- Approximation Algorithms for the Incremental Knapsack Problem via Disjunctive Programming Daniel Bienstock(danocolumbia.edu) Jay Sethuraman(js1353columbia.edu) Chun Ye(cy2214columbia.edu) Abstract: In the \emph{incremental knapsack problem} ($\IK$), we are given a knapsack whose capacity grows weakly as a function of time. There is a time horizon of $T$ periods and the capacity of the knapsack is $B_t$ in period $t$ for $t = 1, \ldots, T$. We are also given a set $S$ of $N$ items to be placed in the knapsack. Item $i$ has a value of $v_i$ and a weight of $w_i$ that is independent of the time period. At any time period $t$, the sum of the weights of the items in the knapsack cannot exceed the knapsack capacity $B_t$. Moreover, once an item is placed in the knapsack, it cannot be removed from the knapsack at a later time period. We seek to maximize the sum of (discounted) knapsack values over time subject to the capacity constraints. We first give a constant factor approximation algorithm for $\IK$, under mild restrictions on the growth rate of $B_t$ (the constant factor depends on the growth rate). We then give a PTAS for $\IIK$, the special case of $\IK$ with no discounting, when $T = O(\sqrt{\log N})$. Keywords: Approximation Algorithms, Integer Programming, Disjunctive Programming Category 1: Combinatorial Optimization (Approximation Algorithms ) Category 2: Integer Programming (0-1 Programming ) Citation: Columbia University, Nov 2013. Download: [PDF]Entry Submitted: 11/18/2013Entry Accepted: 11/18/2013Entry Last Modified: 11/18/2013Modify/Update this entry Visitors Authors More about us Links Subscribe, Unsubscribe Digest Archive Search, Browse the Repository Submit Update Policies Coordinator's Board Classification Scheme Credits Give us feedback Optimization Journals, Sites, Societies Optimization Online is supported by the Mathematical Optmization Society.