The concept of extreme vertex sets, defined in Section 1.5.3, was first introduced by Watanabe and Nakamura [308] to solve the edge-connectivity augmentation problem. The fastest deterministic algorithm currently known for computing all extreme vertex sets was given by Naor, Gusfield, and Martel [259]. Their algorithm first computes the Gomory–Hu cut tree of the graph and then finds all maximal k-edge-connected components for some k, from which all extreme vertex sets are identified by Lemma 1.42, taking O(n(mn log(n2/m)) running time. Bencz&úr and Karger [20] have given a Monte Carlo–type randomized algorithm, which runs in O(n2 log5n) time but is rather involved. Notice that computing all extreme vertex sets is not easier than finding a minimum cut since at least one of the extreme vertex sets is a minimum cut.

In this chapter, we give a simple and efficient algorithm for computing all extreme vertex sets in a given graph, and we show some applications of extreme vertex sets. The algorithm will be used to solve the edge-connectivity augmentation problem in Section 8.3. In Section 6.1,we design a deterministic O(nm + n2 log n) time algorithm for computing the family χ(G) of extreme vertex sets in a given edge-weighted graph G, which is a laminar family, as observed in Lemma 1.41. As a new application of extreme vertex sets, in Section 6.2 we consider a dynamic graph G in which the weight of edges incident to a designated vertex may increase or decrease with time and we give a dynamic minimum cut algorithm that reports a minimum cut of the current G whenever the weight of an edge is updated.

In this chapter, we investigate structures and algorithms of cactus representations, which were introduced in Section 1.5.4 to represent all minimum cuts in an edgeweighted graph G. Throughout this chapter, we assume that λ(G) > 0 for a given graph G, which implies that G is connected. Let C(G) denote the set of all minimum cuts in G. In Section 5.1, we define a canonical form of cactus representations. In Section 5.2, we show that a subset of C(G) that consists of minimum cuts separating two given vertices, s and t, can be represented by a simple cactus structure. In Section 5.3, we design an O(mn + n2 log n) time algorithm for constructing a cactus representation R of C(G).

Canonical Forms of Cactus Representations

In this section, we discuss cactus representations for a subset of minimum cuts, and we prove the existence of two canonical forms, which we call the cycle-type and junction-type normal cactus representations. Such a canonical representation is useful in designing an efficient algorithm that constructs a cactus representation for all the minimum cuts of a given graph [244]. It also helps to efficiently test whether two given graphs have the same “structure” with respect to their minimum cuts, which is based on a planar isomorphism algorithm due to Hopcroft and Tarjan [126].

A cactus representation for a given subset C ⊆ C(G), if one exists, may not be unique unless we impose further structural restrictions.

Recall from Section 1.1 that a mechanism M realizes a mapping F : Θ → ℝk if ζ (μ (θ)) = F (θ) for all θ ∈ Θ, where μ (θ) is the set of equilibrium messages for θ ∈ Θ, and ζ is the outcome mapping of M. The revelation mechanism and the parameter transfer mechanisms discussed in Section 1.1 demonstrate that there always exist mechanisms that realize a given mapping F. These mechanisms, however, require all but at most one of the n agents to transfer their private information to the message space. Given these rather coarse solutions to the problem of realization, the question becomes the following: Is it possible to exploit the particular features of a given mapping F in order to construct a mechanism that economizes on the amount of information that the agents transfer to the message space in order to realize F?

This question originates in the case in which the mapping F specifies a Pareto optimal allocation of private goods for n agents in an economy. The private information of any single agent (e.g., a consumer's preferences, or the cost function of a firm) can be quite complicated, and yet the agents manage to achieve a Pareto optimal allocation by communicating in a market using the relatively small set of signals consisting of prices and proposed trades.

