Amin Saberi

Random walks in peer-to-peer networks

We quantify the effectiveness of random walks for searching and construction of unstructured peer-to-peer (P2P) networks. We have identified two cases where the use of random walks for searching achieves better results than flooding: a) when the overlay topology is clustered, and h) when a client re-issues the same query while its horizon does not change much. For construction, we argue that an expander can he maintained dynamically with constant operations per addition. The key technical ingredient of our approach is a deep result of stochastic processes indicating that samples taken from consecutive steps of a random walk can achieve statistical properties similar to independent sampling (if the second eigenvalue of the transition matrix is hounded away from 1, which translates to good expansion of the network; such connectivity is desired, and believed to hold, in every reasonable network and network model). This property has been previously used in complexity theory for construction of pseudorandom number generators. We reveal another facet of this theory and translate savings in random bits to savings in processing overhead.

AdWords and generalized online matching

How does a search engine company decide what ads to display with each query so as to maximize its revenue? This turns out to be a generalization of the online bipartite matching problem. We introduce the notion of a tradeoff revealing LP and use it to derive two optimal algorithms achieving competitive ratios of 1-1/e for this problem.

Greedy facility location algorithms analyzed using dual fitting with factor-revealing LP

In this article, we will formalize the method of dual fitting and the idea of factor-revealing LP. This combination is used to design and analyze two greedy algorithms for the metric uncapacitated facility location problem. Their approximation factors are 1.861 and 1.61, with running times of O(m log m) and O(n3), respectively, where n is the total number of vertices and m is the number of edges in the underlying complete bipartite graph between cities and facilities. The algorithms are used to improve recent results for several variants of the problem.

A new greedy approach for facility location problems

We present a simple and natural greedy algorithm for the metric uncapacitated facility location problem achieving an approximation guarantee of 1.61. We use this algorithm to find better approximation algorithms for the capacitated facility location problem with soft capacities and for a common generalization of the k-median and facility location problems. We also prove a lower bound of 1+2/e on the approximability of the k-median problem. At the end, we present a discussion about the techniques we have used in the analysis of our algorithm, including a computer-aided method for proving bounds on the approximation factor.

Hybrid search schemes for unstructured peer-to-peer networks

We study hybrid search schemes for unstructured peer-to-peer networks. We quantify performance in terms of number of hits, network overhead, and response time. Our schemes combine flooding and random walks, look ahead and replication. We consider both regular topologies and topologies with supernodes. We introduce a general search scheme, of which flooding and random walks are special instances, and show how to use locally maintained network information to improve the performance of searching. Our main findings are: (a) a small number of supernodes in an otherwise regular topology can offer sharp savings in the performance of search, both in the case of search by flooding and search by random walk, particularly when it is combined with 1-step replication. We quantify, analytically and experimentally, that the reason of these savings is that the search is biased towards nodes that yield more information. (b) There is a generalization of search, of which flooding and random walk are special instances, which may take further advantage of locally maintained network information, and yield better performance than both flooding and random walk in clustered topologies. The method determines edge critically and is reminiscent of fundamental heuristics from the area of approximation algorithms.

Minimizing Effective Resistance of a Graph

The effective resistance between two nodes of a weighted graph is the electrical resistance seen between the nodes of a resistor network with branch conductances given by the edge weights. The effective resistance comes up in many applications and fields in addition to electrical network analysis, including, for example, Markov chains and continuous-time averaging networks. In this paper we study the problem of allocating edge weights on a given graph in order to minimize the total effective resistance, i.e., the sum of the resistances between all pairs of nodes. We show that this is a convex optimization problem and can be solved efficiently either numerically or, in some cases, analytically. We show that optimal allocation of the edge weights can reduce the total effective resistance of the graph (compared to uniform weights) by a factor that grows unboundedly with the size of the graph. We show that among all graphs with

AdWords and generalized on-line matching

n

The spread of innovations in social networks

nodes, the path has the largest value of optimal total effective resistance and the complete graph has the least.

Multi-unit auctions with budget-constrained bidders

How does a search engine company decide what ads to display with each query so as to maximize its revenue? This turns out to be a generalization of the online bipartite matching problem. We introduce the notion of a tradeoff revealing LP and use it to derive two optimal algorithms achieving competitive ratios of 1-1/e for this problem.

On approximately fair allocations of indivisible goods

Which network structures favor the rapid spread of new ideas, behaviors, or technologies? This question has been studied extensively using epidemic models. Here we consider a complementary point of view and consider scenarios where the individuals’ behavior is the result of a strategic choice among competing alternatives. In particular, we study models that are based on the dynamics of coordination games. Classical results in game theory studying this model provide a simple condition for a new action or innovation to become widespread in the network. The present paper characterizes the rate of convergence as a function of the structure of the interaction network. The resulting predictions differ strongly from the ones provided by epidemic models. In particular, it appears that innovation spreads much more slowly on well-connected network structures dominated by long-range links than in low-dimensional ones dominated, for example, by geographic proximity.