Approximate Counting
Subhasree Basu
Rajiv Ratn Shah
Vu Vinh An
Kiran Yedugundla
School of Computing
National University of Singapore
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Agenda
•
•
•
•
•
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Background
The Class #P
Randomized Approximation Schemes
The DNF Counting Problem
Approximating the Permanent
Summary
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Agenda
•
•
•
•
•
•
Background
The Class #P
Randomized Approximation Schemes
The DNF Counting Problem
Approximating the Permanent
Summary
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Solving and Counting
• Suppose a problem ∏
• We check if I is an YES-instance of ∏
• That usually involves finding the solution to ∏ and checking if
I matches it
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Example
If we want to find all possible permutations of
3 different items
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Example
If we want to find all possible permutations of
3 different items
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Solving and Counting
• When we are trying to count, we actually try to find the
number of solutions of ∏ where I is a solution
• Following the same example, in the counting version we want
to know how many such permutations exist
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Approximate Counting
• Approximate Counting – involves two terms Approximate and
Counting
• Counting is getting the number of solutions to a problem
• Approximate because we do not have an exact counting
formula for a vast class of problems
Approximately counting Hamilton paths and cycles in dense graphs
Martin Dyer, Alan Frieze, and Mark Jerrum, SIAM Journal on Computing 27 (1998), 12621272.
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Agenda
•
•
•
•
•
•
Background
The Class #P
Randomized Approximation Schemes
The DNF Counting Problem
Approximating the Permanent
Summary
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The Class #P
• #P is the class of counting problems associated with the NP
decision problems
• Formally a problem ∏ belongs to the #P if there is a nondeterministic polynomial time Turing Machine that, for any
instance I, has a number of accepting computations that is
exactly equal to the number of distinct solutions to instance I
• ∏ is #P-complete if for any problem ∏’ in #P, ∏’ can be
reduced to ∏ by a polynomial time Turing machine.
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Need for Randomization
• #P-complete problems can only be solvable in polynomial
time only if P=NP
• Hence the need for approximate solutions
• Randomization is one such technique to find approximate
answers to counting problems
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Various Applications for Approximate
Counting
•
•
•
•
•
DNF counting problem.
Network reliability.
Counting the number of Knapsack solutions.
Approximating the Permanent.
Estimating the volume of a convex body.
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Various Applications for Approximate
Counting
•
•
•
•
•
DNF counting problem.
Network reliability.
Counting the number of Knapsack solutions.
Approximating the Permanent.
Estimating the volume of a convex body.
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Agenda
•
•
•
•
•
•
Background
The Class #P
Randomized Approximation Schemes
The DNF Counting Problem
Approximating the Permanent
Summary
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Polynomial Approximation Scheme
• Let #(I) be the number of distinct solutions for instance I of
problem ∏
• Let the Approximation algorithm be called A
• It takes an input I and outputs and integer A(I)
• A(I) is supposed to be close to #(I)
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Polynomial Approximation Scheme
DEF 1: A Polynomial Approximation scheme(PAS) for a
counting problem is a deterministic algorithm A that takes and
input instance I and a real number ε > 0, and in time
polynomial in n = |I| produces an output A(I) such that
(1- ε) #(I) < A(I) < (1 + ε) #(I)
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Polynomial Approximation Scheme
DEF 1: A polynomial Approximation scheme(PAS) for a counting problem is a
deterministic algorithm A that takes and input instance I and a real number ε > 0,
and in time polynomial in n = |I| produces an ouput A(I) such that
(1- ε) #(I) < A(I) < (1 + ε) #(I)
DEF 2 :A Fully Polynomial Approximation Scheme (FPAS) is a
polynomial approximation scheme whose running time is
polynomially bounded in both n and 1/ε.
The output A(I) is called an ε-approximation to #(I)
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Polynomial Randomized Approximation
Algorithm
DEF 3: A Polynomial Randomized Approximation
scheme(PRAS) for a counting problem ∏ is a randomized
algorithm A that takes an input instance I and a real number
ε>0, and in time polynomial in n = |I| produces an output
A(I) such that
Pr[(1- ε)#(I) ≤ A(I) ≤ (1 + ε)#(I)] ≥ ¾
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Polynomial Randomized Approximation
Algorithm
DEF 3: A Polynomial Randomized Approximation scheme(PRAS) for a counting
problem ∏ is a randomized algorithm A that takes an input instance I and a real
number ε>0, and in time polynomial in n = |I| produces an output A(I) such that
Pr[(1- ε)#(I) ≤ A(I) ≤ (1 + ε)#(I)] ≥ 3/4
DEF 4: A Fully Polynomial Randomized Approximation Scheme
(FPRAS) is a polynomial randomized approximation scheme
whose running time is polynomially bounded in both n and
1/ε.