The Model of a Mechanism

A fundamental problem in economic theory is to explain how acceptable choices can be made by a group despite the fact that only a small portion of the information that may a priori seem relevant can be taken into account. This problem arises in many settings, ranging from the largest scale problems of macroeconomic systems to the smallest problems of coordination among individuals in an organization. A market economy, for instance, coordinates production by firms and purchases by consumers through prices and quantities. The enormous amount of information held by each firm concerning its production processes and the knowledge of each consumer concerning his own tastes are not communicated among the participants in a market. General equilibrium theory, however, explains a sense in which the production plan selected by each firm and the purchases of each consumer in a market equilibrium are optimal despite the fact that a vast amount of the information known by firms and consumers remains private. A similar phenomenon arises within organizations. Employees cannot communicate all that they know to their manager, and if they could, then the manager could not possibly absorb all of this information. Communication is instead typically limited to conversations and memos. Determining exactly what information should be transmitted to a manager in order to allow him to make good decisions is a fundamental problem in the design of organizations and in the theory of accounting. Firms successfully function, however, despite this limited communication among its layers of management.

This text develops a calculus-based, first-order approach to the construction of economic mechanisms. A mechanism here is informationally decentralized in the sense that it operates in an environment in which relevant information is dispersed among the participating agents. A mechanism thus requires a “language,” or message space, that defines how the agents may communicate with one another. This text focuses on the task of constructing the alternative message spaces that a group of agents may use as languages for communicating with one another and thereby achieve a common objective. The relationship between the language that a group of agents may use and the ends that they may accomplish was identified in Hurwicz (1960); the model of a mechanism that is the main object of study in this text originated in this paper and in the long-term collaboration of Leonid Hurwicz with Stanley Reiter. Whereas constructing the message space is but one aspect of the design of a mechanism, it is fundamental in the sense that other aspects (such as dynamic stability and incentives) revolve around the choice of messages with which agents may communicate.

It is assumed here that the sets in the model of a mechanism are subsets of Euclidean space. Appropriate regularity assumptions are imposed on mappings and correspondences so that it is possible to identify necessary and sufficient differential conditions for the design of an economic mechanism.

This chapter analyzes some additional pricing techniques and marketing tactics intended to further enhance sellers' profits. Section 10.1 links pricing decisions to innovation and product design decisions by computing the profit-maximizing quality levels and service classes to be introduced into the market. The underlining assumption in this analysis is that sellers cannot directly price discriminate among the different consumer groups. Therefore, the seller must devise a price scheme under which consumers belonging to different consumer groups choose to purchase different quality levels.

Section 10.2 examines a special case of the above problem by analyzing markets in which the seller may find it profitable to sell a “damaged good” (such as a version containing less features) to consumers with low willingness to pay, to segment the market between consumers with high and low willingness to pay. The interesting feature of this result is that the low-quality product is more costly to produce than the high-quality product. In the case of a service, the low-quality service is more costly to deliver than the original undamaged high-quality service.

Section 10.3.1 analyzes repeatedly purchased services and identifies the conditions under which a firm should provide a discount to its returning “loyal” consumers, and also the conditions for the polar case under which the firm should discount the price for consumers who switch from competing brands. Section 10.3.2 explores the consequences of having sellers commit to matching the prices offered by competing sellers, and demonstrates why this strategy can be profit enhancing.

The key to a successful implementation of any yield management strategy is getting to know the consumers. Large firms invest tremendous amounts of money on research seeking to characterize their own customers as well as potential consumers. In economic theory, the most useful instrument for characterizing consumer behavior is the demand function. A demand function shows the quantity demanded by an individual, a group, or all the consumers in a given market, as a function of market prices, and some other variables.

Knowing the demand structure is a necessary condition for proper selection of profit-maximizing actions by the firm. But it is not a sufficient condition because the firm must also take cost-of-production considerations into account. Therefore, price decision makers within a firm should properly study the structure of the cost of the service or the product sold by their own firms. They should also distinguish among the different types of costs, particularly between costs associated with a marginal expansion of output and costs associated with investing in infrastructure, research and development (R&D), and capacity.

For this reason, we devote an entire chapter to study the most widely used demand and cost structures. Some readers, particularly readers who took microeconomics courses at a second-year undergraduate level, may be familiar with this material. In general, readers can skip this chapter and use it as a reference for the various concepts whenever necessary.