Mechanism design seeks algorithms whose inputs are provided by selfish agents who would lie if it were to their advantage. Incentive-compatible mechanisms compel the agents to tell the truth by making it in their self-interest to do so. Often, as in combinatorial auctions, such mechanisms involve the solution of NP-hard problems. Unfortunately, approximation algorithms typically destroy incentive compatibility. Randomized rounding is a commonly used technique for designing approximation algorithms. We devise a version of randomized rounding that is incentive-compatible, giving a truthful mechanism for combinatorial auctions with single parameter agents (e.g., "single minded bidders'') that approximately maximizes the social value of the auction. We discuss two orthogonal notions of truthfulness for a randomized mechanism---truthfulness with high probability and in expectation---and give a mechanism that achieves both simultaneously.

We consider combinatorial auctions where multiple copies of many different items are on sale, and each bidder i desires a subset $S_i$. Given a set of bids, the problem of finding the allocation of items that maximizes total valuation is the well-known SetPacking problem. This problem is NP-hard, but for the case of items with many identical copies, the optimum can be approximated very well. To turn this approximation algorithm into a truthful auction mechanism, we overcome two problems: We show how to make the allocation algorithm monotone, and give a method to compute the appropriate payments efficiently.

Using a new recursive technique, we present an explicit construction of an infinite family of N-superconcentrators of density 44. The most economical previously known explicit graphs of this type have density around 60.

Admission control (call control) is a well-studied online problem. We are given a fixed graph with edge capacities, and must process a sequence of calls that arrive over time, accepting some and rejecting others in order to stay within capacity limitations of the network. In the standard theoretical formulation, this problem is analyzed as a benefit problem: The goal is to devise an online algorithm that accepts at least a reasonable fraction of the maximum number of calls that could possibly have been accepted in hindsight. This formulation, however, has the property that even algorithms with optimal competitive ratios (typically $O(\log n)$ where n is the number of nodes) may end up rejecting the vast majority of calls even when it would have been possible in hindsight to reject only very few.

In this paper, we instead consider the goal of approximately minimizing the number of calls rejected. This is much more appropriate for settings in which rejections are intended to be rare events. In order to avoid trivial lower-bounds, we assume preemption is allowed and that calls are given to the algorithm as fixed paths. We show that in a number of cases, we can in fact achieve a competitive ratio of 2 for rejections (so if the optimal in hindsight rejects 0, then we reject 0; if the optimal rejects r, then we reject at most 2r). For other cases, we get worse, but nontrivial, bounds. For the most general case of fixed paths in arbitrary graphs with arbitrary edge capacities, we achieve matching $\Theta(\sqrt{m})$ upper and lower bounds, where m is the number of edges. We also show a connection between these problems and online versions of the vertex-cover and set-cover problems (our factor-2 results give 2-approximations to slight generalizations of the vertex cover problem, much as [Alon et al. 99] show hardness results for the benefit version based on the hardness of approximability of independent set).

We suppose we are given some fixed (but unknown) subset X of a set $\Omega = {\mathbb F}_2^n$, where ${\mathbb F}_2$ denotes the field of two elements. Our goal is to learn as much as possible about the elements of X by asking certain binary questions. Each "question" Q is just some element of $\Omega$, and the "answer" to Q is just the inner product $Q \cdot x \in {\mathbb F}_2$ for some $x \in X$. However, the choice of x is made by a truthful (but possibly malevolent) adversary A, whom we may assume is trying to choose answers so as to yield as little information as possible about X. In this note, we investigate several aspects of this problem. In particular, we are interested in extracting as much information as possible about X from A's answers. Although A can prevent us from learning the identity of any particular element of X, with appropriate questions we can still learn quite a bit about X. We determine the maximum amount of information that can be recovered under these assumptions and describe explicit sets of questions for achieving this goal. For the case that $|X|=2$, we give an $O(n^3)$ algorithm for recovering the desired information. On the other hand, when $|X| \geq 3$, we show that no polynomial-time algorithm can exist for producing a secret set consistent with the answers given, unless $P = NP$.

Several stochastic models were proposed recently to model the dynamic evolution of the web graph. We study the infinite limits of the stochastic processes proposed to model the web graph when time goes to infinity. We prove that deterministic variations of the so-called copying model can lead to several nonisomorphic limits. Some models converge to the infinite random graph R, while the convergence of other models is sensitive to initial conditions or minor changes in the rules of the model. We explain how limits of the copying model of the web graph share several properties with R that seem to reflect known properties of the web graph.

Recently many new "scale-free'' random graph models have been introduced, motivated by the power-law degree sequences observed in many large-scale real-world networks. The most studied of these is the Barabási-Albert growth with "preferential attachment'' model, made precise as the LCD model by the present authors. Here we use coupling techniques to show that in certain ways the LCD model is not too far from a standard random graph; in particular, the fractions of vertices that must be retained under an optimal attack in order to keep a giant component are within a constant factor for the scale-free and classical models.

Recently, I became interested in a current debate over whether file size distributions are best modelled by a power law distribution or a lognormal distribution. In trying to learn enough about these distributions to settle the question, I found a rich and long history, spanning many fields. Indeed, several recently proposed models from the computer science community have antecedents in work from decades ago. Here, I briefly survey some of this history, focusing on underlying generative models that lead to these distributions. One finding is that lognormal and power law distributions connect quite naturally, and hence, it is not surprising that lognormal distributions have arisen as a possible alternative to power law distributions across many fields.