1. An FPTAS for #Knapsack and Related Counting Problems Parikshit Gopalan Adam Klivans Raghu Meka Daniel Štefankovi č Santosh Vempala Eric Vigoda

2. An FPTAS for #Knapsack and Related Counting Problems Parikshit Gopalan, Adam Klivans, Raghu Meka Daniel Štefankovi č, Santosh Vempala, Eric Vigoda

3. 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)

4. 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)

5. Approximate counting (in polynomial-time) deterministic: INPUT  OUT Q   randomized: INPUT  OUT Q   P( )  1- 

6. 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 of size  C (Ajtai, Wigderson’1985) more examples; Monte Carlo, usually using a Markov chain (dependence 1/  2 )

7. 1) is randomness necessary ? Is P = BPP ? Primes  P (Agarwal, Kayal, Saxena 2001) 2) dependence on  ? Monte Carlo  (1/  2 )

8. Knapsack (optimization) INPUT: (w 1,v 1 ),...(w n,v n ), L OUTPUT: S  [n] i  S max  v i i  S  w i  L weights values (integers)

9. Dynamic program #1 (L is small) T[i,M] (optimal solution with items 1,...,i and limit M) T[i,M] = max { T[i-1,M] T[i-1,M-w i ] + v i

10. Dynamic program #2 (v i ’s are small) T[i,V] (smallest weight of a subset of 1,...,i, with value  V) T[i,V] = min { T[i-1,V] T[i-1,V-v i ] + w i approximation algorithm

11. Counting knapsack INPUT: w 1,...,w n, L OUTPUT: S  [n] i  S  w i  L How many with #P-hard are there?

12. Counting knapsack Dyer, Frieze, Kannan, Kapoor, Perkovic, Vazirani’1993 exp(O * (n 1/2 )) /  2 randomized approximation algorithm Morris, Sinclair’1999 O( n c /  2 ) randomized approximation algorithm (MCMC, canonical paths) Dyer’2003 O(n 2.5 + n 2 /  2 ) randomized approximation algorithm (dynamic programming) OURS: O * (n 3 /  )

13. 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-w i ] + rejection sampling approximate counter

14. + rejection sampling approximate counter w i ’ = n 2 w i L w i ’’ =  w i ’  rounding: L’ = n 2 1) get more solutions,  ’’  ’ 2) not too many more, |  ’’|  (n+1)|  ’| S’’   ’’ -  ’, X heaviest in S’’, then S’’ - {X}  ’ Proof:

15. Our dynamic program  (i,A) = smallest M such that knapsack with w 1,...,w i,M has  A solutions  (i,A) = min max  [0,1] {  (i-1,  A)  (i-1,(1-  A)+w i deterministic approximation algorithm

16. Q = 1+  /(n+1) s =  n log Q 2  T(i,j)= min max  [0,1] { T(i-1,  j+ln Q  ) T(i-1,  j+ln Q (1-  )+w i T[0..n,0..s] Lemma 1:  (i,Q j-i )  T[i,j]  (i,Q j )

17. T(i,j)= min max  [0,1] { T(i-1,  j+ln Q  ) T(i-1,  j+ln Q (1-  )+w i Lemma 2: can compute recursion efficiently Q -j,....,Q 0, 1-Q 0,...., 1-Q j only few values of  matter can use binary search TOTAL RUN TIME = O( log(n/  )) n3n3 

18. How to deal with more constraints ? S  [n] i  S  w j,i  L j How many with are there? j  {1,...,k} O( (n/  ) O(k 2 ) log W) algorithm (e.g., contingency tables, multi-dimensional knapsack,...) multi-dimensional knapsack:

19. 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)

20. n layers ROBP for knapsack Problem: width too large Solution: reduce width by approximating

21. Monotone ROBPs ordering: given u,v, is u  v ? midpoint: given u,v, get w s.t. |{x;u  x  w}| = |{x;w  x  v}|  1 transitions: given u, get the outneighbors of u u  v  A(u)  A(v) monotone: given implicitly accepting paths from u u v

22. group the vertices in the layers according to the rough number of accepting paths processing right-left already “shrunk”

23. can be generalized to distributions given by small space sources. n layers p1p1 1-p 1 small space sources = ROBP + probability distributions on outgoing edges More constraints?

24. n layers can be combined to get (S 2,n)-ROBP for intersection additive approximation preserved

25. 1) uniform distribution given by  ’’ can be given by small space source 2) additive approximation  multiplicative approximation

27. Other problems: contingency tables with constant number of rows What other problems are solvable using the technique? Thanks!