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An (ε, δ) FPRAS
DEF 5: An (ε, δ)-FPRAS for a counting problem ∏ is a fully
polynomial randomized approximation scheme that takes an
input instance I and computes an ε-approximation to #(I) with
probability at least 1- δ in time polynomial in n , 1/ ε and
log(1/δ)
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Agenda
•
•
•
•
•
•
Background
The Class #P
Randomized Approximation Schemes
The DNF Counting Problem
Approximating the Permanent
Summary
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Monte Carlo Method
• It is a wide-range of algorithms under one name
• In essence it includes all the techniques that use random
numbers to simulate a problem
• It owes its name to a casino in the principality of Monte Carlo.
• Events in a casino depend heavily on chance of a random
event.
e.g., ball falling into a particular slot on a roulette wheel,
being dealt useful cards from a randomly shuffled deck, or the
dice falling the right way
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DNF Counting - Terminologies
• F(X1, X2,… Xn) is Boolean formula in DNF
–
–
–
–
X1, X2,… Xn are Boolean variables.
F = C1 ∨ C2… ∨ Cm is a disjunction of clauses
Ci = L1 ∧ L2… ∧ Lri is a conjunction of Literals
Li is either variable Xk or Xk’
e.g. F = (x1 ∧ x3’) ∨ (x1 ∧ x2’ ∧ x3) ∨ (x2 ∧ x3)
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Terminologies contd …
• a = (a1, a2, … an ) is truth assignment
• a satisfy F if F(a1, a2, … an ) evaluates to 1 or TRUE
• #F is the number of distinct satisfying assignment of F
Clearly we have here 0 < #F ≤ 2n
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DNF Counting Problem
• The Problem at hand is now to compute the value of #F
• It is known to be #P complete
(We can reduce #SAT to #P complete)
• We will describe an (ε, δ)- FPRAS algorithm for this
• The input size for this is at most nm
• We have to design a approximation scheme that has a running
time polynomial in n, m , 1/ ε and log(1/ δ)
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Some more terminologies
• U is a finite set of known size
• f: U
{0,1} be a Boolean function over U
• Define G = {u є U | f(u) =1} as the pre-image of U
Assume: we can sample uniformly at random from U
We now want to find the size of G i.e., |G|
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Formulation of Our Problem
• Let in our formulation U = {0,1}n , the set of all 2n truth
assignments
• Let f(a) = F(a) for each of aєU
• Hence our G is now the set of all satisfying truth assignments
for F
Our problem thus reduces to finding the size of G
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Monte Carlo method
• Choose N independent samples from U, say, u1, u2, ……. ,uN
• Use the value of f on these samples to estimate the
probability that a random choice will lie in G
• Define the random variable
Yi = 1 if f(ui) = 1 ,
Yi = 0 otherwise
So Yi is 1 if and only if uiєG
• The estimator random variable is
Yi
Z = |U| N
We claim that with high probability Z will be an approximation to
|G|
N
i 1
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An Unsuccessful Attempt (cont.)
• Estimator Theorem:
Let ρ =
|𝐺|
,
|𝑈|
Then Monte Carlo
method yields an 𝜖-approximation to |G| with probability at
least 1 – δ provided N ≥ 𝜖4ρ ln δ2
2
• Why unsuccessful?
– We don’t know the value of ρ
– We can solve this by using a successively refined lower bound
on ρ to determine the number of samples to be chosen
– It has running time of at least N. where N ≥ ρ1
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Agenda
• Background
• Randomized Approximation Schemes
• The DNF Counting Problem
– DNF Counting
– Compute #F
– An Unsuccessful Attempt
– The Coverage Algorithm
• Approximating the Permanent
• Summary
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DNF Counting
•
•
•
•
•
•
•
Let F = C1∨ C2∨ C3∨ C4 = (x1 ∧ x3’) ∨ (x1 ∧ x2’∧ x3) ∨ (x2 ∧ x3) ∨ (x3’)
G be set of all satisfying truth assignment
Hi be set of all satisfying truth assignment for Ci
H1 = {a5, a7}
x1 x2 x3
H2 = {a6}
a1 0 0 0
H2
H3 = {a4, a8}
a2 0 0 1
H4
H4 = {a1, a3, a5, a7}
a3 0 1 0
• H = H1 ∪ H2 ∪ H3 ∪ H4
= {a1, a3, a4, a5, a6, a7, a8}
a4
0
1
1
a5
1
0
0
a6
1
0
1
a7
1
1
0
a8
1
1
1
H1
H3
• It is easy to see that |Hi | = 2 n-ri
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DNF Counting
a1
a2
.
