On a network creation game
We introduce a novel game that models the creation of Internet-like networks by selfish node-agents without central design or coordination. Nodes pay for the links that they establish, and benefit from short paths to all destinations. We study the Nash equilibria of this game, and prove results suggesting that the "price of anarchy"  in this context (the relative cost of the lack of coordination) may be modest. Several interesting: extensions are suggested.

title: On certain connectivity properties of the Internet topology
abstract: We show that random graphs in the preferential connectivity model have constant conductance, and hence have worst-case routing congestion that scales logarithmically with the number of nodes. Another immediate implication is constant spectral gap between the first and second eigenvalues of the random walk matrix associated with these graphs. We also show that the expected frugality (overpayment in the Vickrey-Clarke-Groves mechanism for shortest paths) of a random graph is bounded by a small constant.

title: On the Difficulty of Designing Good Classifiers
abstract: It is a very interesting and well-studied problem, given two point sets W, B\(\subseteq\)ℜn, to design a linear decision tree that classifies them —that is, no leaf subdivision contains points from both B and W — and is as simple as possible, either in terms of the total number of nodes, or in terms of its depth. We show that, unless ZPP=NP, the depth of a classifier cannot be approximated within a factor smaller than 6/5, and that the total number of nodes cannot be approximated within a factor smaller than n1/5. Our proof relies on a simple connection between this problem and graph coloring, and uses recent nonapproximability results for graph coloring. 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.

title: Worst-case Equilibria
abstract: In a system in which noncooperative agents share a common resource, we propose the ratio between the worst possible Nash equilibrium and the social optimum as a measure of the effectiveness of the system. Deriving upper and lower bounds for this ratio in a model in which several agents share a very simple network leads to some interesting mathematics, results, and open problems.

title: Global Synchronization in Sensornets
abstract: Time synchronization is necessary in many distributed systems, but achieving synchronization in sensornets, which combine stringent precision requirements with severe resource constraints, is particularly challenging. This challenge has been met by the recent Reference-Broadcast Synchronization (RBS) proposal, which provides on-demand pairwise synchronization with low overhead and high precision. In this paper we introduce a model of the basic RBS synchronization paradigm. Within the context of this model we characterize the optimally precise clock synchronization algorithm and establish its global consistency. In the course of this analysis we point out unexpected connections between optimal clock synchronization, random walks, and resistive networks, and present a polynomial-time approximation scheme for the problem of calculating the effective resistance in a network based on min-cost flow. We also sketch a polynomial-time algorithm for finding a schedule of data acquisition giving the optimal trade-off between energy consumption and precision of clock synchronization. We also discuss synchronization in the presence of clock skews. In ongoing work we are adapting our synchronization algorithm for execution in a network of seismic sensors that requires global clock consistency.

title: On a conjecture related to geometric routing
abstract: We conjecture that any planar 3-connected graph can be embedded in the plane in such a way that for any nodes s and t, there is a path from s to t such that the Euclidean distance to t decreases monotonically along the path. A consequence of this conjecture would be that in any ad hoc network containing such a graph as a spanning subgraph, two-dimensional virtual coordinates for the nodes can be found for which the method of purely greedy geographic routing is guaranteed to work. We discuss this conjecture and its equivalent forms show that its hypothesis is as weak as possible, and show a result delimiting the applicability of our approach: any 3-connected K3,3-free graph has a planar 3-connected spanning subgraph. We also present two alternative versions of greedy routing on virtual coordinates that provably work. Using Steinitz's theorem we show that any 3-connected planar graph can be embedded in three dimensions so that greedy routing works, albeit with a modified notion of distance; we present experimental evidence that this scheme can be implemented effectively in practice. We also present a simple but provably robust version of greedy routing that works for any graph with a 3-connected planar spanning subgraph.

title: Topological Queries in Spatial Databases
abstract: We studytopological queriesover two-dimensional spatial databases. First, we show that the topological properties of semi-algebraic spatial regions can be completely specified using a classical finite structure, essentially the embedded planar graph of the region boundaries. This provides aninvariantcharacterizing semi-algebraic regions up to homeomorphism. All topological queries on semi-algebraic regions can be answered by queries on the invariant whose complexity is polynomially related to the original. Also, we show that for the purpose of answering topological queries, semi-algebraic regions can always be represented simply as polygonal regions. We then study query languages for topological properties of two-dimensional spatial databases, starting from the topological relationships between pairs of planar regions introduced by Egenhofer. We show that the closure of these relationships under appropriate logical operators yields languages which arecompletefor topological properties. This provides a theoreticala posteriorijustification for the choice of these particular relationships. Unlike the point-based languages studied in previous work on constraint databases, our languages are region based?quantifiers range over regions in the plane. This yields a family of languages, whose complexity ranges fromNCto undecidable. Another type of completeness result shows that the region-based language of complexityNCexpresses precisely the same topological properties as well-known point-based languages.

title: Segmentation Problems

title: Topological Queries

title: On the approximability of trade-offs and optimal access of Web sources
abstract: We study problems in multiobjective optimization, in which solutions to a combinatorial optimization problem are evaluated with respect to several cost criteria, and we are interested in the trade-off between these objectives (the so-called Pareto curve). We point out that, under very general conditions, there is a polynomially succinct curve that /spl epsiv/-approximates the Pareto curve, for any /spl epsiv/>0. We give a necessary and sufficient condition under which this approximate Pareto curve can be constructed in time polynomial in the size of the instance and 1//spl epsiv/. In the case of multiple linear objectives, we distinguish between two cases: when the underlying feasible region is convex, then we show that approximating the multi-objective problem is equivalent to approximating the single-objective problem. If however the feasible region is discrete, then we point out that the question reduces to an old and recurrent one: how does the complexity of a combinatorial optimization problem change when its feasible region is intersected with a hyperplane with small coefficients; we report some interesting new findings in this domain. Finally, we apply these concepts and techniques to formulate and solve approximately a cost-time-quality trade-off for optimizing access to the World-Wide Web, in a model first studied by Etzioni et al. (1996) (which was actually the original motivation for this work).

title: Reducibility among equilibrium problems
abstract: 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.