Branching with a pre-specified finite list of $k$-sparse split sets for binary MILPs

TL;DR

This paper investigates the properties of finite lists of $k$-sparse split sets for binary MILPs, establishing bounds on their covering numbers and dominance relations, especially for sparsity levels 2 and 4.

Contribution

It introduces the concept of covering number for $k$-sparse split sets and characterizes bounds for these numbers, revealing differences between sparsity levels 2 and greater than 2.

Findings

Finite list of 2-sparse disjunctions can dominate all others.

No finite list exists for higher sparsity levels.

Covering number bounds are established for specific coefficient sets.

Abstract

When branching for binary mixed integer linear programs with disjunctions of sparsity level $2$, we observe that there exists a finite list of $2$-sparse disjunctions, such that any other $2$-sparse disjunction is dominated by one disjunction in this finite list. For sparsity level greater than $2$, we show that a finite list of disjunctions with this property cannot exist. This leads to the definition of covering number for a list of splits disjunctions. Given a finite list of split sets $\mathcal{F}$ of $k$-sparsity, and a given $k$-sparse split set $S$, let $\mathcal{F}(S)$ be the minimum number of split sets from the list $\mathcal{F}$, whose union contains $S \cap [0, \ 1]^n$. Let the covering number of $\mathcal{F}$ be the maximum value of $\mathcal{F}(S)$ over all $k$-sparse split sets $S$. We show that the covering number for any finite list of $k$-sparse split sets is at least $\lfloor k/2\rfloor $ for $k \geq 4$. We also show that the covering number of the family of $k$-sparse split sets with coefficients in $\{-1, 0, 1\}$ is upper bounded by $k-1$ for $k \leq 4$.