Bounded Cubic Integer Programming in Fixed Dimension

Abstract: We consider the exact minimization of a rational cubic polynomial over the integer points of a rational polytope in fixed dimension. Del Pia, Hildebrand, Weismantel, and Zemmer proved polynomial-time solvability in dimension two, while quartic polynomial minimization is already NP-hard in dimension two. We show that the bounded cubic result extends to every fixed dimension. The proof combines two exact identities for cubic polynomials with three ingredients from the recent fixed-dimensional algorithms for integer quadratic programming of Ari and Hildebrand: symmetric displacement covers, negative-displacement search for quadratic forms, and integer-query convex feasibility. At an integer query point, a negative Hessian direction yields a linear curvature cut valid for every global minimizer in the current cell, while absence of such a direction yields a linear objective cut through an exact endpoint-Hessian identity. These cuts define an integer-query separation oracle for the convex hull of the curvature-admissible sublevel points. Integer convex feasibility via ellipsoids and lattice algorithms then gives exact optimization without computing lattice centerpoints. A real-algebraic localization bound extends the result to unbounded polyhedra with a bounded real improving sublevel, including all coercive objectives. The argument also explains why degree three is a natural boundary for this approach: the symmetric second difference of a cubic is exactly its directional Hessian, which is affine in the basepoint; for quartics an additional fourth-order term appears.