Approximation algorithms for the generalized incremental knapsack problem

• Published in: Mathematical programming Vol. 198; no. 1; pp. 27 - 83
• Main Authors: Faenza, Yuri, Segev, Danny, Zhang, Lingyi
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.03.2023; Springer; Springer Nature B.V
• Subjects: Algorithms; Approximation; Calculus of Variations and Optimal Control; Optimization; Combinatorics; Dynamic programming; Enumeration; Full Length Paper; Knapsack problem; Mathematical analysis; Mathematical and Computational Physics; Mathematical Methods in Physics; Mathematics; Mathematics and Statistics; Mathematics of Computing; Numerical Analysis; Operations research; Optimization; Polynomials; Theoretical; Time dependence; 90C10; Approximation algorithms; 68W25; Sequencing; Incremental optimization; 90B35
• ISSN: 0025-5610; 1436-4646
• DOI: 10.1007/s10107-021-01755-7

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 - ϵ ) -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.