On a conjecture related to geometric routing
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: Optimization, approximation, and complexity classes
abstract: 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: The serializability of concurrent database updates
abstract: A sequence of interleaved user transactions in a database system may not be ser:ahzable, t e, equivalent to some sequential execution of the individual transactions Using a simple transaction model, it ~s shown that recognizing the transaction histories that are serlahzable is an NP-complete problem. Several efficiently recognizable subclasses of the class of senahzable histories are therefore introduced; most of these subclasses correspond to senahzabdity principles existing in the hterature and used in practice Two new principles that subsume all previously known ones are also proposed Necessary and sufficient conditions are given for a class of histories to be the output of an efficient history scheduler, these conditions imply that there can be no efficient scheduler that outputs all of senahzable histories, and also that all subclasses of senalizable histories studied above have an efficient scheduler Finally, it is shown how these results can be extended to far more general transaction models, to transactions with partly interpreted functions, and to distributed database systems

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: The 1-Steiner Tree Problem
abstract: We present an O ( n 2 ) algorithm for finding the point on the plane which, if added to a given set of n points, will result in the shortest possible spanning tree.

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 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: 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: 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

title: Topological queries in spatial databases
abstract: Handling spatial information is required by many database applications, and each poses different requirements on query languages. In many cases the precise size of the regions is important, while in other applications we may only be interested in the TOPOLOGICAL relations- hips between regions — intuitively, those that pertain to adjacency and connectivity properties of the regions, and are therefore invariant under homeomorphisms. Such differences in scope and emphasis are crucial, as they affect the data model, the query language, and performance. This talk focuses on queries targeted towards topological information for two- dimensional spatial databases, where regions are specified by polynomial inequalities with integer coeficients. We focus on two main aspects: (i) languages for expressing topological queries, and (ii) the representation of topological information. In regard to (i), we study several languages geared towards topological queries, building upon well-known topologi- cal relationships between pairs of planar regions proposed by Egenhofer. In regard to (ii), we show that the topological information in a spatial database can be precisely summarized by a finite relational database which can be viewed as a topological annotation to the raw spatial data. All topological queries can be answered using this annotation, called to- pological invariant. This yields a potentially more economical evaluation strategy for such queries, since the topological invariant is generally much smaller than the raw data. We examine in detail the problem of transla- ting topological queries against the spatial database into queries against the topological invariant. The languages considered are first-order on the spatial database side, and fixpoint and first-order on the topological in- variant side. In particular, it is shown that fixpoint expresses precisely the PTIME queries on topological invariants. This suggests that topolo- gical invariants are particularly well-behaved with respect to descriptive complexity. (Based on joint work with C.H.Papadimitriou, D. Suciu and L. Segoufin.