Tomás Feder

The benefits of relaxing punctuality

Abstract : The most natural, compositional way of modeling real time systems uses a dense domain for time. The satisfiability of real time constraints that are capable of expressing punctual it in this model is, however, known to be undecidable. The authors introduce a temporal language that can constrain the time difference between events only with finite (yet arbitrary) precision and show the resulting logic to be EXPACE-complete. This result allows the authors to develop an algorithm for the verification of timing properties of real time systems with a dense semantics.

Incremental Clustering and Dynamic Information Retrieval

Motivated by applications such as document and image classification in information retrieval, we consider the problem of clustering dynamic point sets in a metric space. We propose a model called incremental clustering which is based on a careful analysis of the requirements of the information retrieval application, and which should also be useful in other applications. The goal is to efficiently maintain clusters of small diameter as new points are inserted. We analyze several natural greedy algorithms and demonstrate that they perform poorly. We propose new deterministic and randomized incremental clustering algorithms which have a provably good performance, and which we believe should also perform well in practice. We complement our positive results with lower bounds on the performance of incremental algorithms. Finally, we consider the dual clustering problem where the clusters are of fixed diameter, and the goal is to minimize the number of clusters.

Optimal algorithms for approximate clustering

In a clustering problem, the aim is to partition a given set of <italic>n</italic> points in <italic>d</italic>-dimensional space into <italic>k</italic> groups, called clusters, so that points within each cluster are near each other. Two objective functions frequently used to measure the performance of a clustering algorithm are, for any <italic>L<subscrpt>4</subscrpt></italic> metric, (a) the maximum distance between pairs of points in the same cluster, and (b) the maximum distance between points in each cluster and a chosen cluster center; we refer to either measure as the cluster size. We show that one cannot approximate the optimal cluster size for a fixed number of clusters within a factor close to 2 in polynomial time, for two or more dimensions, unless P=NP. We also present an algorithm that achieves this factor of 2 in time <italic>&Ogr;</italic>(<italic>n</italic> log <italic>k</italic>), and show that this running time is optimal in the algebraic decision tree model. For a fixed cluster size, on the other hand, we give a polynomial time approximation scheme that estimates the optimal number of clusters under the second measure of cluster size within factors arbitrarily close to 1. Our approach is extended to provide approximation algorithms for the restricted centers, suppliers, and weighted suppliers problems that run in optimal <italic>&Ogr;</italic>(<italic>n</italic> log <italic>k</italic>) time and achieve optimal or nearly optimal approximation bounds.

Incremental clustering and dynamic information retrieval

Motivated by applications such as document and image classification in information retrieval, we consider the problem of clustering dynamic point sets in a metric space. We propose a model called incremental clustering which is based on a careful analysis of the requirements of the information retrieval application, and which should also be useful in other applications. The goal is to efficiently maintain clusters of small diameter as new points are inserted. We analyze several natural greedy algorithms and demonstrate that they perform poorly. We propose new deterministic and randomized incremental clustering algorithms which have a provably good performance, and which we believe should also perform well in practice. We complement our positive results with lower bounds on the performance of incremental algorithms. Finally, we consider the dual clustering problem where the clusters are of fixed diameter, and the goal is to minimize the number of clusters.

Anonymizing tables

We consider the problem of releasing tables from a relational database containing personal records, while ensuring individual privacy and maintaining data integrity to the extent possible. One of the techniques proposed in the literature is k-anonymization. A release is considered k-anonymous if the information for each person contained in the release cannot be distinguished from at least k–1 other persons whose information also appears in the release. In the k-Anonymityproblem the objective is to minimally suppress cells in the table so as to ensure that the released version is k-anonymous. We show that the k-Anonymity problem is NP-hard even when the attribute values are ternary. On the positive side, we provide an O(k)-approximation algorithm for the problem. This improves upon the previous best-known O(klog k)-approximation. We also give improved positive results for the interesting cases with specific values of k — in particular, we give a 1.5-approximation algorithm for the special case of 2-Anonymity, and a 2-approximation algorithm for 3-Anonymity.

