Computational aspects of relaxation complexity: possibilities and limitations

Abstract

The relaxation complexity $${{\,\mathrm{rc}\,}}(X)$$ 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 $${{\,\mathrm{rc}\,}}_\mathbb {Q}(X)$$ rc Q ( X ) , a variant of $${{\,\mathrm{rc}\,}}(X)$$ rc ( X ) in which the polyhedra P are required to be rational, and we show that $${{\,\mathrm{rc}\,}}(X)$$ rc ( X ) can be computed in polynomial time if X is 2-dimensional. Further, we investigate computable lower bounds on $${{\,\mathrm{rc}\,}}(X)$$ rc ( X ) with the particular focus on the existence of a finite set $$Y \subseteq \mathbb {Z}^d$$ Y ⊆ Z d such that separating X and $$Y \setminus X$$ Y \ X allows us to deduce $${{\,\mathrm{rc}\,}}(X) \ge k$$ rc ( X ) ≥ k . In particular, we show for some choices of X that no such finite set Y exists to certify the value of $${{\,\mathrm{rc}\,}}(X)$$ rc ( X ) , providing a negative answer to a question by Weltge (2015). We also obtain an explicit formula for $${{\,\mathrm{rc}\,}}(X)$$ rc ( X ) for specific classes of sets X and present the first practically applicable approach to compute $${{\,\mathrm{rc}\,}}(X)$$ rc ( X ) for sets X that admit a finite certificate.

Efficient MIP Techniques for Computing the Relaxation Complexity

“…We remark that the basic versions of two models have already been used by us in [2] to find rc ε (X, Y ) for X being the integer points in low-dimensional cubes and crosspolytopes. These experiments helped us to prove general formulae for rc(X) in these cases.…”

“…Moreover, hiding sets proposed by Kaibel & Weltge [19] provide a lower bound on rc(X). The bound given by hiding sets can be improved by computing the chromatic number of a graph derived from hiding sets, see [2]. Regarding the computability of rc(X), it has been shown in [3] that there exists a proper subset Obs(X) of Z d \ X such that rc(X) = rc(X, Obs(X)).…”

“…They also provide sufficient conditions on X that guarantee Obs(X) to be finite. Moreover, they establish that rc(X) is computable if d ≤ 3; for d = 2, a polynomial time algorithm to compute rc(X) is discussed in [2]. In general, however, it is an open question whether rc(X) is computable.…”

“…Thus, slightly perturbing a might not separate y anymore. To take care of this, we suggested in [2] to add a safety margin ε > 0 to the separation condition. That is, if a ⊺ x ≤ β is a facet defining inequality of a relaxation of X with a ∞ = 1 that separates y, then we require a ⊺ y ≥ β+ε.…”

“…In this case, we say that y is ε-separated from X. Then, rc ε (X) denotes the smallest number of facets of any relaxation of X that satisfies the safety margin condition 2 . We call such a relaxation an εrelaxation of X. Analogously to rc(X, Y ), we define rc ε (X, Y ) to be the smallest number of inequalities needed to ε-separate X and Y \ X.…”