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An FPTAS for #Knapsack and Related Counting Problems

This paper explores counting problems, such as the number of spanning trees and perfect matchings, and proposes a polynomial-time approximate counting algorithm. It also discusses the necessity of randomness and the dependence on the approximation factor.

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An FPTAS for #Knapsack and Related Counting Problems

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

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

  3. What can be counted? (in polynomial-time) exactly? very little... 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? (in polynomial-time) approximately? a little more... 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: OUT Q 1-   1+ INPUT OUT  randomized: INPUT OUT  P( )1- OUT Q 1-   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) max  vi i  S  wi L i  S weights INPUT: (w1,v1),...(wn,vn), L OUTPUT: (integers) values S [n]

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

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

  11. Counting knapsack  wi L i  S INPUT: w1,...,wn, L OUTPUT: S [n] How many with are there? #P-hard

  12. Counting knapsack Dyer, 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/)

  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-wi] + rejection sampling approximate counter

  14. + rejection sampling approximate counter n2 wi wi’ = L’ = n2 L rounding: wi’’ =  wi’  • get more solutions, ’’  ’ • not too many more, |’’| (n+1)|’| Proof: S’’’’ - ’, X heaviest in S’’, then S’’ - {X}’

  15. Our dynamic program deterministic approximation algorithm smallest M such that knapsack with w1,...,wi,M has  A solutions (i,A) = (i-1, A) { (i,A) = min max [0,1] (i-1,(1-) A)+wi

  16. Q = 1+ /(n+1) s =  n logQ 2 T[0..n,0..s] T(i-1,j+lnQ) { T(i,j)=min max T(i-1,j+lnQ(1-))+wi [0,1] Lemma 1: (i,Qj-i)  T[i,j]  (i,Qj)

  17. T(i-1,j+lnQ) { T(i,j)=min max T(i-1,j+lnQ(1-))+wi [0,1] Lemma 2: can compute recursion efficiently only few values of  matter Q-j,....,Q0, 1-Q0, .... , 1-Qj can use binary search n3 TOTAL RUN TIME = O( log(n/)) 

  18. How to deal with more constraints ?  wj,i Lj i  S (e.g., contingency tables, multi-dimensional knapsack, ...) multi-dimensional knapsack: S [n] How many with j{1,...,k} are there? O( (n/)O(k2) log W) algorithm

  19. Read once branching programs • Layered directed graph • vertices per layer • Edges between consecutive layers • Edges labeled • Input: • Output: Label of final vertex reached n layers Counting the number of accepting paths ? dynamic programming, time = O(nS)

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

  21. Monotone ROBPs accepting paths from u u  v  A(u)  A(v) monotone: u given implicitly • 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 v

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

  23. More constraints? can be generalized to distributions given by small space sources. small space sources = ROBP + probability distributions on outgoing edges p1 1-p1 n layers

  24. More constraints? n layers n layers can be combined to get (S2,n)-ROBP for intersection additive approximation preserved

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

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

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