Branching with a pre-specified finite list of $k$-sparse split sets for binary MILPs
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866914908926377984 |
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| author | Dey, Santanu S. Moran, Diego Xu, Jingye |
| author_facet | Dey, Santanu S. Moran, Diego Xu, Jingye |
| contents | 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 $k \leq 4$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_05392 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Branching with a pre-specified finite list of $k$-sparse split sets for binary MILPs Dey, Santanu S. Moran, Diego Xu, Jingye Optimization and Control Combinatorics 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 $k \leq 4$. |
| title | Branching with a pre-specified finite list of $k$-sparse split sets for binary MILPs |
| topic | Optimization and Control Combinatorics |
| url | https://arxiv.org/abs/2408.05392 |