Lower bounds for cube-ideal set-systems
Ahmad Abdi, Gérard Cornuéjols, Daniel Dadush, Mahsa Dalirrooyfard
TL;DR
This work introduces cube-ideal set-systems, whose convex hulls are defined by capacity and generalized set covering inequalities, and proves exponential lower bounds on their size and linear lower bounds on their VC-dimension via entropy-based analysis. It connects these geometric bounds to graph-theoretic and optimization structures, including strong orientations, dijoins, perfect matchings, and ideal clutters, by showing that key families of combinatorial objects form cube-ideal set-systems or relate to ideal clutters. The paper develops both 3-connected and 2-connected regimes, identifying structural features such as large subcubes contained in the convex hull, faces with exponential point counts, and rainbow-inequality descriptions of cores, and then leverages these to obtain lower bounds on object counts. These results yield new exponential lower bounds for enumerating G-strong orientations, dijoins intersecting minimum cuts, and perfect matchings in r-graphs, with implications for several long-standing questions in combinatorial optimization and polyhedral theory. Overall, the work bridges polyhedral descriptions with combinatorial objects to derive quantitative, architecture-driven guarantees in graph theory and related optimization problems.
Abstract
A set-system $S\subseteq \{0,1\}^n$ is cube-ideal if its convex hull can be described by capacity and generalized set covering inequalities. In this paper, we use combinatorics, convex geometry, and polyhedral theory to give exponential lower bounds on the size of cube-ideal set-systems, and linear lower bounds on their VC dimension. We then provide applications to graph theory and combinatorial optimization, specifically to strong orientations, perfect matchings, dijoins, and ideal clutters.