NP-Hardness of optimizing the sum of Rational Linear Functions over an Asymptotic-Linear-Program

TL;DR

This paper demonstrates that optimizing the sum of rational linear functions over an asymptotic-linear-program is strongly NP-hard by polynomial-time reduction from an NP-complete problem.

Contribution

It introduces a polynomial-time conversion from an NP-complete problem to a real-variable optimization problem constrained by an asymptotic-linear-program, establishing its NP-hardness.

Findings

The real-variable problem has a feasible solution if and only if the original NP-complete problem is feasible.

The coefficients are limited to 0, 1, -1, K, or -K, with K tending to infinity.

The problem is strongly NP-hard, even with polynomially bounded variables and constraints.

Abstract

We convert, within polynomial-time and sequential processing, an NP-Complete Problem into a real-variable problem of minimizing a sum of Rational Linear Functions constrained by an Asymptotic-Linear-Program. The coefficients and constants in the real-variable problem are 0, 1, -1, K, or -K, where K is the time parameter that tends to positive infinity. The number of variables, constraints, and rational linear functions in the objective, of the real-variable problem is bounded by a polynomial function of the size of the NP-Complete Problem. The NP-Complete Problem has a feasible solution, if-and-only-if, the real-variable problem has a feasible optimal objective equal to zero. We thus show the strong NP-hardness of this real-variable optimization problem.