Modeling Combinatorial Disjunctive Constraints via Junction Trees

TL;DR

This paper presents new techniques for constructing small, ideal mixed-integer programming formulations for combinatorial disjunctive constraints using junction trees and the independent branching scheme, improving modeling efficiency.

Contribution

It introduces a novel class of CDCs admit junction trees, provides a combinatorial procedure for MIP formulation, and extends to SOS k constraints with fewer auxiliary variables.

Findings

Developed a combinatorial procedure for MIP formulations

Extended modeling to SOS k constraints

Provided a new ideal extended formulation with fewer variables

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 ($\text{SOS} k$) can be modeled by CDCs admitting junction trees and we also obtain MIP formulations of $\text{SOS} k$. 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.