Reducibility among equilibrium problems
We address the fundamental question of whether the Nash equilibria of a game can be computed in polynomial time. We describe certain efficient reductions between this problem for normal form games with a fixed number of players and graphical games with fixed degree. Our main result is that the problem of solving a game for any constant number of players, is reducible to solving a 4-player game.

title: The complexity of computing a Nash equilibrium
abstract: We resolve the question of the complexity of Nash equilibrium by showing that the problem of computing a Nash equilibrium in a game with 4 or more players is complete for the complexity class PPAD. Our proof uses ideas from the recently-established equivalence between polynomial time solvability of normal form games and graphical games, establishing that these kinds of games can simulate a PPAD-complete class of Brouwer functions.

title: A BGP-based mechanism for lowest-cost routing
abstract: The routing of traffic between Internet domains or Autonomous Systems (ASs), a task known as interdomain routing, is currently handled by the Border Gateway Protocol (BGP). In this paper, we address the problem of interdomain routing from a mechanism-design point of view. The application of mechanism-design principles to the study of routing is the subject of earlier work by Nisan and Ronen  and Hershberger and Suri . In this paper, we formulate and solve a version of the routing-mechanism design problem that is different from the previously studied version in three ways that make it more accurately reflective of real-world interdomain routing: (1) we treat the nodes as strategic agents, rather than the links; (2) our mechanism computes lowest-cost routes for all source-destination pairs and payments for transit nodes on all of the routes (rather than computing routes and payments for only one source-destination pair at a time, as is done in [14, 10]); (3) we show how to compute our mechanism with a distributed algorithm that is a straightforward extension to BGP and causes only modest increases in routing-table size and convergence time (in contrast with the centralized algorithms used in [14, 10]). This approach of using an existing protocol as a substrate for distributed computation may prove useful in future development of Internet algorithms generally, not only for routing or pricing problems. Our design and analysis of a strategyproof, BGP-based routing mechanism provides a new, promising direction in distributed algorithmic mechanism design, which has heretofore been focused mainly on multicast cost sharing.

title: Latent semantic indexing: a probabilistic analysis
abstract: Latent semantic indexing LSI is an information retrieval technique based on the spectral analysis of the term document matrix whose empirical success had heretofore been without rigorous prediction and explanation We prove that under certain conditions LSI does succeed in capturing the underlying semantics of the corpus and achieves improved retrieval performance We also propose the technique of random projection as a way of speeding up LSI We complement our theorems with encouraging experimental results We also argue that our results may be viewed in a more general framework as a theoretical basis for the use of spectral methods in a wider class of applications such as collaborative ltering Computer Science Division U C Berkeley Berkeley CA Email christos cs berkeley edu IBM Almaden Research Center Harry Road San Jose CA Email pragh almaden ibm com Computer Science Department Meiji University Tokyo Japan Email htamaki cs meiji ac jp Department of Mathematics M I T Cambridge MA Email vempala math mit edu

title: Inferring Tree Models for Oncogenesis from Comparative Genome Hybridization Data
abstract: Comparative genome hybridization (CGH) is a laboratory method to measure gains and losses of chromosomal regions in tumor cells. It is believed that DNA gains and losses in tumor cells do not occur entirely at random, but partly through some flow of causality. Models that relate tumor progression to the occurrence of DNA gains and losses could be very useful in hunting cancer genes and in cancer diagnosis. We lay some mathematical foundations for inferring a model of tumor progression from a CGH data set. We consider a class of tree models that are more general than a path model that has been developed for colorectal cancer. We derive a tree model inference algorithm based on the idea of a maximum-weight branching in a graph, and we show that under plausible assumptions our algorithm infers the correct tree. We have implemented our methods in software, and we illustrate with a CGH data set for renal cancer.

title: The Complexity of Games on Highly Regular Graphs
abstract: We present algorithms and complexity results for the problem of finding equilibria (mixed Nash equilibria, pure Nash equilibria and correlated equilibria) in games with extremely succinct description that are defined on highly regular graphs such as the d-dimensional grid; we argue that such games are of interest in the modelling of large systems of interacting agents. We show that mixed Nash equilibria can be found in time exponential in the succinct representation by quantifier elimination, while correlated equilibria can be found in polynomial time by taking advantage of the game’s symmetries. Finally, the complexity of determining whether such a game on the d-dimensional grid has a pure Nash equilibrium depends on d and the dichotomy is remarkably sharp: it is solvable in polynomial time (in fact NL-complete) when d = 1, but it is NEXP-complete for d ≥ 2.

title: On the Difficulty of Designing Good Classifiers
abstract: We consider the problem of designing a near-optimal linear decision tree to classify two given point sets B and W in $\Re^n$. A linear decision tree defines a polyhedral subdivision of space; it is a classifier if no leaf region contains points from both sets. We show hardness results for computing such a classifier with approximately optimal depth or size in polynomial time. In particular, we show that unless NP = ZPP, the depth of a classifier cannot be approximated within any constant factor, and that the total number of nodes cannot be approximated within any fixed polynomial. Our proof uses a simple connection between this problem and graph coloring and uses the result of Feige and Kilian on the inapproximability of the chromatic number. We also study the problem of designing a classifier with a single inequality that involves as few variables as possible and point out certain aspects of the difficulty of this problem.