Richard Steinberg

The State of the Three Color Problem

Abstract The Three Color Problem is: Under what conditions can the regions of a planar map be colored in three colors so that no two regions with a common boundary have the same color? This paper describes the origin of the Three Color Problem and virtually all the major results and conjectures extant in the literature. I am deeply grateful to W.T. Tutte and Horst Sachs for thorough readings of an earlier version of the manuscript; each had subtle corrections and excellent suggestions. I would also like to thank: Noga Alon, Dan Archdeacon, Frank R. Bernhart, Avrim Blum, Vasek Chvatal, Michael Fellows, Steve Fisk, Herbert Fleischner, Branko Grunbaum, David Johnson, Felix Lazebnik, David Petford, Arvind Rajan, Kenneth H. Rosen, Richard E. Stone (who suggested the papers title), Bjarne Toft, loan Tomescu, Craig A. Tovey, Peter Ungar, and two anonymous referees. The accuracy and presentation of the material in this paper is of course the responsibility of the author alone.

Congestion pricing and user adaptation

The problem of sharing bandwidth in a communication network has been the focus of much research aimed at guaranteeing an appropriate quality of service to users. This is particularly challenging in an environment with a great diversity of users and applications, which makes it difficult, if not impossible, to tightly constrain user attributes and requirements. This motivates shifting the burden of rate allocation from the network to the end-systems. We propose a decentralized scheme for user adaptation and study its dynamics. The proposed scheme uses congestion prices as a mechanism for providing both feedback and incentives to end-systems.

A Contract and Balancing Mechanism for Sharing Capacity in a Communication Network

We propose a method for determining how much to charge users of a communication network when they share bandwidth. Our approach can be employed either when a network owner wishes to sell bandwidth for a specified period of time to a number of different users, or when users cooperate to build a network to be shared among themselves. Our proposed contract and balancing mechanism can mediate between rapidly fluctuating prices and the longer time scales over which bandwidth contracts may be traded. An advantage of the process is that it avoids perverse incentives for a capacity provider to increase congestion.

Planar Ramsey numbers

Abstract The planar Ramsey number PR ( k , l ) ( k , l ≥ 2) is the smallest integer n such that any planar graph on n vertices contains either a complete graph on k vertices or an independent set of size l . We find exact values of PR ( k , l ) for all k and l . Included is a proof of a 1976 conjecture due to Albertson, Bollobas, and Tucker that every triangle-free planar graph on n vertices contains an independent set of size ⌊ n /3⌋ + 1.

Chapter 18 Marketing-production joint decision-making

Publisher Summary This chapter describes some tangible benefits that emerge from coexistence and outlines some techniques in management science that have been developed to address this issue. At this point, it seems worthwhile to consider some additional perspectives in order to identify existing gaps that may offer further research opportunities. Few of the models incorporate competition, undoubtedly because the fact that the existence of both production and marketing decisions creates models which are already considerably complex. Despite the daunting nature of competitive formulations, these would be well worth investigating. Another dimension which should be looked into is the case of multiple products. Such models may be quite difficult to analyze, however, with the existing tools.

Congestion Pricing and Noncooperative Games in Communication Networks

We consider congestion pricing as a mechanism for sharing bandwidth in communication networks, and model the interaction among the users as a game. We propose a decentralized algorithm for the users that is based on the history of the price process, where user response to congestion prices is analogous to “fictitious play” in game theory, and show that this results in convergence to the unique Wardrop equilibrium. We further show that the Wardrop equilibrium coincides with the welfare-maximizing capacity allocation.

An overview of combinatorial auctions

An auction is combinatorial when bidders can place bids on combinations of items, called “packages,” rather than just individual items. Computer scientists are interested in combinatorial auctions because they are concerned with the expressiveness of bidding languages, as well as the algorithmic aspects of the underlying combinatorial problem. The combinatorial problem has attracted attention from operations researchers, especially those working in combinatorial optimization and mathematical programming, who are fascinated by the idea of applying these tools to auctions. Auctions have been studied extensively by economists, of course. Thus, the newly emerging field of combinatorial auctions lies at the intersection of computer science, operations research, and economics. In this article, we present a brief introduction to combinatorial auctions, based on our book, Combinatorial Auctions (MIT Press, 2006), in which we look at combinatorial auctions from all three perspectives.

The allocation of value for jointly provided services

Caller I.D. service, whereby the telephone number of the calling party is visually displayed to the called party during ringing, is now available in some areas of the U.S., but it is restricted to calls within a local calling area, and for which the calling and called party are customers of the same local telephone company. If Caller I.D. service is extended nationwide, identification of a long-distance call will, in a typical case, require the participation of three companies: the local exchange carrier originating the call, the long-distance carrier, and the local exchange carrier terminating the call. How shall the revenues from the service be divided among the participating firms? We apply cooperative game theory to address this question.

Multiproduct Competition between Congestible Networks

This paper analyses competition between firms who sell multiple products in the presence of negative externalities. The model involves two networks who each may offer several service classes. Service classes are generated by forming sub-networks differentiated by their congestion levels. The level of congestion on a sub-network is determined by its capacity and the number of users, i.e., quality of demand-dependent. This paper shows that networks will choose to offer only one service class, and thus not to form distinct sub-networks, in equilibrium. In addition to contributing to the theory of multiproduct competition, the paper addresses applied problems. For example, the results suggest that current proposals to implement pricing on the Internet will not be viable under competition.

Tutte's 5-flow conjecture for the projective plane

Heawood proved that every planar graph with no 1-cycles is vertex 5-colorable, which is equivalent to the statement that every planar graph with no 1-bonds has a nowhere-zero 5-flow. Tutte has conjectured that every graph with no 1-bonds has a nowhere-zero 5-flow. We show that Tuttes 5-Flow Conjecture is true for all graphs embeddable in the real projective plane.