Despite some online searching I couldn't find examples of dedicated Integer Linear Programs ($\mathrm{ILP}$s) for determining smallest matchings, that are not contained in a larger one.

It seems that the following could be an example of such an $\mathrm{ILP}$:

$\quad\quad\min\quad \sum\limits_{\lbrace i,j\rbrace\subset \lbrace 1,\dots,n\rbrace}x_{\lbrace i,j\rbrace}$,

s.t.

$x_{\lbrace i,j\rbrace}\in\lbrace 0,1\rbrace$

$x_{\lbrace i,j\rbrace}+\sum\limits_{h\ne i}x_{\lbrace h,j\rbrace}+\sum\limits_{k\ne j}x_{\lbrace i,k\rbrace}\ \in\ \lbrace 1,2\rbrace,\quad\forall \lbrace h,i,j,k\rbrace\subseteq\lbrace 1,\dots,n\rbrace$

Questions:

- has the above $\mathrm{ILP}$ formulation of the MMM problem already appeared in the literature?
- what is know resp. can be said, about the integrality gap of the relaxed $\mathrm{LP}$-formulation, i.e. when

$\quad\quad\min\quad \sum\limits_{\lbrace i,j\rbrace\subset \lbrace 1,\dots,n\rbrace}x_{\lbrace i,j\rbrace}$,$\quad$ s.t.

$\quad x_{\lbrace i,j\rbrace}\in\left[ 0,1\right]$

$\quad x_{\lbrace i,j\rbrace}+\sum\limits_{h\ne i}x_{\lbrace h,j\rbrace}+\sum\limits_{k\ne j}x_{\lbrace i,k\rbrace}\ \in\ \left[1,2\right],\quad\forall \lbrace h,i,j,k\rbrace\subseteq\lbrace 1,\dots,n\rbrace$