.
ai
aj
.
0
1
,
f(a) = F(a)
f: V→ {0, 1}
.
.
.
a2 n
|G| = #F
V
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The Coverage Algorithm
• Importance Sampling
– Want to reduce the size of the sample space
– Ratio ρ is relatively large
– Ensuring that the set G is still completely represented.
– Reformulate the DNF counting problem in a more abstract
framework, called the union of sets problem.
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union of sets problem
• union of sets problem
Let V be a finite Universe. We are given m subsets H1, H2, …, Hm ⊆ V,
such that following assumptions are valid:
1. For all i, |Hi| is computable in polynomial time.
2. It is possible to sample uniformly at random from any Hi
3. For all v ∈ V, it can be determined in polynomial time whether v ∈ Hi
– The Goal is to estimate the size of the Union H = H1 ∪ H2 ∪ … ∪ Hm
– The brute-force approach to compute |H| is inefficient when the
universe and the set Hi are of large cardinality
– The assumption 1-3 turn out to be sufficient to enable the design of
Importance sampling algorithm
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.
DNF Counting
.
.
0,
1
H1
H2
.
.
.
.
H.m
V
𝐻 = ∪𝑖=1…𝑚 Hi
F[v] → {0, 1}
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The Coverage Algorithm
• DNF Counting is special case of union of sets
–
–
–
–
F(X1, X2,… Xn) is Boolean formula in DNF
The universe V corresponds to the space of all 2n truth assignment
Set Hi contains all the truth assignment that satisfy the clause Ci
Easy to sample from Hi by assigning appropriate values to variables
appearing in Ciand choosing rest at random
– Easy to see that |Hi | = 2 n-ri
– Verify in linear timethat some v ∈ V is a member of Hi
– Then the union of sets 𝐻 = ∪𝑖=1…𝑚 Hi gives the set of satisfying
assignments for F
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The Coverage Algorithm
• Solution to union of sets problem.
– Define a multiset U = H1 ⊎ H2 … ⊎ Hm
– Multiset union contains as many copies of v ∈ V as the number of
Hi`sthat contain that v.
– U = {(v,i) | (v,i)∈ Hi}
– Observe that |U| = 𝑚
𝑖=1 |Hi| ≥ |H|
– For all v ∈ H, cov(v) = {(v,i) | (v,i) ∈ U}
– In DNF problem, for a truth assignment a, the set cov(a) is the set of
clauses satisfied by a
v1
v2
v3
v4
…
…
vs
H1
1
0
1
0
…
…
0
H2
1
1
0
0
…
…
1
…
…
..
…
…
…
…
…
Hm
0
0
1
1
…
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…
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The Coverage Algorithm
• The following observations are immediate
–
–
–
–
The number of coverage set is exactly |H|
U =∪𝑣 ∈𝐻 COV(v)
|U| = 𝑣 ∈𝐻 |𝐶𝑂𝑉(𝑣)|
For all v ∈ H, |COV(v)| ≤ m
H1
H2
H3
.
.
.
Hm
v1 v2
v3
*
x
0
.
.
.
x
*
0
x
.
.
.
0
0
*
0
.
.
.
0
…
vs
…
…
…
0
0
*
.
.
.
.
. .
…
x
– Define function f((v, i)) = {1 if i = min{ j |v ∈Hj}; 0 otherwise}
– Define set G = {(v, i) ∈ U | f((v, i)) = 1}
– |G| = |H|
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The Coverage Algorithm
• Algorithm
– for j = 1 to N do
• pick (v, i) uniformly at random from U
• Set Yj =1 if f((v, i)) = 1 ; else 0
// if((v, i)) = 1, call it special pair
– Output(𝑁1
– E[Yj] = 𝜇=
𝑁
𝑗=1 Yj).|U|
= Y .|U|
𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑠𝑝𝑒𝑐𝑖𝑎𝑙 𝑝𝑎𝑖𝑟𝑠
|𝑈|
=
|∪𝑖=1…𝑚 Hi|
|𝑈|
=
|𝐻|
|𝑈|
– So the algorithm is an unbiased estimator for |∪𝑖=1…𝑚 Hi| as desired
R.M. Karp and M. Luby, “Monte-carlo algorithms for enumeration and reliability problems,” In
Proceedings of the 15th Annual ACM Symposium on Theory of Computing, 1983, pp. 56–64.
