Realignment in the NHL, MLB, the NFL, and the NBA

Sports leagues consist of conferences subdivided into divisions. Teams play a number of games within their divisions and fewer games against teams in different divisions and conferences. Usually, a league structure remains stable from one season to the next. However, structures change when growth or contraction occurs, and realignment of the four major professional sports leagues in North America has occurred more than twenty-five times since 1967. In this paper, we describe a method for realigning sports leagues that is flexible, adaptive, and that enables construction of schedules that minimize travel while satisfying other criteria. We do not build schedules; we develop league structures which support the subsequent construction of efficient schedules. Our initial focus is the NHL, which has an urgent need for realignment following the recent move of the Atlanta Thrashers to Winnipeg, but our methods can be adapted to virtually any situation. We examine a variety of scenarios for the NHL, and apply our methods to the NBA, MLB, and NFL. We find the biggest improvements for MLB and the NFL, where adopting the best solutions would reduce league travel by about 20%.Comment: 20 figures, 1 tabl

Exact arborescences, matchings and cycles

AbstractSuppose we are given a graph in which edge has an integral weight. An ‘exact’ problem is to determine whether a desired structure exists for which the sum of the edge weights is exactly k for some prescribed k.We consider the special case of the problem in which all costs are zero or one for arborescences and show that a ‘continuity’ property is prossessed similar to that possessed by matroids. This enables us to determine in polynomial time the complete set of values of k for which a solution exists. We also give a minmax theorem for the maximum possible value of k, in terms of a packing of certain directed cuts in the graph.We also show how enumerative techniques can be used to solve the general exact problem for arborescences (implying spanning trees), perfect matchings in planar graphs and sets of disjoint cycles in a class of planar directed graphs which includes those of degree three. For these problems, we thereby obtain polynomial algorithms provided that the weights are bounded by a constant or encoded in unary

FACES OF MATCHING POLYHEDRA

Let G = (V, E, ~) be a finite loopless graph, let b=(bi:ieV) be a vector of positive integers. A feasible matching is a vector X = (x.: j e: E) J of nonnegative integers such that for each node i of G, the sum of the over the edges j of G incident with i is no greater than bi. The matching polyhedron P(G, b) is the convex hull of the set of feasible matchings. In Chapter 3 we describe a version of Edmonds' blossom algorithm which solves the problem of maximizing C • X over P (G, b) where c =. (c.: j e: E) J is an arbitrary real vector. This algorithm proves a theorem of Edmonds which gives a set of linear inequalities sufficient to define P(G, b). In Chapter 4 we prescribe the unique subset of these inequalities which are necessary to define P(G, b), that is, we characterize the facets of P(G, b). We also characterize the vertices of P(G, b), thus describing the structure possessed by the members of the minimal set X of feasible matchings of G such that for any real vector c = (c.: j e: E), c • x is maximized over P(G, b) J member of X. by a In Chapter 5 we present a generalization of the blossom algorithm which solves the problem: maximize c • x over a face F of P(G, b) for any real vector c = (c.: j e: E). J In other words, we find a feasible matching x of G which satisfies the constraints obtained by replacing an arbitrary subset of the inequalities which define P(G, b) by equations and which maximizes c • x subject to this restriction. We also describe an application of this algorithm to matching problems having a hierarchy of objective functions, so called ''multi-optimization'' problems. In Chapter 6 we show how the blossom algorithm can be combined with relatively simple initialization algorithms to give an algorithm which solves the following postoptimality problem. Given that we know a matching 0 x £ P(G, b) maximizes c · x over P(G, b), we wish to utilize 0 X which to find a feasible matching x' £ P(G, b') which maximizes c • x over P(G, b'), where b' = (b!: i £ V) ]_ vector of positive integers and arbitrary real vector. c=(c.:j£E) J is a is an In Chapter 7 we describe a computer implementation of the blossom algorithm described herein

Graphic loans: East Asia and beyond

The national languages of East Asia (Chinese, Japanese, Korean and Vietnamese) have made extensive use of a type of linguistic borrowing sometimes referred to as a 'graphic loan'. Such loans have no place in the conventional classification of loans based on Haugen (1950) or Weinreich (1953), and research on loan word theory and phonology generally overlooks them. The classic East Asian phenomenon is discussed and a framework is proposed to describe its mechanism. It is argued that graphic loans are more than just 'spelling pronunciations', because they are a systematic and widespread process, independent of but not inferior to phonological borrowing. The framework is then expanded to cover a range of other cases of borrowing between languages to show that graphic loans are not a uniquely East Asian phenomenon, and therefore need to be considered as a major category of loan

Computing, Business, and Operations Research: The Next Challenges

Dr. William Pulleyblank, the Vice President for the Center for Business Optimization of IBM Global Business Services, presented a lecture on April 17, 2008, 3 pm, at the Smithgall Student Services Building, in room 117 on the Georgia Tech campusRuntime: 73:14 minutesThere have been two consistent drivers over the last sixty years of the evolution of computing: Computer power and price/performance improve by a factor of two every eighteen months; the problems that we wish to solve require this growth in capability and more. We seem to be reaching inflection points with both of these drivers. High performance systems are turning to massive parallelism to continue the required growth in performance. New challenges are arising in business and industry that require the solution of fundamentally different problems as well as the development of new approaches to old problems. Moreover, the rapid growth of a global economy has given an unprecedented urgency to dealing with these changes. I will review these subjects and some approaches that are being applied, with varying degrees of success. In particular, I will discuss five technical problems that must be solved to enable us to successfully meet the business challenges that we will face in the future

A LINEAR PROGRAMMING RELAXATION OF THE NODE PACKING PROBLEM OR 2-BICRITICALGRAPHS AND NODE COVERS

The problem of finding a minimum cardinality set of nodes in a graph which meet every edge is of considerable theoretical as well as practical interest. Because of the difficulty of this problem, a linear relaxation of an integer programming model is sometimes used as a heuristic. In fact Nemhauser and Trotter showed that any variables which receive integer values in an optimal solution to the relaxation can retain the same values in an optimal solution to the integer program. We define 2-bicritical graphs and give several characterizations of them. One characterization is that they are precisely the graphs for which an optimal solution to the linear relaxation will have no integer valued variables. Then we show that almost all graphs are 2-bicritical, and hence the linear relaxation almost never helps.We are currently acquiring citations for the work deposited into this collection. We recognize the distribution rights of this item may have been assigned to another entity, other than the author(s) of the work.If you can provide the citation for this work or you think you own the distribution rights to this work please contact the Institutional Repository Administrator at [email protected]

The perfectly matchable subgraph polytope of an arbitrary graph

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