Signed Permutation Polytope Packing in VLSI Design

Three mathematical models of a placement problem in VLSI design is offered, where modules are squares, octagons, and rhombuses, and 45°-rotation is permitted. For that, the generalized placement problem of signed permutation polytopes within a signed permutation polytope is constructed based on a concept of a circular-like object. The way to formalize 45°-rotation is presented, which applies the theory of continuous functional representations of discrete sets.