Search

With recent technological advances in workstations, graphics, graphical user interfaces, and object oriented programming languages, a significant number of researchers are developing general-purpose software and integrated software systems for domains in discrete mathematics, including graph theory, combinatorics, combinatorial optimization, and sets. This software aims to provide effective computational tools for research, applications prototyping, and teaching. In March 1992, DIMACS sponsored a workshop on Computational Support for Discrete Mathematics in order to facilitate interactions between the researchers, developers, and educators who work in these areas. Containing refereed papers based on talks presented at the workshop, this volume documents current and past research in these areas and should provide impetus for new interactions.

This book, the first dedicated to quadratic assignment problems, contains refereed papers presented at the Workshop on Quadratic Assignment and Related Problems, held at DIMACS in May 1993. Bringing together researchers from academia and industry, the workshop focused on recent computational approaches and applications. The methods described here include eigenvalue estimates and reduction techniques for lower bounds, parallelization, genetic algorithms, polyhedral approaches, greedy and adaptive search algorithms. The applications include graph bandwidth problems, telecommunications network design, load balancing, VLSI design, data association problems, and multidimensional assignment problems. In addition, this book contains a survey article with an extensive bibliography.

This volume, published jointly with the Association for Computing Machinery, comprises a collection of research articles celebrating the occasion of Victor Klee's sixty-fifth birthday in September 1990. During his long career, Klee has made contributions to a wide variety of areas, such as discrete and computational geometry, convexity, combinatorics, graph theory, functional analysis, mathematical programming and optimization, and theoretical computer science. In addition, Klee made important contributions to mathematics education, mathematical methods in economics and the decision sciences, applications of discrete mathematics in the biological and social sciences, and the transfer of knowledge from applied mathematics to industry. In honor of Klee's achievements, this volume presents more than forty papers on topics related to Klee's research. While the majority of the papers are research articles, a number of survey articles are also included. Mirroring the breadth of Klee's mathematical contributions, this book shows how different branches of mathematics interact. It is a fitting tribute to one of the foremost leaders in discrete mathematics.

This book comprises a collection of articles stemming from a DIMACS Working Group and DIMACS Workshop on Theoretical Advances in Information Recording held at Rutgers University, Piscataway, NJ. Written by leading researchers in information theory and data storage technology, the articles address problems related to the efficient and reliable storage of information in devices based upon novel optical, magnetic, and biological recording mechanisms. The primary focus of the articles is on signal processing and coding techniques applicable to exploratory technologies being considered for future generations of storage devices, including two-dimensional optical storage (TwoDOS), heat-assisted magnetic recording (HAMR), and volumetric macro-molecular data storage. Specific topics addressed include channel equalization, timing recovery, data detection, modulation coding, and error control coding. Several articles explore the emerging connections between data storage, information theory, and the storage and processing of genetic information in living cells. Articles in the volume also illustrate the broader applicability of fundamental advances in information theory that have arisen in the context of information storage technology. The volume is suitable for graduate students and research scientists interested in applications of information theory, communication theory, and coding theory to man-made and natural data storage systems.

This volume presents selected papers from a three-day workshop held during the DIMACS special years on Mathematical Support for Molecular Biology. Participants from the world over attended, giving the workshop an important international component. The study of discrete mathematics and optimization with medical applications is emerging as an important new research area. Significant applications have been found in medical research, for example in radiosurgical treatment planning, virtual endoscopy, and more. This volume presents a substantive cross-section of active research topics ranging from medical imaging to human anatomy modeling, from gamma knife treatment planning to radiation therapy, and from epileptic seizures to DNA screening. This book is an up-to-date resource reflecting current research directions.

This volume contains the proceedings from the first DIMACS meeting on discrete mathematical chemistry held at Rutgers University (New Brunswick, NJ). The contributions reflect the presentations and spotlight the breadth of current research on the topic—from the Benzenoid Clar problem to the Wulff-shape of sphere packings. Much of the volume reflects the combined mathematical and physical interest in the new molecules, fullerenes. This DIMACS conference highlighted the range of opportunities for fruitful and informed collaboration across the mathematics-chemistry boundaries. The interdisciplinary nature of the contributions pays testament to the fact that “real” chemistry and “real” mathematics do indeed interact.

