Approximation for knapsack problems with multiple constraints

Abstract

In this paper, the approximation for four kinds of knapsack problems with multiple constraints is studied: 0/1 Multiple Constraint Knapsack Problem (0/1 MCKP), Integer Multiple Constraint Knapsack Problem (Integer MCKP), 0/1k-Constraint Knapsack Problem (0/1k-CKP) and Integerk-Constraint Knapsack Problem (Integerk-CKP). The following results are obtained:

• 1) UnlessNP=co−R, no polynomial time algorithm approximates 0/1 MCKP or Integer MCKP within a factork ^(1/2)−σ for any σ>0; unlessNP=P, no polynomial time algorithm approximates 0/1 MCKP or Integer MCKP within a factork ^(1/4)−σ for any σ>0, wherek stands for the number of constraints.

• 2) For any fixed positive integerk, 0/1k-CKP has a fully polynomial time approximation scheme (FPTAS).

• 3) For any fixed positive integerk, Integerk-CKP has a fast FPTAS which has time complexity$O\left( {n + \frac{1}{{\varepsilon ^3 }} + \frac{1}{{\varepsilon ^{2^{k + 1} - 2} }}} \right)$ and space complexity$O(n + (1/\varepsilon ^3 ))$, and finds an approximate solution to within ε of the optimal solution.