A Faster FPTAS for Knapsack Problem With Cardinality Constraint

02/03/2019
by   Wenxin Li, et al.
0

We study the K-item knapsack problem (, 1.5-dimensional KP), which is a generalization of the famous 0-1 knapsack problem (, 1-dimensional KP) in which an upper bound K is imposed on the number of items selected. This problem is of fundamental importance and is known to have a broad range of applications in various fields such as computer science and operation research. It is well known that, there is no FPTAS for the d-dimensional knapsack problem when d≥ 2, unless P = NP. While the K-item knapsack problem is known to admit an FPTAS, the complexity of all existing FPTASs have a high dependency on the cardinality bound K and approximation error ε, which could result in inefficiencies especially when K and ε^-1 increase. The current best results are due to mastrolilli2006hybrid, in which two schemes are presented exhibiting a space-time tradeoff--one scheme with time complexity O(n+Kz^2/ε^2) and space complexity O(n+z^3/ε), while another scheme requires O(n+(Kz^2+z^4)/ε^2) run-time but only needs O(n+z^2/ε) space, where z={K,1/ε}. In this paper we close the space-time tradeoff exhibited in mastrolilli2006hybrid by designing a new FPTAS with a run-time of O(n+z^2/ε^2), while simultaneously reaching the O(n+z^2/ε) space bound. Our scheme provides O(K) and O(z) improvements on the long-established state-of-the-art algorithms in time and space complexity respectively, and is the first scheme that achieves a run-time that is asymptotically independent of cardinality bound K under fixed ε. Another salient feature of our scheme is that it is the first FPTAS, which achieves better time and space complexity bounds than the very first standard FPTAS over all parameter regimes.

READ FULL TEXT
research
02/03/2019

Knapsack Problem With Cardinality Constraint: A Faster FPTAS Through the Lens of Numerical Analysis and Applications

We study the K-item knapsack problem (, 1.5-dimensional knapsack problem...
research
11/30/2022

(No) Quantum space-time tradeoff for USTCON

Undirected st-connectivity is important both for its applications in net...
research
08/06/2023

Knapsack with Small Items in Near-Quadratic Time

The Bounded Knapsack problem is one of the most fundamental NP-complete ...
research
02/13/2013

Topological Parameters for Time-Space Tradeoff

In this paper we propose a family of algorithms combining tree-clusterin...
research
07/06/2021

Space Efficient Two-Dimensional Orthogonal Colored Range Counting

In the two-dimensional orthogonal colored range counting problem, we pre...
research
07/17/2017

On Treewidth and Stable Marriage

Stable Marriage is a fundamental problem to both computer science and ec...
research
05/11/2019

Cost-Based Approach to Complexity: A Common Denominator?

Complexity remains one of the central challenges in science and technolo...

Please sign up or login with your details

Forgot password? Click here to reset