Forbidden minor characterizations for low-rank optimal solutions to semidefinite programs over the elliptope

We study a new geometric graph parameter \egd(G), defined as the smallest integer $r\ge 1$ for which any partial symmetric matrix which is completable to a correlation matrix and whose entries are specified at the positions of the edges of $G$, can be completed to a matrix in the convex hull of correlation matrices of \rank at most $r$. This graph parameter is motivated by its relevance to the problem of finding low rank solutions to semidefinite programs over the elliptope, and also by its relevance to the bounded rank Grothendieck constant. Indeed, \egd(G)\le r if and only if the rank-$r$ Grothendieck constant of $G$ is equal to 1. We show that the parameter \egd(G) is minor monotone, we identify several classes of forbidden minors for \egd(G)\le r and we give the full characterization for the case $r=2$. We also show an upper bound for \egd(G) in terms of a new tree-width-like parameter \sla(G), defined as the smallest $r$ for which $G$ is a minor of the strong product of a tree and $K_r$. We show that, for any 2-connected graph $G\ne K_{3,3}$ on at least 6 nodes, \egd(G)\le 2 if and only if \sla(G)\le 2.Comment: 33 pages, 8 Figures. In its second version, the paper has been modified to accommodate the suggestions of the referees. Furthermore, the title has been changed since we feel that the new title reflects more accurately the content and the main results of the pape

Knotenfärbungen mit Abstandsbedingungen

Knotenfärbungen mit Abstandsbedingungen sind graphentheoretische Konzepte, motiviert durch das praktische Problem der Frequenzzuweisung in Mobilfunknetzen. In der Arbeit werden verschiedene Varianten solcher Färbungen vorgestellt. Für (Listen-)Färbungen mit einer beliebigen Anzahl r von Abstandsbedingungen werden allgemeine Eigenschaften und Schranken für die benötigte Anzahl von Farben bewiesen. Anschließend wird der Spezialfall r=2 behandelt. Färbungen mit zwei Abstandsbedingungen - die sogenannten L(d,s)-Labellings - werden für eine Reihe von Graphenklassen untersucht, u.a. für reguläre Parkettierungen, Weg- und Kreispotenzen und Graphen mit Durchmesser 2. Die Listenversion dieser Färbungen - die sogenannten L(d,s)-List Labellings - werden für Wege, Sterne, Kreise und Kakteen betrachtet. Ferner werden Untersuchungen zum Zusammenhang von L(2,1)-Labellings und L(2,1)-List Labellings bei speziellen Bäumen durchgeführt

Traveling salesman path problems

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2005.Includes bibliographical references (p. 153-155).In the Traveling Salesman Path Problem, we are given a set of cities, traveling costs between city pairs and fixed source and destination cities. The objective is to find a minimum cost path from the source to destination visiting all cities exactly once. The problem is a generalization of the Traveling Salesman Problem with many important applications. In this thesis, we study polyhedral and combinatorial properties of a variant we call the Traveling Salesman Walk Problem, in which the minimum cost walk from the source to destination visits all cities at least once. Using the approach of linear programming, we study properties of the polyhedron corresponding to a linear programming relaxation of the traveling salesman walk problem. Our results relate the structure of the underlying graph of the problem instance with polyhedral properties of the corresponding fractional walk polyhedron. We first characterize traveling salesman walk perfect graphs, graphs for which the convex hull of incidence vectors of traveling salesman walks can be described by linear inequalities. We show these graphs have a description by way of forbidden minors and also characterize them constructively.(cont.) We extend these results to relate the underlying graph structure to the integrality gap of the corresponding fractional walk polyhedron. We present several graph operations which preserve integrality gap; these operations allow us to find the integrality gap of graphs built from smaller bricks, whose integrality gaps can be found by computational or other methods.by Fumei Lam.Ph.D

Rigidity Percolation in Disordered Fiber Systems: Theory and Applications

Nanocomposites, particularly carbon nanocomposites, find many applications spanning an impressive variety of industries on account of their impressive properties and versatility. However, the discrepancy between the performance of individual nanoparticles and that of nanocomposites suggests continued technological development and better theoretical understanding will provide much opportunity for further property enhancement. Study of computational renderings of disordered fiber systems has been successful in various nanocomposite modeling applications, particularly toward the characterization of electrical properties. Motivated by these successes, I develop an explanatory model for `mechanical' or `rheological percolation,' terms used by experimentalists to describe a nonlinear increase in elastic modulus/strength that occurs at particle inclusion volume fractions well above the electrical percolation threshold. Specifically, I formalize a hypothesis given by \\citet*{penu}, which states that these dramatic gains result from the formation of a `rigid CNT network.' Idealizing particle interactions as hinges, this amounts to the network property of \\emph{rigidity percolation}---the emergence of a giant component (within the inclusion contact network) that is not only connected, but furthermore the inherent contacts are patterned to constrain all internal degrees of freedom in the component. Rigidity percolation has been studied in various systems (particularly the characterization of glasses and proteins) but has never been applied to disordered systems of three-dimensional rod-like particles. With mathematically principled arguments from \\emph{rigidity matroid theory}, I develop a scalable algorithm (\\emph{Rigid Graph Compression}, or \\emph{RGC}), which can be used to detect rigidity percolation in such systems by iteratively compressing provably rigid subgraphs within the rod contact networks. Prior to approaching the 3D system, I confirm the usefulness of \\emph{RGC} by using it to accurately approximate the rigidity percolation threshold in disordered systems of 2D fibers---achieving $<1\\%$ error relative to a previous exact method. Then, I develop an implementation of \\emph{RGC} in three dimensions and determine an upper bound for the rigidity percolation threshold in disordered 3D fiber systems. More work is required to show that this approximation is sufficiently accurate---however, this work confirms that rigidity in the inclusion network is a viable explanation for the industrially useful mechanical percolation. Furthermore, I use \\emph{RGC} to quantitatively characterize the effects of interphase growth and spatial CNT clustering in a real polymer nanocomposite system of experimental interest.Doctor of Philosoph

Algorithmic Graph Theory

The main focus of this workshop was on mathematical techniques needed for the development of efficient solutions and algorithms for computationally difficult graph problems. The techniques studied at the workshhop included: the probabilistic method and randomized algorithms, approximation and optimization, structured families of graphs and approximation algorithms for large problems. The workshop Algorithmic Graph Theory was attended by 46 participants, many of them being young researchers. In 15 survey talks an overview of recent developments in Algorithmic Graph Theory was given. These talks were supplemented by 10 shorter talks and by two special sessions