Approximation Algorithms for The Generalized Incremental Knapsack Problem

TL;DR

This paper introduces the generalized incremental knapsack problem, a multi-period extension of the classical problem, and provides a polynomial-time approximation algorithm with a (1/2 - epsilon) ratio, along with a quasi-polynomial time scheme.

Contribution

It presents the first polynomial-time approximation algorithm and a QPTAS for the generalized incremental knapsack problem, a strongly NP-hard extension of classical knapsack.

Findings

Achieved a (1/2 - epsilon)-approximation algorithm.

Developed a QPTAS for the problem.

Proved the problem is unlikely to be APX-hard.

Abstract

We introduce and study a discrete multi-period extension of the classical knapsack problem, dubbed generalized incremental knapsack. In this setting, we are given a set of $n$ items, each associated with a non-negative weight, and $T$ time periods with non-decreasing capacities $W_1 \leq \dots \leq W_T$. When item $i$ is inserted at time $t$, we gain a profit of $p_{it}$; however, this item remains in the knapsack for all subsequent periods. The goal is to decide if and when to insert each item, subject to the time-dependent capacity constraints, with the objective of maximizing our total profit. Interestingly, this setting subsumes as special cases a number of recently-studied incremental knapsack problems, all known to be strongly NP-hard. Our first contribution comes in the form of a polynomial-time $(\frac{1}{2}-\epsilon)$-approximation for the generalized incremental knapsack problem. This result is based on a reformulation as a single-machine sequencing problem, which is addressed by blending dynamic programming techniques and the classical Shmoys-Tardos algorithm for the generalized assignment problem. Combined with further enumeration-based self-reinforcing ideas and newly-revealed structural properties of nearly-optimal solutions, we turn our basic algorithm into a quasi-polynomial time approximation scheme (QPTAS). Hence, under widely believed complexity assumptions, this finding rules out the possibility that generalized incremental knapsack is APX-hard.