Support-based lower bounds for the positive semidefinite rank of a nonnegative matrix

The positive semidefinite rank of a nonnegative $(m\times n)$-matrix~$S$ is the minimum number~$q$ such that there exist positive semidefinite $(q\times q)$-matrices $A_1,\dots,A_m$, $B_1,\dots,B_n$ such that S(k,\ell) = \mbox{tr}(A_k^* B_\ell). The most important, lower bound technique for nonnegative rank is solely based on the support of the matrix S, i.e., its zero/non-zero pattern. In this paper, we characterize the power of lower bounds on positive semidefinite rank based on solely on the support.Comment: 9 page

A PROBLEM OF CUTTING OFF THE LAMINATED SEMIS TYPE PLATE

A problem often coped on many domains such as wood manufacturing, glass, plastics and metallic platework industry, is the shaping or cutting off a big plate in many pieces. With this purpose there are algorithms of optimizing for positioning the parts following to be cut off from a row plate. From mathematical point of view, in positioning the parts on a raw plate the number of solutions increase four times evrey time a new part is added, and in case of finding the best solution for about few hundreds of pieces or parts would require years of processing on the most performant computers nowadays – for an analogy remember the famous story with the rice beads which the King had to pay to the master teaching him the chess: twice more for each square of the chessboard; for the total quantity assessment, King ascertained that the crops in his whole life wouldn’t have been enough