Inverse Problems of Some NP-Complete Problems

Abstract

Abstract. The Knapsack problem and integer programming are NPcomplete problems. In this paper we show that the inverse problem of Knapsack problem can be solved with a pseudo-polynomial algorithm. We also show that the inverse problem of integer programming with fixed number of constraints is pseudo-polynomial.

Inverse multi-objective combinatorial optimization

“…First, as stressed in the next section, it provides intuitions for solving inverse combinatorial optimization problems. Second, it is required for proving the incorrectness of the method proposed by Huang (2005) for solving the inverse knapsack problem.…”

“…For example, the inverse integer linear programming problem by Huang (2005), Schaefer (2009), Wang (2009, the inverse {0,1}-knapsack problem by Huang (2005), and combinatorial optimization problems by Yang and Zhang (1999). However, the said approaches are either computationally expensive, or fail in the search of inverse solutions.…”

“…The method proposed by Huang (2005) for the inverse knapsack problem is based on the reduction from the knapsack problem to the shortest path problem. However, as it will be explained in what follows, this approach is incorrect.…”

“…An incorrect approach for solving the inverse {0,1}-knapsack problem Based on this reduction, Huang (2005) concluded that the inverse {0,1}-knapsack can be reduced to solve the inverse shortest path on the knapsack's 5.5. THE INVERSE KNAPSACK PROBLEM 41 network model.…”

“…Therefore, the length of many arcs such as ((2, 1), (3, 3)) and ((2, 2), (3, 4)) must also be modified in such a way that the network remains correct. In other words, the inverse {0,1}-knapsack problem can be solved as the inverse shortest path problem, but only with additional constraints, which was not the case in the work by Huang (2005). Due to the lack of an ad hoc approach for solving the inverse knapsack problem, the next chapter is dedicated to the study of this problem.…”