating problem.

2.

Markov Chain and converging speed.

3.

Solution

4.

Conclusions

UNIFORM GENERATOR AND RANDOMIZED COUNTER

 Counting the number of perfect matchings is self-

• reducible. Let be an arbitrary edge of G.

 Claim: “If there exist a fully polynomial almost

uniform generator of perfect matchings then there exists a fully polynomial randomized algorithm to compute the number of perfect matchings.”

( ) ( \ ) { { } : ( \ { , })} M G M G e M e M M G u v   

( , ) e u v 

UNIFORM GENERATOR AND RANDOMIZED COUNTER

Aim:

 Estimate M(G1)  Estimate ration

M(G1)/M(G)

 Sample t elements

• f M(G) to count

number of elements in M(G1)

 t is big enough and

choose the best G1 which yield maximum ratio

select e M(G) not select e M(G1)

OUTLINE

1.

Reduction from counting problem to generating problem.

2.

Markov Chain and converging speed.

3.

Solution

4.

Conclusions

MARKOV CHAIN AND CONVERGING SPEED

 Given a finite state space N, Markov Chain on N is

a series of states with transition matrix P.

 If the chain is ergodic, denote be the

stationary distribution, the unique vector satisfying and .

 The relative point wise distance (r.p.d.) after t steps:  An ergodic Markov chain is said to be time-

reversible if it satisfies the detailed balance condition:

( )

t t

X

 

( )

i i N

 





P   

1

i i N









,

| | ( )

max

t ij j i j N j

p t  



  

,

ij i ji j

p p i j N     

MARKOV CHAIN (CONTINUE)

 A time-reversible chain is represented by graph H

• f |N| vertices with the weight

 The conductance of a time-reversible chain is

defined by:

OUTLINE

1.

Reduction from counting problem to generating problem.

2.

Markov Chain and converging speed.

3.

Solution

4.

Conclusions

UNIFORM GENERATOR AND MARKOV CHAIN

 Suppose the Markov Chain on the set of elements

S is ergodic and uniform distribution when the number of step go to infinite

 Build a polynomial time almost uniform generator

by simulating the chain with large enough t steps. t is poly(|S|)

 What is the set of elements here? Set of perfect

matching?

 What is the transition matrix?

SOLUTION:

S is the set of perfect matchings and near-perfect matchings (two vertices are not matched). In any state , choose an edge uniform random and then:

 (i) If M is a perfect matching and , move to state  (ii) If M is a near-perfect matching and u, v are

unmatched in M, move to

 (iii) If M is a near-perfect matching, u is matched to

w in M and v is unmatched in M, move to ; symmetrically, if v is matched to w and u is unmatched, move to

 (iv) In all other cases, do nothing.

With probability ½, the state moves to itself.

( ) M M G 

( , ) e u v E  

e M 

' M M e  

' M M e  

' ( , ) M M e u w    ' ( , ) M M e w v   

RESULTS

 The process is ergodic and time-reversal with

uniform stationary distribution.

 In dense graph i.e. degree of each vertex is at least

n/2, the process converges rapidly and at least 1/n2 elements of S are perfect matching.

CONCLUSIONS

 Show the relationship between generating and

counting problems.

 The relationship between the conductance of

underlying graph and the converging speed.

 The technique to prove the lower bound of the

conductance.

 Further: computer the permanent of all 0-1 matrix,

matrix with non-negative elements.