The role of rationality in integer-programming relaxations

Abstract

AbstractFor a finite set $$X \subset \mathbb {Z}^d$$ X ⊂ Z d that can be represented as $$X = Q \cap \mathbb {Z}^d$$ X = Q ∩ Z d for some polyhedron Q, we call Q a relaxation of X and define the relaxation complexity $${\text {rc}}(X)$$ rc ( X ) of X as the least number of facets among all possible relaxations Q of X. The rational relaxation complexity $${\text {rc}}_\mathbb {Q}(X)$$ rc Q ( X ) restricts the definition of $${\text {rc}}(X)$$ rc ( X ) to rational polyhedra Q. In this article, we focus on $$X = \Delta _d$$ X = Δ d , the vertex set of the standard simplex, which consists of the null vector and the standard unit vectors in $$\mathbb {R}^d$$ R d . We show that $${\text {rc}}(\Delta _d) \le d$$ rc ( Δ d ) ≤ d for every $$d \ge 5$$ d ≥ 5 . That is, since $${\text {rc}}_\mathbb {Q}(\Delta _d)=d+1$$ 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 $${\text {rc}}(\Delta _d) \in O(\nicefrac {d}{\sqrt{\log (d)}})$$ rc ( Δ d ) ∈ O ( d log ( d ) ) , which shows that the ratio $$\nicefrac {{\text {rc}}(\Delta _d)}{{\text {rc}}_\mathbb {Q}(\Delta _d)}$$ rc ( Δ d ) rc Q ( Δ d ) goes to 0, as $$d \rightarrow \infty $$ d → ∞ .