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 study the inefficiency of equilibrium outcomes in bottleneck congestion games. These games model situations in which strategic players compete for a limited number of facilities. Each player allocates his weight to a (feasible) subset of the facilities with the goal to minimize the maximum (weight-dependent) latency that he experiences on any of these facilities. We derive upper and (asymptotically) matching lower bounds on the (strong) price of anarchy of linear bottleneck congestion games for a natural load balancing social cost objective (i.e., minimize the maximum latency of a facility). We restrict our studies to linear latency functions. Linear bottleneck congestion games still constitute a rich class of games and generalize, for example, load balancing games with identical or uniformly related machines with or without restricted assignments.

/pdf/on-the-inefficiency-of-equilibria-in-linear-bottleneck-1y5mzkf53w.pdf

On the inefficiency of equilibria in linear bottleneck congestion games

We consider the single-source many-targets shortest-path (SSMTSP) problem in directed graphs with non-negative edge weights. A source node s and a target set T is specified and the goal is to compute a shortest path from s to a node in T . Our interest in the shortest path problem with many targets stems from its use in weighted bipartite matching algorithms. A weighted bipartite matching in a graph with n nodes on each side reduces to n SSMTSP problems, where the number of targets varies between n and 1 . The SSMTSP problem can be solved by Dijkstra's algorithm. We describe a heuristic that leads to a significant improvement in running time for the weighted matching problem; in our experiments a speed-up by up to a factor of 12 was achieved. We also present a partial analysis that gives some theoretical support for our experimental findings.

A Heuristic for Dijkstra's Algorithm With Many Targets and its Use in Weighted Matching Algorithms

We study the inefficiency and computation of pure Nash equilibria in unweighted congestion games, where the strategies of each player i are given implicitly by the binary vectors of a polytope $P_i$. Given these polytopes, a strategy profile naturally corresponds to an integral vector in the aggregation polytope PN = ∑i Pi. We identify two general properties of the aggregation polytope $P_N$ that are sufficient for our results to go through, namely the integer decomposition property (IDP) and the box-totally dual integrality property (box-TDI). Intuitively, the IDP is needed to decompose a load profile in PN into a respective strategy profile of the players, and box-TDI ensures that the intersection of a polytope with an arbitrary integer box is an integral polytope. Examples of polytopal congestion games which satisfy IDP and box-TDI include common source network congestion games, symmetric totally unimodular congestion games, non-symmetric matroid congestion games and symmetric matroid intersection congestion games (in particular, r-arborescences and strongly base-orderable matroids). Our main contributions for polytopal congestion games satisfying IDP and box-TDI are as follows: We derive tight bounds on the price of stability for these games. This extends a result of Fotakis (2010) on the price of stability for symmetric network congestion games to the larger class of polytopal congestion games. Our bounds improve upon the ones for general polynomial congestion games obtained by Christodoulou and Gairing (2016). We show that pure Nash equilibria can be computed in strongly polynomial time for these games. To this aim, we generalize a recent aggregation/decomposition framework by Del Pia et al. (2017) for symmetric totally unimodular and non-symmetric matroid congestion games, both being a special case of our polytopal congestion games. Finally, we generalize and extend results on the computation of strong equilibria in bottleneck congestion games studied by Harks, Hoefer, Klimm and Skopalik (2013). In particular, we show that strong equilibria can be computed efficiently for symmetric totally unimodular bottleneck congestion games. In general, our results reveal that the combination of IDP and box-TDI gives rise to an efficient approach to compute a pure Nash equilibrium whose inefficiency is better than in general congestion games.

/pdf/potential-function-minimizers-of-combinatorial-congestion-upek4a18hu.pdf

Potential Function Minimizers of Combinatorial Congestion Games: Efficiency and Computation

We consider metrical task systems, a general framework to model online problems. Borodin, Linial and Saks [3] presented a deterministic work function algorithm (WFA) for metrical task systems having a tight competitive ratio of 2n-1. We present a smoothed competitive analysis of WFA. Given an adversarial task sequence, we smoothen the request costs by means of a symmetric additive smoothing model and analyze the competitive ratio of WFA on the smoothed task sequence. We prove upper and matching lower bounds on the smoothed competitive ratio of WFA. Our analysis reveals that the smoothed competitive ratio of WFA is much better than O(n) and that it depends on several topological parameters of the underlying graph G, such as the maximum degree D and the diameter. For example, already for small perturbations the smoothed competitive ratio of WFA reduces to O(log n) on a clique or a complete binary tree and to $O(\sqrt{n})$ on a line. We also provide the first average case analysis of WFA showing that its expected competitive ratio is O(log(D)) for various distributions.

/pdf/topology-matters-smoothed-competitiveness-of-metrical-task-3b6w327gg8.pdf

Topology Matters: Smoothed Competitiveness of Metrical Task Systems

We introduce a new measure of the discrepancy in strategic games between the social welfare in a Nash equilibrium and in a social optimum, that we call selfishness level. It is the smallest fraction of the social welfare that needs to be offered to each player to achieve that a social optimum is realized in a pure Nash equilibrium. The selfishness level is unrelated to the price of stability and the price of anarchy and is invariant under positive linear transformations of the payoff functions. Also, it naturally applies to other solution concepts and other forms of games.

We study the selfishness level of several well-known strategic games. This allows us to quantify the implicit tension within a game between players' individual interests and the impact of their decisions on the society as a whole. Our analyses reveal that the selfishness level often provides a deeper understanding of the characteristics of the underlying game that influence the players' willingness to cooperate.

In particular, the selfishness level of finite ordinal potential games is finite, while that of weakly acyclic games can be infinite. We derive explicit bounds on the selfishness level of fair cost sharing games and linear congestion games, which depend on specific parameters of the underlying game but are independent of the number of players. Further, we show that the selfishness level of the n-players Prisoner's Dilemma is c/(b(n-1)-c), where b and c are the benefit and cost for cooperation, respectively, that of the n-players public goods game is (1 - c/n)/(c - 1), where c is the public good multiplier, and that of the Traveler's Dilemma game is 1/2 (b - 1), where b is the bonus. Finally, the selfishness level of Cournot competition (an example of an infinite ordinal potential game), Tragedy of the Commons, and Bertrand competition is infinite.

/pdf/selfishness-level-of-strategic-games-1poo06ffoh.pdf

Guido Schäfer

Papers

On the inefficiency of equilibria in linear bottleneck congestion games

A Heuristic for Dijkstra's Algorithm With Many Targets and its Use in Weighted Matching Algorithms

Potential Function Minimizers of Combinatorial Congestion Games: Efficiency and Computation

Topology Matters: Smoothed Competitiveness of Metrical Task Systems

Selfishness level of strategic games