Optimization, approximation, and complexity classes
We define a natural variant of NP, MAX NP, and also a subclass called MAX SNP. These are classes of optimization problems, and in fact contain several natural, well-studied ones. We show that problems in these classes can be approximated with some bounded error. Furthermore, we show that a number of common optimization problems are complete under a kind of careful transformation (called L-reduction) that preserves approximability. It follows that such a complete problem has a polynomial-time approximation scheme iff the whole class does. These results may help explain the lack of progress on the approximability of a host of optimization problems.

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 Complexity of Database Queries
abstract: 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: 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: 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: 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: 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: Optimum Grip of a Polygon
abstract: It has been shown by Baker, Fortune and Grosse that any two-dimensional polygonal object can be prehended stably with three fingers, so that its weight (along the third dimension) is balanced. Besides, in this paper we show that form closure of a polygon object can be achieved by four fingers (previous proofs were not complete). We formulate and solve the problem of finding the optimum stable grip or form closure of any given polygon. For stable grip it is most natural to minimize the forces needed to balance through friction the object''s weight along the third dimension. For form closure, we minimize the worst-case forces needed to balance any unit force acting on the center of gravity of the object. The mathematical techniques used in the two instances are an interesting mix of Optimization and Euclidean geometry. Our results lead to algorithms for the efficient computation of the optimum grip in each case.

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.