  We propose a general algorithm of constructing an extended formulation for
any given set of linear constraints with integer coefficients. Our algorithm
consists of two phases: first construct a decision diagram $(V,E)$ that somehow
represents a given $m \times n$ constraint matrix, and then build an equivalent
set of $|E|$ linear constraints over $n+|V|$ variables. That is, the size of
the resultant extended formulation depends not explicitly on the number $m$ of
the original constraints, but on its decision diagram representation.
Therefore, we may significantly reduce the computation time for optimization
problems with integer constraint matrices by solving them under the extended
formulations, especially when we obtain concise decision diagram
representations for the matrices. We can apply our method to $1$-norm
regularized hard margin optimization over the binary instance space
$\{0,1\}^n$, which can be formulated as a linear programming problem with $m$
constraints with $\{-1,0,1\}$-valued coefficients over $n$ variables, where $m$
is the size of the given sample. Furthermore, introducing slack variables over
the edges of the decision diagram, we establish a variant formulation of soft
margin optimization. We demonstrate the effectiveness of our extended
formulations for integer programming and the $1$-norm regularized soft margin
optimization tasks over synthetic and real datasets.