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Analysis
• The Value of N:
– E[Yj] = µ ≥
𝑠
𝑚𝑠
=
1
𝑚
– By Estimator Theorem N =
4
ϵ 2μ
≤
4𝑚
𝜖2
= # trials
v1 v2
v3
H1
H2
H3
*
x
*
.
.
.
.
.
.
x
• Complexity
–
–
–
–
Computing |U| requires O(nm)
Checking (v, i) is special requires O(nm)
Generating a random pair requires O(n+m)
Total running time per trial = O(nm)
– Total running time =
𝑛𝑚2
O( 2 )
𝜖
Hm
*
.
.
.
…
vs
…
…
x …
*
. .
.
. .
.
.
. .
…
x
Karp, R., Luby, M., Madras, N., “Monte-Carlo Approximation Algorithms for Enumeration Problems", J.
𝑛𝑚
of Algorithms, Vol. 10, No. 3, Sept. 1989, pp. 429-448. improved the running time to O( 2 )
𝜖
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(𝜖, δ)-FPRAS for DNF Counting
• Our Goal is to estimate the size of G ⊆ U such
that G = f-1(1)
– Apply Estimator Theorem based on naïve Monte Carlo
sampling technique
We claim that the naïve Monte Carlo sampling algorithm
gives an (𝝐, δ)-FPRAS for estimating the size of G.
Lemma: In the union of sets problem ρ =
|𝐺|
|𝑈|
≥
1
𝑚
Proof: relies on the observations made above
• |U| =
𝑣 ∈𝐻
|𝐶𝑂𝑉(𝑣)| ≤
𝑣 ∈𝐻
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𝑚 ≤ m|H| = m|G|
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The Coverage Algorithm
• The Monte Carlo Sampling technique gives as
(𝜖, δ)-FPRAS for |G|, hence also for |H|
Estimator Theorem:The Monte Carlo method yields an 𝜖approximation to |G| with probability at least 1 - δ provided
2
N ≥ 4𝑚
ln
𝜖
δ
2
The running time is polynomial in N
Proof: We need to show:
(1) fcan be calculated in polynomial time
(2) It is possible to sample uniformly from U.
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The Coverage Algorithm
Proof(1): Compute f((v,i)) in O(mn) by checking whether truth
assignment v satisfies Ci but none of the clauses Cj for j <i.
Proof(2): It is possible to sample uniformly from U.
Sampling an element (v,i) uniformly from U:
I.
choose is.t. 1 ≤ i ≤ m and Pr[i] =
|𝐻𝑖|
|𝑈|
=
|𝐻𝑖|
𝑚 |𝐻 |
𝑖=1
𝑖
=
2𝑛
_
𝑘𝑖
𝑚 2𝑛 _ 𝑘𝑖
𝑖=1
II. Set the Variables in Hi so that Hi is satisfied
III. choose a truth assignment for the remaining variables
uniformly at random
IV. output iand the resulting truth assignment v
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Agenda
•
•
•
•
Background
Randomized Approximation Schemes
The DNF Counting Problem
Approximating the Permanent
– Number of perfect matchings in bipartite graph
– Near-uniform generation
– The canonical path argument
• Summary
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Matrix Permanent
• The permanent of a square matrix in linear algebra is a function of the
matrix similar to the determinant. The permanent, as well as the
determinant, is a polynomial in the entries of the matrix.
• Permanent formula:
Let 𝑄 = (𝑄𝑖𝑗 ) be an 𝑛 × 𝑛 matrix, the permanent of the matrix is defined as:
𝑛
𝑝𝑒𝑟 𝑄 =
𝑄𝑖,𝜋𝑖
𝜋∈𝑆𝑛 𝑖=1
Where 𝑆𝑛 𝑖𝑠 𝑡ℎ𝑒 𝑠𝑦𝑚𝑚𝑒𝑡𝑟𝑖𝑐 𝑔𝑟𝑜𝑢𝑝 𝑜𝑓 𝑝𝑒𝑟𝑚𝑢𝑡𝑎𝑡𝑖𝑜𝑛𝑠 𝑜𝑓 𝑠𝑖𝑧𝑒 𝑛
• Example:
𝑝𝑒𝑟
𝑎
𝑐
𝑏
= 𝑎𝑑 + 𝑏𝑐
𝑑
• Best running time: Ο(𝑛2𝑛 )
A Z Broder. 1986. How hard is it to marry at random? (On the approximation of the permanent). In
Proceedings of the eighteenth annual ACM symposium on Theory of computing (STOC '86). ACM, New
York, NY, USA, 50-58. DOI=10.1145/12130.12136
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Agenda
•
•
•
•
Background
Randomized Approximation Schemes
The DNF Counting Problem
Approximating the Permanent
– Number of perfect matchings in bipartite graph
– Near-uniform generation
– The canonical path argument
• Summary
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Bipartite Graph
• Definition: a bipartite graph is a set of graph vertices decomposed into
two disjoint sets such that no two graph vertices within the same set
are adjacent.
