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Approximation Algorithms for the Incremental Knapsack Problem via Disjunctive Programming

Daniel Bienstock(dano***at***columbia.edu)
Jay Sethuraman(js1353***at***columbia.edu)
Chun Ye(cy2214***at***columbia.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: 11/18/2013
Entry Accepted: 11/18/2013
Entry Last Modified: 11/18/2013

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