An easily computable upper bound on the Hoffman constant for homogeneous inequality systems

\(\)Let $A\in \mathbb{R}^{m\times n}\setminus \{0\}$ and $P:=\{x:Ax\le 0\}$. This paper provides a procedure to compute an upper bound on the following {\em homogeneous Hoffman constant} \[ H_0(A) := \sup_{u\in \mathbb{R}^n \setminus P} \frac{\text{dist}(u,P)}{\text{dist}(Au, \mathbb{R}^m_-)}. \] In sharp contrast to the intractability of computing more general Hoffman constants, the procedure described in this paper is entirely … Read more

Minimum-Link Covering Trails for any Hypercubic Lattice

\(\) In 1994, Kranakis et al. published a conjecture about the minimum link-length of every rectilinear covering path for the \(k\)-dimensional grid \(P(n,k) := \{0,1, \dots, n-1\} \times \{0,1, \dots, n-1\} \times \cdots \times \{0,1, \dots, n-1\}\). In this paper we consider the general, NP-complete, Line-Cover problem, where the edges are not required to be … Read more