We will review some of the major results in random graphs and some of the more challenging open problems. We will cover algorithmic and structural questions. We will touch on newer models, including those related to the WWW.

Random graphs

tions. Bootstrap has found many applications in engineering field, including artificial neural networks, biomedical engineering, environmental engineering, image processing, and radar and sonar signal processing. Basic concepts of the bootstrap are summarized in each section as a step-by-step algorithm for ease of implementation. Most of the applications are taken from the signal processing literature. The principles of the bootstrap are introduced in Chapter 2. Both the nonparametric and parametric bootstrap procedures are explained. Babu and Singh (1984) have demonstrated that in general, these two procedures behave similarly for pivotal (Studentized) statistics. The fact that the bootstrap is not the solution for all of the problems has been known to statistics community for a long time; however, this fact is rarely touched on in the manuscripts meant for practitioners. It was first observed by Babu (1984) that the bootstrap does not work in the infinite variance case. Bootstrap Techniques for Signal Processing explains the limitations of bootstrap method with an example. I especially liked the presentation style. The basic results are stated without proofs; however, the application of each result is presented as a simple step-by-step process, easy for nonstatisticians to follow. The bootstrap procedures, such as moving block bootstrap for dependent data, along with applications to autoregressive models and for estimation of power spectral density, are also presented in Chapter 2. Signal detection in the presence of noise is generally formulated as a testing of hypothesis problem. Chapter 3 introduces principles of bootstrap hypothesis testing. The topics are introduced with interesting real life examples. Flow charts, typical in engineering literature, are used to aid explanations of the bootstrap hypothesis testing procedures. The bootstrap leads to second-order correction due to pivoting; this improvement in the results due to pivoting is also explained. In the second part of Chapter 3, signal processing is treated as a regression problem. The performance of the bootstrap for matched filters as well as constant false-alarm rate matched filters is also illustrated. Chapters 2 and 3 focus on estimation problems. Chapter 4 introduces bootstrap methods used in model selection. Due to the inherent structure of the subject matter, this chapter may be difficult for nonstatisticians to follow. Chapter 5 is the most impressive chapter in the book, especially from the standpoint of statisticians. It provides real data bootstrap applications to illustrate the theory covered in the earlier chapters. These include applications to optimal sensor placement for knock detection and land-mine detection. The authors also provide a MATLAB toolbox comprising frequently used routines. Overall, this is a very useful handbook for engineers, especially those working in signal processing.

Probability and Random Processes

In this chapter, you will see how the simplex method simplifies when it is applied to a class of optimization problems that are known as “network flow models.” You will also see that if a network flow model has “integer-valued data,” the simplex method finds an optimal solution that is integer-valued.

Flows in Networks

This chapter is devoted to a more detailed examination of game theory. Game theory is an important tool for analyzing strategic behavior, is concerned with how individuals make decisions when they recognize that their actions affect, and are affected by, the actions of other individuals or groups. Strategic behavior recognizes that the decision-making process is frequently mutually interdependent. Game theory is the study of the strategic behavior involving the interaction of two or more individuals, teams, or firms, usually referred to as players. Two game theoretic scenarios were examined in this chapter: Simultaneous-move and multi-stage games. In simultaneous-move games the players effectively move at the same time. A normal-form game summarizes the players, possible strategies and payoffs from alternative strategies in a simultaneous-move game. Simultaneous-move games may be either noncooperative or cooperative. In contrast to noncooperative games, players of cooperative games engage in collusive behavior. A Nash equilibrium, which is a solution to a problem in game theory, occurs when the players’ payoffs cannot be improved by changing strategies. Simultaneous-move games may be either one-shot or repeated games. One-shot games are played only once. Repeated games are games that are played more than once. Infinitely-repeated games are played over and over again without end. Finitely-repeated games are played a limited number of times. Finitely-repeated games have certain or uncertain ends.

An Introduction to Game Theory

Discrete optimization problems are everywhere, from traditional operations research planning problems, such as scheduling, facility location, and network design; to computer science problems in databases; to advertising issues in viral marketing. Yet most such problems are NP-hard. Thus unless P = NP, there are no efficient algorithms to find optimal solutions to such problems. This book shows how to design approximation algorithms: efficient algorithms that find provably near-optimal solutions. The book is organized around central algorithmic techniques for designing approximation algorithms, including greedy and local search algorithms, dynamic programming, linear and semidefinite programming, and randomization. Each chapter in the first part of the book is devoted to a single algorithmic technique, which is then applied to several different problems. The second part revisits the techniques but offers more sophisticated treatments of them. The book also covers methods for proving that optimization problems are hard to approximate. Designed as a textbook for graduate-level algorithms courses, the book will also serve as a reference for researchers interested in the heuristic solution of discrete optimization problems.

