On the Properties of Interval Linear Programs with a Fixed Coefficient Matrix
Interval programming is a modern tool for dealing with uncertainty in practical optimization problems. In this paper, we consider a special class of interval linear programs with interval coefficients occurring only in the objective function and the right-hand-side vector, i.e. programs with a fixed (real) coefficient matrix. The main focus of the paper is on the complexity-theoretic properties of interval linear programs. We study the problems of testing weak and strong feasibility, unboundedness and optimality of an interval linear program with a fixed coefficient matrix. While some of these hard decision problems become solvable in polynomial time, many remain (co-)NP-hard even in this special case. Namely, we prove that testing strong feasibility, unboundedness and optimality remains co-NP-hard for programs described by equations with non-negative variables, while all of the weak properties are easy to decide. For inequality-constrained programs, the (co-)NP-hardness results hold for the problems of testing weak unboundedness and strong optimality. However, if we also require all variables of the inequality-constrained program to be non-negative, all of the discussed problems are easy to decide.
KeywordsInterval linear programming Computational complexity
The first two authors were supported by the Czech Science Foundation under the project P402/13-10660S and by the Charles University, project GA UK No. 156317. The work of the first author was also supported by the grant SVV-2017-260452. The work of the third author was supported by the Czech Science Foundation Project no. 17-13086S.
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