Anastasios Sidiropoulos

Convergence and approximation in potential games

We study the speed of convergence to approximately optimal states in two classes of potential games. We provide bounds in terms of the number of rounds, where a round consists of a sequence of movements, with each player appearing at least once in each round. We model the sequential interaction between players by a best-response walk in the state graph, where every transition in the walk corresponds to a best response of a player. Our goal is to bound the social value of the states at the end of such walks. In this paper, we focus on two classes of potential games: selfish routing games, and cut games (or party affiliation games (Fabrikant et al. 2004 [12])). Other than bounding the price of anarchy of selfish routing games (Roughgarden and Tardos, 2002 [25], Awerbuch et al. 2005 [2], Christodoulou and Koutsoupias, 2005 [9]), there are many interesting problems about game dynamics in these games. It is known that exponentially long best-response walks may exist to pure Nash equilibria (Fabrikant et al. 2004 [12]), and random best-response walks converge to solutions with good approximation guarantees after polynomially many best responses (Goemans et al. 2005 [17]). In this paper, we study the speed of convergence on deterministic best-response walks in these games and prove that starting from an arbitrary configuration, after one round of best responses of players, the resulting configuration is a @Q(n)-approximate solution. Furthermore, we show that starting from an empty configuration, the solution after any round of best responses is a constant-factor approximation. We also provide a lower bound for the multi-round case, where we show that for any constant number of rounds t, the approximation guarantee cannot be better than n^@e^(^t^), for some @e(t)>0. We also study cut games, that provide an illustrative example of potential games. The convergence of potential games to locally optimum solutions has been studied in the context of local search algorithms (Johnson et al. 1988 [19], Schaffer and Yannakakis, 1991 [28]). In these games, we consider two social functions: the cut (defined as the weight of the edges in the cut), and the total happiness (defined as the weight of the edges in the cut, minus the weight of the remaining edges). For the cut social function, we prove that the expected social value after one round of a random best-response walk is at least a constant factor approximation to the optimal social value. We also exhibit exponentially long best-response walks with poor social value. For the unweighted version of this cut game, we prove @W(n) and O(n) lower and upper bounds on the number of rounds of best responses to converge to a constant-factor cut. In addition, we suggest a way to modify the game to enforce a fast convergence in any fair best-response walk. For the total happiness social function, we show that for unweighted graphs of sufficiently large girth, starting from a random configuration, greedy behavior of players in a random order converges to an approximate solution after one round.

On distributing symmetric streaming computations

A common approach for dealing with large data sets is to stream over the input in one pass, and perform computations using sublinear resources. For truly massive data sets, however, even making a single pass over the data is prohibitive. Therefore, streaming computations must be distribued over many machines. In practice, obtaining significant speedups using distributed computations has numerous challenges including synchronization, load balancing, overcoming processor failures, and data distribution. Successful Systems in practice such as Googles MapReduce and Apaches Hadoop address these problems by only allowing a certain class of highly distributable tasks defined by local computations that can be applied in any order to the input. The fundamental question that arises is: How does the class of computational tasks supported by these systems differ from the class for which streaming solutions exist? We introduce a simple algorithmic model for massive, unordered, distributed (mud) computation, as implemented by these systems. We show that in principle, mud algorithms are equivalent in power to symmetric streaming algorithms. More precisely, we show that any symmetric (order-invariant) function that can be computed by a steraming algorithm can also be computed by a mud algorithym, with comparable space and communication complexity. Our simulation uses Savitchs theorem and therefore has superpolynomial time complexity. We extend our simulation result to some natural classes of approximate and randomized steraming algorithms. We also give negative results, using communication complexity arguments to prove that extensions to private randomness, promise problems and indeterminate functions are impossible. We also introduce an extension of the mud model to multiple keys and multiple rounds.

Circular partitions with applications to visualization and embeddings

We introduce a hierarchical partitioning scheme of the Euclidean plane, called circular partitions. Such a partition consists of a hierarchy of convex polygons, each having small aspect ratio, and satisfying specified volume constraints. We apply these partitions to obtain a natural extension of the popular Treemap visualization method. Our proposed algorithm is not constrained in using only rectangles, and can achieve provably better guarantees on the aspect ratio of the constructed polygons. Under relaxed conditions, we can also construct circular partitions in higher-dimensional spaces. We use these relaxed partitions to obtain improved approximation algorithms for embedding ultrametrics into d-dimensional Euclidean space. In particular, we give a polylog(Delta)-approximation algorithm for embedding n-point ultrametrics into R^d with minimum distortion (Delta denotes the spread of the metric). The previously best-known approximation ratio for this problem was polynomial in n [Badoiu et al. SoCG 2006]. This is the first algorithm for embedding a non-trivial family of weighted graph metrics into a space of constant dimension that achieves polylogarithmic approximation ratio.

