Papers › Computational Aspects of Relaxation Complexity: Possibilities and Limitations
Computational Aspects of Relaxation Complexity: Possibilities and Limitations
Gennadiy Averkov, Christopher Hojny, Matthias Schymura
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The relaxation complexity rc(X) of the set of integer points X contained in a polyhedron is the smallest number of facets of any polyhedron P such that the integer points in P coincide with X. It is a useful tool to investigate the existence of compact linear descriptions of X. In this article, we derive tight and computable upper bounds on rc_ℚ(X), a variant of rc(X) in which the polyhedra P are required to be rational, and we show that rc(X) can be computed in polynomial time if X is 2-dimensional. Further, we investigate computable lower bounds on rc(X) with the particular focus on the existence of a finite set Y ⊆ℤᵈ such that separating X and Y ∖X allows us to deduce rc(X) ≥k. In particular, we show for some choices of X that no such finite set Y exists to certify the value of rc(X), providing a negative answer to a question by Weltge (2015). We also obtain an explicit formula for rc(X) for specific classes of sets X and present the first practically applicable approach to compute rc(X) for sets X that admit a finite certificate.
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