The role of rationality in integer-programming relaxations
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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 (Lower bounds on the size of integer programs without additional variables, Mathematical Programming, 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→∞.
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