An FPTAS for #Knapsack and Related Counting Problems

Parikshit GopalanAdam KlivansRaghu Meka

Daniel ŠtefankovičSantosh Vempala

Eric Vigoda

An FPTAS for #Knapsack and Related Counting Problems

Parikshit Gopalan, Adam Klivans, Raghu MekaDaniel Štefankovič, Santosh Vempala, Eric Vigoda

What can be counted?

exactly? very little...

(in polynomial-time)

number of spanning trees (using determinant),Kirchoff’1847.

perfect matchings in planar graphs (using Pfaffians), Kasteleyn’1960.

(rest: usually #P-hard)

What can be counted?

approximately? a little more...

(in polynomial-time)

perfect matchings in bipartite graphs (permanent of non-negative matrices), Jerrum, Sinclair, Vigoda’2001.

Ferromagnetic Ising model, Jerrum, Sinclair’1989.

Independent sets (5), Weitz’2004.

k-colorings (k (11/6)), Vigoda’1999.

....

(approximate counting random sampling, Jerrum, Valiant, Vazirani’1986)

Approximate counting(in polynomial-time)

deterministic:

INPUT OUT

OUT Q

randomized:INPUT

OUT

OUT Q

P( )1-

not too many examples: independent sets in degree 5 graphs (Weitz’2004), matchings in bounded degree graphs (Bayati, Gamarnik, Katz, Nair, Tetali’2007),satisfying assignments of DNF formulas with terms ofsize C (Ajtai, Wigderson’1985)

more examples; Monte Carlo, usuallyusing a Markov chain (dependence 1/2)

1) is randomness necessary ?

Is P = BPP ?

Primes P (Agarwal, Kayal, Saxena 2001)

2) dependence on ?

Monte Carlo (1/2)

Knapsack (optimization)INPUT: (w1,v1),...(wn,vn), L

OUTPUT: S [n]

i Smax vi

i S wi L

weights

values

(integers)

Dynamic program #1(L is small)

T[i,M](optimal solution withitems 1,...,i and limit M)

T[i,M] = max {T[i-1,M]

T[i-1,M-wi] + vi

Dynamic program #2(vi’s are small)

T[i,V](smallest weight ofa subset of 1,...,i,with value V)

T[i,V] = min{T[i-1,V]

T[i-1,V-vi] + wi

approximation algorithm

Counting knapsackINPUT: w1,...,wn, L

OUTPUT: S [n]

i S wi L

How many

with

#P-hard

are there?

Counting knapsackDyer, Frieze, Kannan, Kapoor, Perkovic, Vazirani’1993 exp(O*(n1/2)) / 2

randomized approximation algorithm

Morris, Sinclair’1999 O( nc / 2 )

randomized approximation algorithm (MCMC, canonical paths)

Dyer’2003 O(n2.5 + n2/2)

randomized approximation algorithm (dynamic programming)

OURS: O*(n3/)

Dyer’2003:

T[i,M](number of solutions with items 1,...,i and limit M)

T[i,M] = T[i-1,M] + T[i-1,M-wi]

+ rejection sampling approximate counter

+ rejection sampling

approximate counter

wi’ =n2 wi

L

wi’’ = wi’ rounding:

L’ = n2

1) get more solutions, ’’ ’2) not too many more, |’’| (n+1)|’|

S’’ ’’ - ’, X heaviest in S’’, then S’’ - {X}’ Proof:

Our dynamic program

(i,A) = smallest M such that knapsackwith w1,...,wi,M has A solutions

(i,A) = min max[0,1]

{(i-1, A)

(i-1,(1- A)+wi

deterministic approximation algorithm

Q = 1+ /(n+1)s = n logQ 2

T(i,j)=min max[0,1]

{T(i-1,j+lnQ )

T(i-1,j+lnQ(1-)+wi

T[0..n,0..s]

Lemma 1: (i,Qj-i) T[i,j] (i,Qj)

T(i,j)=min max[0,1]

{T(i-1,j+lnQ )

T(i-1,j+lnQ(1-)+wi

Lemma 2: can compute recursion efficiently

Q-j,....,Q0, 1-Q0, .... , 1-Qj

only few values of matter

can use binary search

TOTAL RUN TIME = O( log(n/))n3

How to deal with more constraints ?

S [n]

i S wj,i Lj

How many with

are there?

j{1,...,k}

O( (n/)O(k2) log W) algorithm

(e.g., contingency tables,multi-dimensional knapsack, ...)

multi-dimensional knapsack:

Read once branching programs

n layers

• Layered directed graph

• vertices per layer• Edges between

consecutive layers• Edges labeled • Input: • Output: Label of

final vertex reached

Counting the number of accepting paths ? dynamic programming, time = O(nS)

n layers

ROBP for knapsack

Problem: width too large

Solution: reduce width by approximating

Monotone ROBPs

• ordering: given u,v, is u v ?

• midpoint: given u,v, get w s.t. |{x;uxw}| = |{x;wxv}| 1

• transitions: given u, get the outneighbors of u

u v A(u) A(v)monotone:

given implicitly

accepting paths from u

u

v

group the vertices in the layers according to the rough number of accepting paths processing right-left

already “shrunk”

can be generalized to distributions given by small space sources.

n layers

p1

1-p1

small space sources = ROBP + probability distributions on outgoing edges

More constraints?

More constraints?

n layers

n layers

can be combined to get (S2,n)-ROBP forintersection

additive approximation preserved

1) uniform distribution given by ’’ can be given by small space source

2) additive approximation multiplicative approximation

Other problems:

contingency tableswith constant number of rows

What other problems are solvable using the technique?

Thanks!