On the Complexity of Database Queries
We revisit the issue of the complexity of database queries, in the light of the recent parametric refinement of complexity theory. We show that, if the query size (or the number of variables in the query) is considered as a parameter, then the relational calculus and its fragments (conjunctive queries, positive queries) are classified at appropriate levels of the so-called W hierarchy of Downey and Fellows. These results strongly suggest that the query size is inherently in the exponent of the data complexity of any query evaluation algorithm, with the implication becoming stronger as the expressibility of the query language increases. On the positive side, we show that this exponential dependence can be avoided for the extension of acyclic queries with ? (but not <) inequalities.

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: On a network creation game
abstract: 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: Algorithms, Games, and the Internet
abstract: Over the past fifty years, researchers in Theoretical Computer Science have sought and achieved a productive foundational understanding of the von Neumann computer and its software, employing the mathematical tools of Logic and Combinatorics. The next half century appears now much more confusing (half- centuries tend to look like that in the beginning). What computational artifact will be the object of the next great modeling adventure of our field? And what mathematical tools will be handy in this endeavor?

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: 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: The complexity of recognizing polyhedral scenes
abstract: Given a drawing of straight lines on the plane, we wish to decide whether it is the projection of the visible part of a set of opaque polyhedra. Although there is an extensive literature and reports on empirically succesful algorithm: for this problem, there has been no definite result concerning its complexity. In this paper we show that, rather surprisingly, this problem is NP-complete. This is true even in the relatively simple case of trihedral scenes (no four planes share a point) without shadows or cracks. Despite this negative result, we present a fast algorithm for the important special case of orthohedral scenes (all planes are perpendicular to one of the three axes) with a fixed number of "possible" objects.

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: 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.

title: An approximation scheme for planar graph TSP
abstract: We consider the special case of the traveling salesman problem (TSP) in which the distance metric is the shortest-path metric of a planar unweighted graph. We present a polynomial-time approximation scheme (PTAS) for this problem.

title: Topological Queries