• Notations: G(U, V, E) where:
– 𝑈 = 𝑢1 , … , 𝑢𝑛 𝑎𝑛𝑑 𝑉 = {𝑣1 , … , 𝑣𝑛 } are disjoint sets of vertices in
which n is number of vertices in each set
– E is set of graph edges
• Bipartite graph G could be represented in 0-1 matrix A(G) as following:
U1
V1
V1
U1
U2
V2
U2
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V2
1 1
1 0
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Graph Perfect Matchings
• A matching is a collection of edges 𝑀 ⊆ 𝐸 such that vertex
occurs at most once in M
• A perfect matching is a matching of size n
U1
V1
U1
V1
U2
V2
U2
V2
Rajeev Motwani and Prabhakar Raghavan. 1995. Randomized Algorithms. Cambridge University Press,
New York, NY, USA. Chapter 11.
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Graph Perfect Matchings
• Let #(𝐺) denote the number of perfect matchings in the bipartite graph
G: #(𝐺) = 𝑝𝑒𝑟(𝐴 𝐺 )
• Computing the number of perfect matchings in a given bipartite graph is
#P-complete  computing permanent of 0-1 matrix is also #P-complete
• Propose a 𝜖, 𝛿 − 𝐹𝑃𝑅𝐴𝑆 for counting number of perfect matchings
problem
U1
V1
V1
U1
U2
U2
V2
#(𝐺) = 1
𝑝𝑒𝑟 𝐴 𝐺
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V2
1 1
1 0
= 1×0 + 1×1 = 1
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Introduction to Monte Carlo Method
• A method which solves a problem by generating
suitable random numbers and observing that
fraction of the numbers obeying some properties
• Monte Carlo methods tend to follow a particular
pattern:
– Define a domain of possible inputs.
– Generate inputs randomly from a
probability distribution over the
domain.
– Perform a deterministic
computation on the inputs.
– Aggregate the results.
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Estimator Theorem
• Estimator Theorem:
Let ρ =
|𝐺|
,
|𝑈|
Then Monte Carlo
method yields an 𝜖-approximation to |G| with probability at
least 1 – δ provided
4
2
𝑁 ≥ 2 ln
𝜖 𝜌 𝛿
where N ≥
1
ρ
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Agenda
•
•
•
•
Background
Randomized Approximation Schemes
The DNF Counting Problem
Approximating the Permanent
– Number of perfect matchings in bipartite graph
– Near-uniform generation
– The canonical path argument
• Summary
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Uniform Generation
• Let 𝑀𝑘 denote the set of distinct matchings of size k in G and
𝑚𝑘 = 𝑀𝑘
• A uniform generator for 𝑀𝑘 is a randomized polynomial time
algorithm 𝑈𝑘 that:
– Takes G as input
– Returns a matching 𝑚 ∈ 𝑀𝑘 such that m is uniformly
distributed over 𝑀𝑘
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Near-uniform generation for 𝑀𝑘
• Definition: Given a sample space omega, a generator
𝑈 is said to be a near-uniform generator for Ω with
error 𝜌 if, for all 𝜔 ∈ Ω:
Pr 𝑈 = 𝜔 − 1/|Ω|
≤𝜌
1/|Ω|
• A near-uniform generation for 𝑀𝑘 ∪ 𝑀𝑘−1 is the one
in which the sample space is 𝑀𝑘 ∪ 𝑀𝑘−1
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Near-uniform generation 𝑈𝑘 for 𝑀𝑘 ∪
𝑀𝑘−1
• Rationale:
– Use 𝑈𝑘 and Monte Carlo method to estimate ratios 𝑟𝑘 = 𝑚𝑘 /𝑚𝑘−1
(estimator theorem)
– 𝑟1 = 𝑚1 = 𝑛𝑜. 𝑜𝑓 𝑒𝑑𝑔𝑒𝑠 𝑖𝑛 𝑡ℎ𝑒 𝑖𝑛𝑝𝑢𝑡 𝑔𝑟𝑎𝑝ℎ