Discrete mathematics stands among the leading disciplines of mathematics and theoretical computer science. This is due primarily to its increasing role in university curricula and its growing importance in applications ranging from optimization to molecular biology. An inaugural conference was held cooperatively by DIMATIA and DIMACS to focus on the versatility, width, and depth of current progress in the subject area. This volume offers a well-balanced blend of research and survey papers reflecting the exciting, attractive topics in contemporary discrete mathematics. Discussed in the book are topics such as graph theory, partially ordered sets, geometrical Ramsey theory, computational complexity issues and applications.

Time division multiplexing (TDM) has been the fundamental basis for adding capacity to digital telecommunications networks for decades. However, within the past two years, wavelength division multiplexing (WDM) has been emerging as an important and widely deployed complement to TDM. Sales of systems based on the new technology have risen at breathtaking speed. The driving force behind this sales explosion was the unexpected rapid exhaustion of long distance fiber network capacity. This fiber exhaust, combined with favorable economics for WDM, led to the use of this technology over other alternatives. The WDM deployment raises fundamental and challenging problems that require novel and innovative solutions. This volume presents papers from an interdisciplinary workshop held at DIMACS on multichannel optical networks. Leading computer science theorists and practitioners discussed admissions control, routing and channel assignment, multicasting and protection, and fault-tolerance. The book features application of theoretical and/or algorithmical results to practical problems and addresses the influence of practical problems to theoretical/algorithmic studies. The volume can serve as a text for an advanced course in computer science, networking, and operations research.

This volume is a collection of articles written by experienced primary, secondary, and collegiate educators. The book explains why discrete mathematics should be taught in K–12 classrooms and offers practical guidance on how to do so. In this book, teachers at all levels will find a great deal of valuable material to help them introduce discrete mathematics in their classrooms. One main article provides a comprehensive and detailed view of discrete mathematics for K–12. Another surveys the resources that are available for teachers. School and district curriculum leaders will find material that addresses how discrete mathematics can be introduced into their curricula. College faculty members will find ideas and topics that can be incorporated into a variety of courses. Features: Classroom activities and an annotated list of resources. Authors who are directors of innovative programs and who are well known for their work. A description of discrete mathematics providing the opportunity for a fresh start for students who have been previously unsuccessful in mathematics. Discussion on discrete mathematics as it is used to achieve the goals of the current effort to improve mathematics education. Guidance on topics, resources and teaching; a valuable guide for both pre-service and in-service professional development.

This book contains papers presented at the Workshop on Parallel Processing of Discrete Optimization Problems held at DIMACS in April 1994. The contents cover a wide spectrum of the most recent algorithms and applications in parallel processing of discrete optimization and related problems. Topics include parallel branch and bound algorithms, scalability, load balancing, parallelism and irregular data structures and scheduling task graphs on parallel machines. Applications include parallel algorithms for solving satisfiability problems, location problems, linear programming, quadratic and linear assignment problems. This book would be suitable as a textbook in advanced courses on parallel algorithms and combinatorial optimization.

This book grew out of the fourth Special Year at DIMACS, which was devoted to the subject of combinatorial optimization. During the special year, a number of workshops, small and large, dealt with various aspects of this theme. Organizers of the workshops and selected participants were asked to write surveys about the hottest results and ideas in their fields. Therefore, this book is not a set of conference proceedings but rather a carefully refereed collection of invited survey articles written by outstanding researchers. Aimed at researchers in discrete mathematics, operations research, and the theory of computing, this book offers an in-depth look at many topics not treated in textbooks.

