Faster FPTASes for counting and random generation of Knapsack solutions

In the #P-complete problem of counting 0/1 Knapsack solutions, the input consists of a sequence of n nonnegative integer weights w(1), ..., w(n) and an integer C, and we have to find the number of subsequences (subsets of indices) with total weight at most C. We give faster and simpler fully polynomial-time approximation schemes (FPTASes) for this problem, and for its random generation counterpart. Our method is based on dynamic programming and discretization of large numbers through floating-point arithmetic. We improve both deterministic counting FPTASes from Gopalan et al. (2011) [9], Stefankovic et al. (2012) [6] and the randomized counting and random generation algorithms in Dyer (2003) [5].Our method is general, and it can be directly applied on top of combinatorial decompositions (such as dynamic programming solutions) of various problems. For example, we also improve the complexity of the problem of counting 0/1 Knapsack solutions in an arc-weighted DAG. (C) 2019 The Authors. Published by Elsevier Inc.