𝑚𝑛
𝑚𝑛−1
𝑚2
𝑚𝑛 =
×
× ⋯×
× 𝑚1 =
𝑚𝑛−1 𝑚𝑛−2
𝑚1
𝑛
𝑟𝑘
𝑘=1
1
– For bipartite graph G, try to find 𝑈𝑘 with error 𝜌 ≤ 4 and running
𝑛
time polynomially bounded in n
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School of
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Design 𝑈𝑘
• Devise Markov chain 𝐶𝑘 , each state is one element of
𝑀𝑘 ∪ 𝑀𝑘−1
• Simulate 𝐶𝑘 by executing a random walk on the
underlying graph of 𝐶𝑘 . Each vertex in the graph is
one element of 𝑀𝑘 ∪ 𝑀𝑘−1
• After 𝜏 steps (𝜏 is not too large), 𝐶𝑘 will approach its
stationary distribution. The stationary probability of
1
each state is equal to
𝑀𝑘 ∪𝑀𝑘−1
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School of
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Structure of 𝐶𝑛
• Underlying graph:
– Nodes: element of 𝑀𝑛 ∪ 𝑀𝑛−1
– Edges: let E denote the set of edges
in G. In any state m of 𝐶𝑛 , with
probability ½, remain at the state
and do nothing. Otherwise, choose
an edge e=(u,v):
•
Reduce: If 𝑚 ∈ 𝑀𝑛 and 𝑒 ∈ 𝑚, move
to state m’=m-e
•
Augment: If 𝑚 ∈ 𝑀𝑛−1 , with u and v
unmatched in m, move to m’=m+e
•
Rotate: If 𝑚 ∈ 𝑀𝑛−1 , with u
matched to w and v unmatched,
move to m’=(m+e)-f where f=(u,w)
•
Reduce
Augmen
t
Rotate
Idle: otherwise, stay at current place
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School of
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𝐶2 for 𝐺 = 𝐾22
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School of
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Properties of 𝐶𝑛
• Each transition has an associated probability
1
of
2|𝐸|
• If a transition exists from m to m’ then the
reverse transition also exists and has the same
probability
Approximating the Permanent Mark Jerrum and Alistair Sinclair, SIAM Journal
on Computing 18 (1989), 1149-1178.
.
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School of
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𝐶𝑛 Transition Probability Matrix
• 𝜋: the stationary distribution of 𝐶𝑛
• 𝑋𝑡 : the state of 𝐶𝑛 after t steps of simulation with 𝑋0 is the
beginning state
• 𝑝𝑖𝑗 : the probability of transition i  j then:
(𝑡)
1
𝑝𝑖𝑗 =
2𝐸
1
𝑝𝑖𝑖 ≥
2
•
𝑝𝑖𝑗
•
𝑃 = 𝑝𝑖𝑗 : transition probability matrix
•
𝑃𝑡 = [𝑝𝑖𝑗
Conditional probability that 𝑋𝑡 = 𝑗 given that 𝑋0 = 𝑖
𝑡
]
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School of
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Prove 𝜖, 𝛿 − 𝐹𝑃𝑅𝐴𝑆 𝑜𝑓 𝑈𝑘
• Proof sketch:
– Prove the stationary distribution of 𝐶𝑛 is a uniform
distribution after finite steps
– The approximated stationary distribution does not
depend on the starting state when number of
simulation steps is polynomially bounded in n and
1
the error is less than 4
𝑛
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School of
Computing
Uniform stationary distribution of 𝐶𝑛
• The underlying graph of 𝐶𝑛 :
– Irreducible: we can go from any matching in a graph to any
other matching
– Aperiodic: self-loop probabilities are positive
𝐶𝑛 is ergodic
Any finite, ergodic Markov Chain converges to a unique
stationary distribution after finite steps
• Transition probability matrix P:
