Adaptive and Cost-Optimal Parallel Algorithm for the 0-1 Knapsack Problem

Abstract

The 0-1 knapsack problem is well known to be NP-complete problem. In the past two decades, much effort has been done in order to find techniques that could lead to algorithms with a reasonable running time. This paper proposes a new parallel algorithm for the 0-1 knapsack problem where the optimal merging algorithm is adopted. Based on an EREW PRAM machine with shared memory, the proposed algorithm utilizes O((2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n/4</sup> ) <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1-e</sup> ) processors, 0 ≤ ε ≤ 1, and O(2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n/2</sup> ) memory to find a solution for the n-element 0-1 knapsack problem in time O(2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n/4</sup> (2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n/4</sup> ) <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">e</sup> ). Thus the cost of the proposed parallel algorithm is O(2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n/2</sup> ), which is both the lowest upper-bound time and without memory conflicts if only quantity of objects is considered in the complexity analysis for the 0-1 knapsack problem. Thus it is an improvement result over the past researches.