Deterministic network coding by matrix completion

Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2005.Includes bibliographical references (leaves 81-85).Network coding is a new field of research that addresses problems of transmitting data through networks. Multicast problems are an important class of network coding problems where there is a single sender and all data must be transmitted to a set of receivers. In this thesis, we present a new deterministic algorithm to construct solutions for multicast problems that transmit data at the maximum possible rate. Our algorithm easily generalizes to several variants of multicast problems. Our approach is based on a new algorithm for maximum-rank completion of mixed matrices-taking a matrix whose entries are a mixture of numeric values and symbolic variables, and assigning values to the variables so as to maximize the resulting matrix rank. Our algorithm is faster than existing deterministic algorithms and can operate over smaller fields. This algorithm is extended to handle collections of matrices that can share variables. Over sufficiently large fields, the algorithm can compute a completion that simultaneously maximizes the rank of all matrices in the collection. Our simultaneous matrix completion algorithm requires working over a field whose size exceeds the number of matrices in the collection. We show that this algorithm is best-possible, in the sense that no efficient algorithm can operate over a smaller field unless P=NP.by Nicholas James Alexander Harvey.S.M

Combinatorics of Grassmannian Decompositions

This thesis studies several combinatorially defined families of subsets of the Grassmannian. We introduce and study a family of subsets called “basis shape loci” associated to transversal matroids. Additionally, we study the Deodhar and positroid decompositions of the Grassmannian. A basis shape locus takes as input data a zero/nonzero pattern in a matrix, which is equivalent to a specific presentation of a transversal matroid. The locus is defined to be the set of points in the Grassmannian which are the row spaces of matrices with the prescribed zero/nonzero pattern. We show that this locus depends only on the transversal matroid, not on the specific presentation. When a transversal matroid is a positroid, the closure of its basis shape locus is exactly the positroid variety labelled by the matroid. We give a sufficient, and conjecturally necessary, condition for when a transversal matroid is a positroid. Components in the Deodhar decomposition are indexed by Go-diagrams, certain fillings of Ferrers shapes with white stones, black stones, and pluses. Le-diagrams are a common combinatorial object indexing positroids; all Le-diagrams are Go-diagrams. We give a system of local flips on fillings of Ferrers shapes which may be used to turn arbitrary diagrams into Go-diagrams. When a Go-diagram is a Le-diagram, these flips are exactly the previously studied Le-moves. Using these local flips, we conjecture a combinatorial condition describing when one Deodhar component is contained in the closure of another within a Schubert cell. We define a variety containing and conjecturally equal to the closure of a Deodhar component and prove that this combinatorial criterion implies a containment of these varieties. We further show that there is no reasonable description of Go-diagrams in terms of forbidden subdiagrams by providing an injection from the set of valid Go- diagrams into the set of minimal forbidden subdiagrams. In lieu of such a description, we give an algorithmic characterization of Go-diagrams. Finally, we use the above results to prove several corollaries about Wilson loop cells, which arise in the study of scattering amplitudes in N=4 supersymmetric Yang-Mills theory. Notably, it was previously known that the matroid represented by a generic point in a Wilson loop cell is a positroid. We show that the closure of the Wilson loop cell agrees with the positroid variety labelled by this positroid