On the feasibility of generalized inverse linear programs

TL;DR

This paper examines the computational complexity of feasibility problems in generalized inverse linear programs, revealing conditions under which these problems are tractable or NP-hard, depending on the problem's structure and target set.

Contribution

It provides a detailed complexity analysis of inverse LP feasibility, identifying tractable cases and establishing NP-hardness results for various problem formulations.

Findings

Tractable for standard form LPs with singleton targets

NP-hard for natural form LPs and basis targets in optimistic scenarios

Almost immediately NP-hard for partially fixed target solutions

Abstract

We investigate the feasibility problem for generalized inverse linear programs. Given an LP with affinely parametrized objective function and right-hand side as well as a target set Y, the goal is to decide whether the parameters can be chosen such that there exists an optimal solution that belongs to Y (optimistic scenario) or such that all optimal solutions belong to Y (pessimistic scenario). We study the complexity of this decision problem and show how it depends on the structure of the set Y, the form of the LP, the adjustable parameters, and the underlying scenario. For a target singleton Y={y}, we show that the problem is tractable if the given LP is in standard form, but NP-hard if the LP is given in natural form. If instead we are given a target basis B, the problem in standard form becomes NP-complete in the optimistic case, while remaining tractable in the pessimistic case. For partially fixed target solutions, the problem gets almost immediately NP-hard, but for particular cases we prove tractability if the number of free target variables is fixed. Moreover, we give a rigorous proof of membership in NP for any polyhedral target set, and discuss how this property can be extended to more general target sets using an oracle-based approach.