Approximation algorithms for the generalized incremental knapsack problem

• 成果类型: Article
• 署名作者: Faenza, Yuri; Segev, Danny; Zhang, Lingyi
• 署名单位: Columbia University; Tel Aviv University
• 刊物名称: MATHEMATICAL PROGRAMMING
• ISSN/ISSBN: 0025-5610
• DOI: 10.1007/s10107-021-01755-7
• 发表日期: 2023
• 页码: 27-83
• 关键词: assignment; FLOW

摘要

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 <= ... <= 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 (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 new structural properties of nearly-optimal solutions, we turn our 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.