Fast Approximation Algorithms for the Knapsack and Sum of Subset Problems

Abstract

Given a positive integer M and n pairs of positive integers (p~, cD, , (p. , c.), maximize the sum~ ~p~ subject to the constramts~ ~c, < M and ~, = 0 or 1 This is the well-known 0/1 knapsack problem An algorithm is presented which finds for any 0 < e < 1 an approximate solution P satisfying (P* -P)/P* < ~, where P* is the desired optimal sum Moreover, for any fixed e, the algorithm has time complexity 0(n log n) and space complexity O(n) Modification of the algorithm for the unbounded knapsack problem where the ~,'s can be any nonnegative integer results in a O(n) computing time A hnear-time algorithm is also obtained for a special class of 0/1 knapsack problems having the property that p,/c, is the same for all 1 < z < n KEY WORDS AND PHRASES.