The paper presents fast approximation algorithms for the 0/1 knapsack problem and its variants, including the unbounded knapsack problem and a special class of 0/1 knapsack problems with uniform profit density. The main algorithm, named KNAPSACK, finds an approximate solution \(\hat{P}\) such that \((P^* - \hat{P}) / P^* \leq \epsilon\) for any \(0 < \epsilon < 1\), where \(P^*\) is the optimal solution. The algorithm has a time complexity of \(O(n \log n) + O(u(\epsilon) n)\) and space complexity of \(O(n) + O(v(\epsilon))\), where \(u(\epsilon)\) and \(v(\epsilon)\) are functions depending only on \(\epsilon\). For the unbounded knapsack problem, the algorithm can be modified to achieve a time complexity of \(O(u(\epsilon) n)\). Additionally, a linear-time algorithm is derived for the special class of 0/1 knapsack problems with uniform profit density. The paper also discusses the application of these algorithms to other combinatorial problems, such as the sum of subset problem, and provides error analysis to validate the approximation quality.The paper presents fast approximation algorithms for the 0/1 knapsack problem and its variants, including the unbounded knapsack problem and a special class of 0/1 knapsack problems with uniform profit density. The main algorithm, named KNAPSACK, finds an approximate solution \(\hat{P}\) such that \((P^* - \hat{P}) / P^* \leq \epsilon\) for any \(0 < \epsilon < 1\), where \(P^*\) is the optimal solution. The algorithm has a time complexity of \(O(n \log n) + O(u(\epsilon) n)\) and space complexity of \(O(n) + O(v(\epsilon))\), where \(u(\epsilon)\) and \(v(\epsilon)\) are functions depending only on \(\epsilon\). For the unbounded knapsack problem, the algorithm can be modified to achieve a time complexity of \(O(u(\epsilon) n)\). Additionally, a linear-time algorithm is derived for the special class of 0/1 knapsack problems with uniform profit density. The paper also discusses the application of these algorithms to other combinatorial problems, such as the sum of subset problem, and provides error analysis to validate the approximation quality.