We compare the relative strength of valid inequalities for the integer hull of the feasible region of mixed integer linear programs with two equality constraints, two unrestricted integer variables and any number of nonnegative continuous variables. In particular, we prove that the closure of Type~2 triangle (resp. Type~3 triangle; quadrilateral) inequalities, are all within a factor of $1.5$ of the integer hull, and provide examples showing that the approximation factor is not less than $1.125$. There is no fixed approximation ratio for split or Type~1 triangle inequalities however.

## Citation

Department of Combinatorics and Optimization, Faculty of Mathematics, University of Waterloo, Ontario N2L 3G1, Canada, March 2013.