Modeling Combinatorial Disjunctive Constraints via Junction Trees

Abstract

We introduce techniques to build small ideal mixed-integer programming (MIP) formulations of combinatorial disjunctive constraints (CDCs) via the independent branching scheme. We present a novel pairwise IB-representable class of CDCs, CDCs admitting junction trees, and provide a combinatorial procedure to build MIP formulations for those constraints. Generalized special ordered sets (SOS k) can be modeled by CDCs admitting junction trees and we also obtain MIP formulations of SOS k. Furthermore, we provide a novel ideal extended formulation of any combinatorial disjunctive constraints with fewer auxiliary binary variables with an application in planar obstacle avoidance.

Maximal Clique and Edge-Ranking Bounds of Biclique Cover Number

“…Biclique edge cover graph is also studied by [16] for confluent drawings, which is a field studying how to draw nonplanar graphs on a plane. Huchette and Vielma [17] and Lyu et al [22] solved MBCP in order to find small and strong mixed-integer programming (MIP) formulations of disjunctive constraints.…”

“…We want to further tighten the upper bound of mc(G c ) − 1. We want to note that the effort of reducing the gap has been attempted in Algorithm 1 of [22] in the setting of combinatorial disjunctive constraints. However, Lyu et al [22] only provided a greedy biclique merging procedure and no theoretical improvements on the upper bound of the biclique cover number than mc(G c ) − 1.…”