https://www.designofapproxalgs.com/book.pdf

The Design of Approximation Algorithms

. We consider budget feasible mechanisms for procurement auctions with additive valuation functions. For the divisible case, where agents can be allocated fractionally, there exists an optimal mechanism with approximation guarantee e/ ( e − 1) under the small bidder assumption. We study the divisible case without the small bidder assumption, but assume that the true costs of the agents are bounded by the budget. This setting lends itself to modeling economic situations in which the goods represent time and the agents’ true costs are not necessar-ily small compared to the budget. Non-trivially, we give a mechanism with an approximation guarantee of 2.62, improving the optimal result of 3 for the indivisible case. Additionally, we give a lower bound on the approximation guarantee of 1.25. We then study the problem in more competitive markets and assume that the agents’ value over cost eﬃciencies are bounded by some θ . For θ ≤ 2 , we give a mechanism with an approximation guarantee of 2 and a lower bound of 1.18. Finally, we extend our results to diﬀerent agent types with concave additive valuation functions.

Budget Feasible Mechanisms for Procurement Auctions with Divisible Agents

We consider clustering games in which the players are embedded in a network and want to coordinate (or anti-coordinate) their strategy with their neighbors. The goal of a player is to choose a strategy that maximizes her utility given the strategies of her neighbors. Recent studies show that even very basic variants of these games exhibit a large Price of Anarchy: A large inefficiency between the total utility generated in centralized outcomes and equilibrium outcomes in which players selfishly try to maximize their utility. Our main goal is to understand how structural properties of the network topology impact the inefficiency of these games. We derive topological bounds on the Price of Anarchy for different classes of clustering games. These topological bounds provide a more informative assessment of the inefficiency of these games than the corresponding (worst-case) Price of Anarchy bounds. As one of our main results, we derive (tight) bounds on the Price of Anarchy for clustering games on Erdős-Renyi random graphs (where every possible edge in the network is present with a fixed probability), which, depending on the graph density, stand in stark contrast to the known Price of Anarchy bounds.

/pdf/topological-price-of-anarchy-bounds-for-clustering-games-on-4q0l2jf5ll.pdf

Topological Price of Anarchy Bounds for Clustering Games on Networks

/pdf/the-enriched-median-routing-problem-and-its-usefulness-in-8w4q10ta.pdf

The enriched median routing problem and its usefulness in practice

Packing problems constitute an important class of optimization problems. However, despite the large number of variants that have been studied in the literature, most packing problems encompass a single tier of capacity restrictions only. For example, in the Multiple Knapsack Problem, we want to assign a selection of items to multiple knapsacks such that their capacities are not exceeded. But what if these knapsacks are partitioned into clusters, each imposing an additional (aggregated) capacity restriction on the knapsacks contained in that cluster?

Maximum Coverage with Cluster Constraints: An LP-Based Approximation Technique

The vertex cover problem is a fundamental and widely studied combinatorial optimization problem. It is known that its standard linear programming relaxation is integral for bipartite graphs and half-integral for general graphs. As a consequence, the natural rounding algorithm based on this relaxation computes an optimal solution for bipartite graphs and a 2-approximation for general graphs. This raises the question of whether one can obtain improved bounds on the approximation ratio, depending on how close the graph is to being bipartite. In this paper, we consider a round-and-bipartize algorithm that exploits the knowledge of an induced bipartite subgraph to attain improved approximation ratios. Equivalently, we suppose that we have access to a subset of vertices S whose removal bipartizes the graph. If S is an independent set, we prove a tight approximation ratio of 1+1 /ρ , where 2 ρ − 1 denotes the odd girth of the contracted graph ˜ G := G /S and thus satisﬁes ρ ≥ 2. We show that this is tight for any graph and independent set by providing a family of weight functions for which this bound is attained. In addition, we give tight upper bounds for the fractional chromatic number and the integrality gap of such graphs, both of which also depend on the odd girth.If S is an arbitrary set, we prove a tight approximation ratio of (1 + 1 /ρ ) (1 − α ) + 2 α , where α ∈ [0 , 1] denotes the total normalized dual sum of the edges lying inside of the set S . As an algorithmic application, we show that for any eﬃciently k -colorable graph with k ≥ 4 we can ﬁnd a bipartizing set satisfying α ≤ 1 − 4 /k . This provides an approximation algorithm recovering the bound of 2 − 2 /k in the worst case (i.e., when ρ = 2), which is best possible for this setting when using the standard relaxation.

Guido Schäfer

Papers

Budget Feasible Mechanisms for Procurement Auctions with Divisible Agents

Topological Price of Anarchy Bounds for Clustering Games on Networks

The enriched median routing problem and its usefulness in practice

Maximum Coverage with Cluster Constraints: An LP-Based Approximation Technique

A Round and Bipartize Approximation Algorithm for Vertex Cover