Split Cuts in Fixed Integer Dimensions

Abstract

We study how many split cuts, which include the Gomory mixed-integer cuts, are needed to reach the integer hull, and how large split closures can be. In every fixed dimension, a rational pure-integer polytope admits a polynomial-time constructible split proof of its integer hull with polynomial total encoding length; consequently, its full split rank is polynomial in the input length. This establishes the split-rank conjecture of Basu, Conforti, Di Summa, and Jiang for bounded pure-integer polyhedra in fixed dimension. Central to our approach is a preservation result: in arbitrary pure-integer dimension, an inequality whose violating points map, under an integer-affine map, into the interior of a planar lattice-free polyhedron has a polynomial split proof retaining every point already satisfying it. Whereas Chvátal--Gomory closures have polynomial-size inequality descriptions in fixed dimension, split closures need not: tetrahedra of input length $Θ(\log N)$ can have split closures with at least $2N-1$ facets, yet extension complexity $Θ(\log N)$. For the two-row system $x=f+Rs$, $s\ge0$, $x\in\mathbb Z^2$ of input length $L$, the split closure has an exact extended formulation with $O(L)$ inequalities and auxiliary variables; this bound is tight. Separation over the split closure is polynomial-time in fixed total dimension. Finally, for mixed-integer feasible rational polytopes with fixed integer dimension $p$ and any continuous dimension, we maximize a primitive integral objective on the integer variables with $O_p(G+1)$ split cuts, each removing a current optimal LP vertex, until an optimal vertex is mixed-integer feasible; here $G$ is the integrality gap of the LP relaxation, rounded down.