In this paper the authors prove the following results (via a unified approach) for all sufficiently large $n$: (i) [$1$-factorization conjecture] Suppose that $n$ is even and $D\geq 2\lceil n/4\rceil -1$. Then every $D$-regular graph $G$ on $n$ vertices has a decomposition into perfect matchings. Equivalently, $\chi'(G)=D$. (ii) [Hamilton decomposition conjecture] Suppose that $D \ge \lfloor n/2 \rfloor $. Then every $D$-regular graph $G$ on $n$ vertices has a decomposition into Hamilton cycles and at most one perfect matching. (iii) [Optimal packings of Hamilton cycles] Suppose that $G$ is a graph on $n$ vertices with minimum degree $\delta\ge n/2$. Then $G$ contains at least ${\rm reg}_{\rm even}(n,\delta)/2 \ge (n-2)/8$ edge-disjoint Hamilton cycles. Here ${\rm reg}_{\rm even}(n,\delta)$ denotes the degree of the largest even-regular spanning subgraph one can guarantee in a graph on $n$ vertices with minimum degree $\delta$. (i) was first explicitly stated by Chetwynd and Hilton. (ii) and the special case $\delta= \lceil n/2 \rceil$ of (iii) answer questions of Nash-Williams from 1970. All of the above bounds are best possible.

This is a collection of translations of a variety of papers on discrete mathematics by members of the Moscow Seminar on Discrete Mathematics. This seminar, begun in 1972, was marked by active participation and intellectual ferment. Mathematicians in the USSR often encountered difficulties in publishing, so many interesting results in discrete mathematics remained unknown in the West for some years, and some are unknown even to the present day. To help fill this communication gap, this collection offers papers that were obscurely published and very hard to find. Among the topics covered here are: graph theory, network flow and multicommodity flow, linear programming and combinatorial optimization, matroid theory and submodular systems, matrix theory and combinatorics, parallel computing, complexity of algorithms, random graphs and statistical mechanics, coding theory, and algebraic combinatorics and group theory.

This book is a collection of papers from the proceedings of the first symposium of the Japan Association for Mathematical Sciences. Topics covered center around problems of geometric analysis in relation to heat kernels, random walks, and Poisson boundaries on discrete groups, graphs, and other combinatorial objects. The material is suitable for graduate students and research mathematicians interested in heat kernels and random works on groups and graphs.

This volume contains nineteen survey papers describing the state of current research in discrete and computational geometry as well as a set of open problems presented at the 2006 AMS-IMS-SIAM Summer Research Conference'Discrete and Computational Geometry—Twenty Years Later', held in Snowbird, Utah, in June 2006. Topics surveyed include metric graph theory, lattice polytopes, the combinatorial complexity of unions of geometric objects, line and pseudoline arrangements, algorithmic semialgebraic geometry, persistent homology, unfolding polyhedra, pseudo-triangulations, nonlinear computational geometry, $k$-sets, and the computational complexity of convex bodies. Discrete and computational geometry originated as a discipline in the mid-1980s when mathematicians in the well-established field of discrete geometry and computer scientists in the (then) nascent field of computational geometry began working together on problems of common interest. The combined field has experienced a huge growth in the past twenty years, which the present volume attests to.

For the mathematician interested in discrete mathematics, from the senior undergraduate to the professional level, this volume provides first-rate surveys of the important combinatorics themes in ordered sets. These expository lectures, given at a 1985 Joint Summer Research Conference, cover a wide range of topics, which include: the three-machine problem to illustrate the order-theoretic aspects of scheduling theory; the techniques used in settling the “matching conjecture”; the decomposition of ordered sets into few chains; the reorientation of graphs; the varied occurrences of the meet-distribution property; surveys techniques used in settling binary sorting problems; the formulation of a general viewpoint for retraction; the survey of cutsets; and the role played by subdiagrams in ordered sets.

This volume is a collection of refereed expository and research articles in discrete and computational geometry written by leaders in the field. Articles are based on invited talks presented at the AMS-IMS-SIAM Summer Research Conference, “Discrete and Computational Geometry: Ten Years Later”, held in 1996 at Mt. Holyoke College (So. Hadley, MA). Topics addressed range from tilings, polyhedra, and arrangements to computational topology and visibility problems. Included are papers on the interaction between real algebraic geometry and discrete and computational geometry, as well as on linear programming and geometric discrepancy theory.