Approximating infeasible 2VPI-systems

Abstract

It is a folklore result that testing whether a given system of equations with two variables per inequality (a 2VPI system) of the form x i ∈-∈x j ∈=∈c ij is solvable, can be done efficiently not only by Gaussian elimination but also by shortest-path computation on an associated constraint graph. However, when the system is infeasible and one wishes to delete a minimum weight set of inequalities to obtain feasibility (MinFs2 ∈=), this task becomes NP-complete. Our main result is a 2-approximation for the problem MinFs2 ∈= for the case when the constraint graph is planar using a primal-dual approach. We also give an α-approximation for the related maximization problem MaxFs2 ∈= where the goal is to maximize the weight of feasible inequalities. Here, α denotes the arboricity of the constraint graph. Our results extend to obtain constant factor approximations for the case when the domains of the variables are further restricted. © 2012 Springer-Verlag Berlin Heidelberg.