A fully polynomial-time approximation scheme for approximating a sum of random variables

Abstract Given n independent integer-valued random variables X 1 , X 2 , … , X n and an integer C , we study the fundamental problem of computing the probability that the sum X = X 1 + X 2 + ⋯ + X n is at most C . We assume that each random variable X i is implicitly given by an oracle O i , which given two input integers n 1 , n 2 returns the probability of n 1 ≤ X i ≤ n 2 . We give the first deterministic fully polynomial-time approximation scheme (FPTAS) to estimate the probability up to a relative error of 1 ± ϵ . Our algorithm is based on the technique for approximately counting knapsack solutions, developed in Gopalan et al. (2011).

[1]  Eric Vigoda,et al.  An FPTAS for #Knapsack and Related Counting Problems , 2011, 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science.

[2]  Ashish Goel,et al.  Stochastic load balancing and related problems , 1999, 40th Annual Symposium on Foundations of Computer Science (Cat. No.99CB37039).

[3]  Eric Vigoda,et al.  A Deterministic Polynomial-Time Approximation Scheme for Counting Knapsack Solutions , 2010, SIAM J. Comput..

[4]  Evdokia Nikolova,et al.  Approximation Algorithms for Reliable Stochastic Combinatorial Optimization , 2010, APPROX-RANDOM.

[5]  G. Bennett.  Probability Inequalities for the Sum of Independent Random Variables , 1962 .

[6]  Matthew Brand,et al.  Stochastic Shortest Paths Via Quasi-convex Maximization , 2006, ESA.

[7]  S. Rice,et al.  Saddle point approximation for the distribution of the sum of independent random variables , 1980, Advances in Applied Probability.

[8]  Jingxian Wu,et al.  Approximating a Sum of Random Variables with a Lognormal , 2007, IEEE Transactions on Wireless Communications.

[9]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[10]  Jian Li,et al.  Maximizing Expected Utility for Stochastic Combinatorial Optimization Problems , 2010, 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science.

[11]  Kunhiraman Nair,et al.  On stochastic spanning tree problem , 1993, Networks.

[12]  Hiroaki Ishii,et al.  Stochastic spanning tree problem , 1981, Discret. Appl. Math..

[13]  L. V. Osipov.  On Probabilities of Large Deviations for Sums of Independent Random Variables , 1973 .

[14]  Yuval Rabani,et al.  Allocating bandwidth for bursty connections , 1997, STOC '97.

[15]  R. Ravi,et al.  A PTAS for the chance-constrained knapsack problem with random item sizes , 2010, Oper. Res. Lett..

[16]  Vasileios Vasaitis.  Approximate Counting by Dynamic Programming , 2005 .

[17]  Ashish Goel,et al.  Improved approximation results for stochastic knapsack problems , 2011, SODA '11.

[18]  Jian Li,et al.  Stochastic combinatorial optimization via poisson approximation , 2012, STOC '13.

[19]  V. V. Petrov.  On the Probabilities of Large Deviations for Sums of Independent Random Variables , 1965 .

[20]  H. E. Daniels,et al.  Tail Probability Approximations , 1987 .

[21]  Eli Upfal,et al.  Probability and Computing: Randomized Algorithms and Probabilistic Analysis , 2005 .

[22]  Alistair Sinclair,et al.  Random walks on truncated cubes and sampling 0-1 knapsack solutions , 1999, 40th Annual Symposium on Foundations of Computer Science (Cat. No.99CB37039).