Complexity of Integer Programming in Reverse Convex Sets via Boundary Hyperplane Cover

TL;DR

This paper investigates the computational complexity of determining integer feasibility in reverse convex sets, introducing a Boundary Hyperplane Cover structure that enables polynomial-time solutions in fixed dimensions under certain conditions.

Contribution

It introduces the Boundary Hyperplane Cover structure, providing a polynomial-time algorithm for fixed-dimension cases with bounded reverse convex constraints.

Findings

Complexity varies between NP-Hard and polynomial-time depending on the setting.

Boundary Hyperplane Cover enables efficient solutions in fixed dimensions.

The approach applies to bounded reverse convex constraints with polyhedral domains.

Abstract

We study the complexity of identifying the integer feasibility of reverse convex sets. We present various settings where the complexity can be either NP-Hard or efficiently solvable when the dimension is fixed. Of particular interest is the case of bounded reverse convex constraints with a polyhedral domain. We introduce a structure, \emph{Boundary Hyperplane Cover}, that permits this problem to be solved in polynomial time in fixed dimension provided the number of nonlinear reverse convex sets is fixed.