We make a case for sub-polyhedral scheduling using (Unit-)TwoVariable-Per-Inequality or (U)TVPI Polyhedra. We empirically show that using general convex polyhedra leads to a scalability problem in a widely used polyhedral scheduler. We propose methods in which polyhedral schedulers can beat the scalability challenge by using sub-polyhedral under-aproximations of the polyhedra resulting from the...

A polyhedral map is called {p, q}-equivelar if each face has p edges and each vertex belongs to q faces. In [12], it was shown that there exist infinitely many geometrically realizable {p, q}-equivelar polyhedral maps if q > p = 4, p > q = 4 or q − 3 > p = 3. It was shown in [6] that there exist infinitely many {4, 4}and {3, 6}-equivelar polyhedral maps. In [1], it was shown that {5, 5}and {6, ...

In this thesis I study reconstruction of orthogonal polyhedral surfaces and orthogonal polyhedra from partial information about their boundaries. There are three main questions for which I provide novel results. The first question is “Given the dual graph, facial angles and edge lengths of an orthogonal polyhedral surface or polyhedron, is it possible to reconstruct the dihedral angles?” The se...

We characterise finite and infinitesimal rigidity for bar-joint frameworks in R with respect to polyhedral norms (i.e. norms with closed unit ball P a convex d-dimensional polytope). Infinitesimal and continuous rigidity are shown to be equivalent for finite frameworks in R which are well-positioned with respect to P. An edge-labelling determined by the facets of the unit ball and placement of ...

Let   G Aut and  E G   denote the automorphism group and the edge set of a graph  G, respectively. Weinberg’s Theorem states that 4 is a constant sharp upper bound on the ratio     G E G Aut over planar (or spherical) 3-connected graphs  G. We have obtained various analogues of this theorem for nonspherical graphs, introducing two Weinberg-type bounds for an arbitrary closed surfa...

We prove that two polygons A and B have a reversible hinged dissection (a chain hinged dissection that reverses inside and outside boundaries when folding between A and B) if and only if A and B are two non-crossing nets of a common polyhedron. Furthermore, monotone hinged dissections (where all hinges rotate in the same direction when changing from A to B) correspond exactly to non-crossing ne...

Guaranteed upper bounds on the length of a shortest cycle through k ≤ 5 prescribed vertices of a polyhedral graph or plane triangulation are proved.

In this note we prove an explicit formula for the lower semicontinuous envelope of some functionals defined on real polyhedral chains. More precisely, denoting by H : R → [0,∞) an even, subadditive, and lower semicontinuous function with H(0) = 0, and by ΦH the functional induced by H on polyhedral m-chains, namely

We analyze the asymptotic convergence of all infinite products of matrices taken in a given finite set by looking only at finite or periodic products. It is known that when the matrices of the set have a common nonincreasing polyhedral norm, all infinite products converge to zero if and only if all infinite periodic products with periods smaller than a certain value converge to zero. Moreover, ...

A binary three-dimensional (3D) image I is well-composed if the boundary surface of its continuous analog is a 2D manifold. In this paper, we present a method to locally “repair” the cubical complex Q(I) (embedded in R3) associated to I to obtain a polyhedral complex P (I) homotopy equivalent to Q(I) such that the boundary surface of P (I) is a 2D manifold (and, hence, P (I) is a well-composed ...