– Symmetric, doubly stochastic: each of rows and columns
sum to 1
The stationary distribution of 𝐶𝑛 must be the uniform
distribution on its states
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School of
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Approximated stationary distribution
• Relative point-wise distance:
𝑝𝑖𝑗 (𝑡) − 𝜋𝑗
∆ 𝑡 = 𝑚𝑎𝑥
𝜋𝑗
• Need to prove that ∆(𝜏) reduces below
polynomial in n
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1
𝑛4
for 𝜏
School of
Computing
Proof ∆ 𝑡 <
1
𝑛4
• 𝜆1 > 𝜆2 ≥ ⋯ 𝜆𝑛 are eigenvalues of P
• 𝑁 = 𝑀𝑛 ∪ 𝑀𝑛−1 is number of states in 𝐶𝑛
ln 𝑁 = Ο(𝑛2 )
• By theorem 6.21:
∆(𝑡) ≤ 𝑛15 (𝜆2 )𝑡
• By Proposition B.3:
𝜆2 ≤ 𝑒 −
• Choose t = 𝜏 =
(4 ln 𝑛)(ln 𝑁)
1−𝜆2
15
∆(𝑡) ≤ 𝑛
×
1−𝜆2
then when 𝑛 > 2:
(4 ln 𝑛)(ln 𝑁)
1−𝜆2
(𝑒 − 1−𝜆2 )
64
1
≤ 4
𝑛
School of
Computing
Proof 𝜏 is polynomially bounded in n
• 𝑤𝑖𝑗 = 𝜋𝑖 𝑝𝑖𝑗 the stationary probability of transition i  j
𝑤𝑖𝑗 = 𝑤𝑗𝑖
• Let S be a subset of the set of states of 𝐶𝑛 such that: ∅ ⊂ 𝑆 ⊂
𝑀𝑛 ⋃𝑀𝑛−1 and 𝑆 is its complement:
– Capacity of S: ∁𝑆 = 𝑖∈𝑆 𝜋𝑖
– Ergodic flow out of S: 𝐹𝑠 =
𝑖∈𝑆,𝑗∈𝑆 𝑤𝑖𝑗
– Conductance of S: Φ𝑆 = 𝐹𝑆 /∁𝑆
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School of
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Proof 𝜏 is polynomially bounded in n
• From theorems 6.16, 6.17 and 6.19:
Φ2
𝜆2 ≤ 1 −
2

1
1−𝜆2
≤
2
Φ2
• By Canonical path argument: Φ ≥
1
12𝑛6
1

1−𝜆2
is polynomially upper-bounded in n
𝜏 =
(4 ln 𝑛)(ln 𝑁)
1−𝜆2
is also polynomially upper-bounded in n
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School of
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Agenda
•
•
•
•
Background
Randomized Approximation Schemes
The DNF Counting Problem
Approximating the Permanent
– Approximating the number of perfect matchings
in bipartite graph
– The canonical path argument
• Summary
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School of
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Conductance
• Let S be a subset of the set of states of 𝐶𝑛 such that: ∅ ⊂ 𝑆 ⊂
𝑀𝑛 ⋃𝑀𝑛−1 and 𝑆 is its complement:
– Capacity of S: ∁𝑆 = 𝑖∈𝑆 𝜋𝑖
– Ergodic flow out of S: 𝐹𝑠 =
𝑖∈𝑆,𝑗∈𝑆 𝑤𝑖𝑗
– Conductance of S: Φ𝑆 = 𝐹𝑆 /∁𝑆
– Conductance of Markov chain with state space 𝑄,
is defined as
Φ=
min
∅⊂𝑆⊂𝑄,∁𝑆 ≤1/2
Φ𝑆
S
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𝑆
School of
Computing
Canonical path
For every pair of states 𝑋, 𝑌 ∈ 𝑄 define a path
𝑋, 𝑌 = 𝑋 = 𝑃0, 𝑃1, 𝑃2, , … , 𝑃𝑚, = 𝑌
where the probability of going from 𝑃𝑖 to 𝑃𝑖+1 is nonzero.
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School of
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Canonical path
Symmetric difference XY
• This consists of a disjoint collection of paths in H (some of
which may be closed cycles), each of which has edges that
belong alternately to X and to Y.
• Fix some arbitrary ordering on all simple paths in H, and
designated in each of them a so-called “start vertex”,
which is arbitrary if the path is a closed cycle but must be
an endpoint otherwise.
• This ordering induces a unique ordering P1, P2, . . . , Pm on
the paths appearing in XY.
