Approximate algorithms for the knapsack problem on parallel computers

Computing an optimal solution to the knapsack problem is known to be NP-hard. Consequently, fast parallel algorithms for finding such a solution without using an exponential number of processors appear unlikely. An attractive alternative is to compute an approximate solution to this problem rapidly using a polynomial number of processors. In this paper, we present an efficient parallel algorithm for finding approximate solutions to the 0–1 knapsack problem. Our algorithm takes an ε, 0 < ε < 1, as a parameter and computes a solution such that the ratio of its deviation from the optimal solution is at most a fraction ε of the optimal solution. For a problem instance having n items, this computation uses O(n 5 2 /ε 3 2 ) processors and requires O(log 3n + log 2nlog( 1 ε )) time. The upper bound on the processor requirement of our algorithm is established by reducing it to a problem on weighted bipartite graphs. This processor complexity is a significant improvement over that of other known parallel algorithms for this problem.

Authors

• P.S. Gopalakrishnan
• I.V. Ramakrishnan
• L.N. Kanal

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