List Homomorphisms and Circular Arc Graphs

G, H, and lists , a list homomorphism of G to Hwith respect to the listsL is a mapping , such that for all , and for all . The list homomorphism problem for a fixed graph H asks whether or not an input graph G together with lists , , admits a list homomorphism with respect to L. We have introduced the list homomorphism problem in an earlier paper, and proved there that for reflexive graphs H (that is, for graphs H in which every vertex has a loop), the problem is polynomial time solvable if H is an interval graph, and is NP-complete otherwise. Here we consider graphs H without loops, and find that the problem is closely related to circular arc graphs. We show that the list homomorphism problem is polynomial time solvable if the complement of H is a circular arc graph of clique covering number two, and is NP-complete otherwise. For the purposes of the proof we give a new characterization of circular arc graphs of clique covering number two, by the absence of a structure analogous to Gallais asteroids. Both results point to a surprising similarity between interval graphs and the complements of circular arc graphs of clique covering number two.

Clique Partitions, Graph Compression and Speeding-Up Algorithms

We first consider the problem of partitioning the edges of a graph G into bipartite cliques such the total order of the cliques is minimized, where the order of a clique is the number of vertices in it. It is shown that the problem is NP-complete. We then prove the existence of a partition of small total order in a sufficiently dense graph and devise an efficient algorithm to compute such a partition and the running time. Next, we define the notion of a compression of a graph G and use the result on graph partitioning to efficiently compute an optimal compression for graphs of a given size. An interesting application of the graph compression result arises from the fact that several graph algorithms can be adapted to work with the compressed representation of the input graph, thereby improving the bound on their running times, particularly on dense graphs. This makes use of the trade-off result we obtain from our partitioning algorithm. The algorithms analyzed include those for matchings, vertex connectivity, edge connectivity, and shortest paths. In each case, we improve upon the running times of the best-known algorithms for these problems.

Balanced matroids

Dalancea

List Homomorphisms to Reflexive Graphs

LetHbe a fixed graph. We introduce the following list homomorphism problem: Given an input graphGand for each vertexvofGa “list”L(v)?V(H), decide whether or not there is a homomorphismf:G?Hsuch thatf(v)?L(v) for eachv?V(G). We discuss this problem primarily in the context of reflexive graphs, i.e., graphs in which each vertex has a loop. We give a polynomial time algorithm to solve the problem whenHis an interval graph and prove that whenHis not an interval graph the problem isNP-complete. If the lists are restricted to induce connected subgraphs ofH, we give a polynomial time algorithm whenHis a chordal graph and prove that whenHis not chordal the problem is againNP-complete. We also argue that the complexity of certain other modifications of the problem (including the retract problem) are likely to be difficult to classify. Finally, we mention some newer results on irreflexive and general graphs.

Clique partitions, graph compression and speeding-up algorithms

We first consider the problem of partitioning the edges of a graph ~ into bipartite cliques such that the total order of the cliques is minimized, where the order of a clique is the number of vertices in it. It is shown that the problem is NP-complete. We then prove the existence of a partition of small total order in a sufficiently dense graph and devise an efilcient algorithm to compute such a partition. It turns out that our algorithm exhibits a trade-off between the total order of the partition and the running time. Next, we define the notion of a compression of a graph ~ and use the result on graph partitioning to efficiently compute an optimal compression for graphs of a given size. An interesting application of the graph compression result arises from the fact that several graph algorithms can be adapted to work with the compressed rep~esentation of the input graph, thereby improving the bound on their running times particularly on dense graphs. This makes use of the trade-off result we obtain from our partitioning algorithm. The algorithms analyzed include those for matchings, vertex connectivity, edge connectivity and shortest paths. In each case, we improve upon the running times of the best-known algorithms for these problems.