Why solving Knapsack problem is not considered as linear programming?

问题

Why isn't the knapsack problem included under the category of linear programming algorithms in spite of the fact that the Knapsack problem statement seems similar to the problems in linear programming?

回答1:

Knapsack can be written as an integer linear programming program. Unlike normal linear programming, this problem requires that variables in the solution are integers. Linear programming is known to be solvable in polynomial time, while integer linear programming is NP-complete.

Exercise for the reader: show that 3SAT can be reduced to integer linear programming.

Trivia: there are approximation algorithms for problems such as MAX-3SAT (a variant of 3SAT where we want to maximize the number of satisfied clauses). First you write MAX-3SAT as an integer linear program. Then, you relax it to linear program, by removing the integer restriction. Then, you solve the linear program in polynomial time. Finally, given real x_i ∈ [0,1], you round them to integers, or generate random integer solution y_i where probability of y_i = 1 is x_i.

来源：https://stackoverflow.com/questions/11600524/why-solving-knapsack-problem-is-not-considered-as-linear-programming