Approximate minimization algorithms for the 0/1 Knapsack and Subset-Sum Problem

An employee transporting problem

An employee transporting problem is described and a set partitioning model is developed. An investigation of the model leads to a knapsack problem as a surrogate problem. Finding a partition corresponding to the knapsack problem provides a solution to the problem. An exact algorithm is proposed to obtain a partition (subset-vehicle combination) corresponding to the knapsack solution. It require...

Average-Case Performance of Rollout Algorithms for Knapsack Problems

Rollout algorithms have demonstrated excellent performance on a variety of dynamic and discrete optimization problems. Interpreted as an approximate dynamic programming algorithm, a rollout algorithm estimates the value-to-go at each decision stage by simulating future events while following a heuristic policy, referred to as the base policy. While in many cases rollout algorithms are guarantee...

Reducing a Target Interval to a Few Exact Queries

Many combinatorial problems involving weights can be formulated as a so-called ranged problem. That is, their input consists of a universe U , a (succinctly-represented) set family F ⊆ 2 , a weight function ω : U → {1, . . . , N}, and integers 0 ≤ l ≤ u ≤ ∞. Then the problem is to decide whether there is an X ∈ F such that l ≤ ∑ e∈X ω(e) ≤ u. Well-known examples of such problems include Knapsac...

Exponentially Small Bounds on the Expected Optimum

In this paper we are interested in behavior of this problem when the xi are i.i.d. random variables. Under fairly general conditions, the median of the solution for the subset sum problem has been shown to be exponentially small when t is near E [∑n i=1 xi ] [Luek82]; this result has found application in the probabilistic analysis of approximation algorithms for the 0-1 Knapsack problem [Luek82...

An O(nr) algorithm for the Subset-sum Problem - and other balanced knapsack algorithms