The role of rationality in integer-programming relaxations.

For a finite set X ⊂ℤ^d that can be represented as X = Q ∩ℤ^d for some polyhedron Q , we call Q a relaxation of X and define the relaxation complexity rc(X) of X as the least number of facets among all possible relaxations Q of X . The rational relaxation complexity rc_ℚ(X) restricts the definition of rc(X) to rational polyhedra Q . In this article, we focus on X = Δ _d , the vertex set of the standard simplex, which consists of the null vector and the standard unit vectors in ℝ^d . We show that rc(Δ _d) ≤ d for every d ≥ 5 . That is, since rc_ℚ(Δ _d)=d+1 , irrationality can reduce the minimal size of relaxations. This answers an open question posed by Kaibel and Weltge (Math Program 154(1):407–425, 2015). Moreover, we prove the asymptotic statement rc(Δ _d) ∈ O(d/√(log (d))) , which shows that the ratio rc(Δ _d)/rc_ℚ(Δ _d) goes to 0, as d →∞ .