• The canonical path from X to Y involves “unwinding” each
of the Pi
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School of
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Canonical path
• There are two cases to consider:
– Pi is not a cycle
• If Pi starts with a X-edge, remove it (↓-transition -reduce)
• Perform a sequence of ↔-transitions (removes X-edge, inserts Y-edge - Rotate)
• If Pi’s length is odd (we remain with one Y-edge), insert it (↑-transition - Augment)
– Pi is a cycle
• Let Pi=(v0,v1,. . .,v2l+1)
– V0 is the start vertex,
,
• Remove (v0,v1) – an X-edge (↓-transition-reduce)
• We are left with a path with endpoints v0, v1, one of which must be the start vertex
v
2j
, v2 j 1   X
v
2 j 1
, v2 j  2   Y
• Continue as above, but:
– If v0 is the start vertex, use v1 as the start vertex
– If v1 is the start vertex, use v0 as the start vertex
– (This trick serves to distinguish paths from cycles)
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School of
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Unwinding a path
X
Y
X
↔-transition
↓-transition
X
Y
X
X
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School of
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X
Unwinding a cycle
Y
Y
X
X
Y
Y
X
X
↓-transition
X
X
↔-transition
X
X
X
Y
X
X
X
↔-transition
↑-transition
Y
Y
Y
Y
X
Y
Y
Y
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Y
Y
School of
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Example
X:
Y:
XY:
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School of
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Canonical path Argument
Theorem: For the Markov chain 𝐶𝑛 ,
Conductance Φ ≥ 1/12𝑛6
Proof: Let H be the graph underlying 𝐶𝑛
The transition probabilities along all the oriented edges of H are
exactly 1/(2 𝐸 ) where E is the set of edges in G.
We can show that,
1
for any subset S of the vertices of H with capacity ∁𝑆 ≤ , the
2
number of edges between 𝑆 and 𝑆 is large.
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School of
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Canonical path Argument
A Canonical path between every pair of vertices
of H, such that no oriented edge of H occurs in
more than bN of these paths.
For a subset S of the vertices of H, the number
of such canonical paths crossing the cut from S
to 𝑆 is 𝑆 𝑁 − 𝑆 ≥ |𝑆|𝑁/2
we assume |𝑆| ≤ 𝑁/2
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School of
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Canonical path Argument
Since at most bN edges pass through each of the
edges between S and 𝑆 , the number of such
edges must be at least |S|/2b, so that the
conductance of Cn is at least
1/(4𝑏|𝐸|) ≥ 1/(4𝑏𝑛2 )
Now we have to prove b = 3𝑛4 to get the
desired lower bound for conductance
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School of
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Canonical path Argument
Let m be any matching in Mn
Define k(m) – set of all nodes 𝑠 ∈ 𝑀𝑛 ∪ 𝑀𝑛−1
such that 𝑠 = 𝑚 𝑎𝑛𝑑 𝑠 ≠ 𝑚
Lemma: For any m ∈ 𝑀𝑛 , 𝑘 𝑚 < 𝑛2
The only perfect matching that chooses m as its partner
is m itself. It requires n Reduce operations to reach any near
near perfect matching adjacent to m. The number of near
perfect matchings at distance exactly 2 from m is at most n(n-1).
Thus, there are at most n+n(n-1) different near perfect
matchings within distance of two of m.
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School of
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Canonical path Argument
• Associate a unique node 𝑠 ∈ 𝑀𝑛 with every node s ∈ 𝑀𝑛−1 ∪
𝑀𝑛
• Choose a canonical path between s and 𝑠, If s is in 𝑀𝑛 , then
we set s = 𝑠.
• Canonical paths between nodes s, 𝑡 ∈ 𝑀𝑛−1 ∪ 𝑀𝑛 consists of
three consecutive segments, s to 𝑠, 𝑠 to 𝑡 and 𝑡 to t .
• Two types of Segments in paths:
Type A : paths between a node s ∈ 𝑀𝑛−1 ∪ 𝑀𝑛 and its partners 𝑠 ∈
𝑀𝑛
Type B : paths between pairs of nodes in 𝑀𝑛
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School of
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𝐶2 for 𝐺 = 𝐾22
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School of
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Canonical path argument
Lemma: Any transition T in H lies on at most
3𝑛4 N distinct canonical paths.
Fix a transaction T=(u,v) in H
Total number of type A segments canonical paths that T has is
bounded by (k(v)+k(u))N < 2𝑛2 N
Type B segments is atmost 𝑛2
× 𝑛𝑜. 𝑜𝑓 𝑝𝑒𝑓𝑒𝑐𝑡𝑚𝑎𝑡𝑐ℎ𝑖𝑛𝑔𝑠 𝑤𝑖𝑡ℎ 𝑇 × 𝑛2
= 𝑛2 × N × 𝑛2 = 𝑛4 𝑁
Total paths with T is bounded by 3 𝑛4 N
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School of
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Summary
•
•
•
•
Background
Randomized Approximation Schemes
The DNF Counting Problem
Approximating the Permanent
– Approximating the number of perfect matchings
in bipartite graph
– The canonical path argument
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School of
Computing