On the geometry of graphs with a forbidden minor

We study the topological simplification of graphs via random embeddings, leading ultimately to a reduction of the Gupta-Newman-Rabinovich-Sinclair (GNRS) L1 embedding conjecture to a pair of manifestly simpler conjectures. The GNRS conjecture characterizes all graphs that have an O(1)-approximate multi-commodity max-flow/min-cut theorem. In particular, its resolution would imply a constant factor approximation for the general Sparsest Cut problem in every family of graphs which forbids some minor. In the course of our study, we prove a number of results of independent interest.

Ordinal Embedding: Approximation Algorithms and Dimensionality Reduction

This paper studies how to optimally embed a general metric, represented by a graph, into a target space while preserving the relative magnitudes of most distances. More precisely, in an ordinal embedding, we must preserve the relative order between pairs of distances (which pairs are larger or smaller), and not necessarily the values of the distances themselves. The relaxation of an ordinal embedding is the maximum ratio between two distances whose relative order is inverted by the embedding. We develop polynomial-time constant-factor approximation algorithms for minimizing the relaxation in an embedding of an unweighted graph into a line metric and into a tree metric. These two basic target metrics are particularly important for representing a graph by a structure that is easy to understand, with applications to visualization, compression, clustering, and nearest-neighbor searching. Along the way, we improve the best known approximation factor for ordinally embedding unweighted trees into the line down to 2. Our results illustrate an important contrast to optimal-distortion metric embeddings, where the best approximation factor for unweighted graphs into the line is O(n1/2), and even for unweighted trees into the line the best is

Theory research at Google

\tilde O(n^{1/3})

How to walk your dog in the mountains with no magic leash

. We also show that Johnson-Lindenstrauss-type dimensionality reduction is possible with ordinal relaxation and i¾? 1 metrics (and i¾? p metrics with 1 ≤ p≤ 2), unlike metric embedding of i¾? 1 metrics.

Fat polygonal partitions with applications to visualization and embeddings

Through the history of Computer Science, new technologies have emerged and generated fundamental problems of interest to theoretical computer scientists. From the era of telecommunications to computing and now, the Internet and the web, there are many such examples. This article is derived from the emergence of web search and associated technologies, and focuses on the problems of research interest to theoretical computer scientists that arise, in particular at Google.

Optimal Stochastic Planarization

We describe a O(log n)-approximation algorithm for computing the homotopic Frechet distance between two polygonal curves that lie on the boundary of a triangulated topological disk. Prior to this work, algorithms where known only for curves on the Euclidean plane with polygonal obstacles. A key technical ingredient in our analysis is a O(log n)-approximation algorithm for computing the minimum height of a homotopy between two curves. No algorithms were previously known for approximating this parameter. Surprisingly, it is not even known if computing either the homotopic Frechet distance, or the minimum height of a homotopy, is in NP.

Inapproximability for metric embeddings into Rd

LetT be a rooted and weighted tree, where the weight of any node is equal to the sum of the weights of its children. The popular Treemap algorithm visualizes such a tree as a hierarchical partition of a square into rectangles, where the area of the rectangle corresponding to any node inT is equal to the weight of that node. The aspect ratio of the rectangles in such a rectangular partition necessarily depends on the weights and can become arbitrarily high. We introduce a new hierarchical partition scheme, called a polygonal partition, which uses convex polygons rather than just rectangles. We present two methods for constructing polygonal partitions, both having guarantees on the worst-case aspect ratio of the con- structed polygons; in particular, both methods guarantee a bound on the aspect ratio that is independent of the weights of the nodes. We also consider rectangular partitions with slack, where the areas of the rectangles may dier slightly from the weights of the corresponding nodes. We show that this makes it possible to obtain partitions with constant aspect ratio. This result generalizes to hyper- rectangular partitions in R d . We use these partitions with slack for embedding ultrametrics into d-dimensional Euclidean space: we give a polylog()-approx