Eigenvector Synchronization, Graph Rigidity and the Molecule Problem

The graph realization problem has received a great deal of attention in recent years, due to its importance in applications such as wireless sensor networks and structural biology. In this paper, we extend on previous work and propose the 3D-ASAP algorithm, for the graph realization problem in $\mathbb{R}^3$, given a sparse and noisy set of distance measurements. 3D-ASAP is a divide and conquer, non-incremental and non-iterative algorithm, which integrates local distance information into a global structure determination. Our approach starts with identifying, for every node, a subgraph of its 1-hop neighborhood graph, which can be accurately embedded in its own coordinate system. In the noise-free case, the computed coordinates of the sensors in each patch must agree with their global positioning up to some unknown rigid motion, that is, up to translation, rotation and possibly reflection. In other words, to every patch there corresponds an element of the Euclidean group Euc(3) of rigid transformations in $\mathbb{R}^3$, and the goal is to estimate the group elements that will properly align all the patches in a globally consistent way. Furthermore, 3D-ASAP successfully incorporates information specific to the molecule problem in structural biology, in particular information on known substructures and their orientation. In addition, we also propose 3D-SP-ASAP, a faster version of 3D-ASAP, which uses a spectral partitioning algorithm as a preprocessing step for dividing the initial graph into smaller subgraphs. Our extensive numerical simulations show that 3D-ASAP and 3D-SP-ASAP are very robust to high levels of noise in the measured distances and to sparse connectivity in the measurement graph, and compare favorably to similar state-of-the art localization algorithms.Comment: 49 pages, 8 figure

DisC Diversity: Result Diversification based on Dissimilarity and Coverage

Recently, result diversification has attracted a lot of attention as a means to improve the quality of results retrieved by user queries. In this paper, we propose a new, intuitive definition of diversity called DisC diversity. A DisC diverse subset of a query result contains objects such that each object in the result is represented by a similar object in the diverse subset and the objects in the diverse subset are dissimilar to each other. We show that locating a minimum DisC diverse subset is an NP-hard problem and provide heuristics for its approximation. We also propose adapting DisC diverse subsets to a different degree of diversification. We call this operation zooming. We present efficient implementations of our algorithms based on the M-tree, a spatial index structure, and experimentally evaluate their performance.Comment: To appear at the 39th International Conference on Very Large Data Bases (VLDB), August 26-31, 2013, Riva del Garda, Trento, Ital

Burning a Graph is Hard

Graph burning is a model for the spread of social contagion. The burning number is a graph parameter associated with graph burning that measures the speed of the spread of contagion in a graph; the lower the burning number, the faster the contagion spreads. We prove that the corresponding graph decision problem is \textbf{NP}-complete when restricted to acyclic graphs with maximum degree three, spider graphs and path-forests. We provide polynomial time algorithms for finding the burning number of spider graphs and path-forests if the number of arms and components, respectively, are fixed.Comment: 20 Pages, 4 figures, presented at GRASTA-MAC 2015 (October 19-23rd, 2015, Montr\'eal, Canada

Clustering with diversity

We consider the {\em clustering with diversity} problem: given a set of colored points in a metric space, partition them into clusters such that each cluster has at least $\ell$ points, all of which have distinct colors. We give a 2-approximation to this problem for any $\ell$ when the objective is to minimize the maximum radius of any cluster. We show that the approximation ratio is optimal unless $\mathbf{P=NP}$, by providing a matching lower bound. Several extensions to our algorithm have also been developed for handling outliers. This problem is mainly motivated by applications in privacy-preserving data publication.Comment: Extended abstract accepted in ICALP 2010. Keywords: Approximation algorithm, k-center, k-anonymity, l-diversit

Wireless Scheduling with Power Control

We consider the scheduling of arbitrary wireless links in the physical model of interference to minimize the time for satisfying all requests. We study here the combined problem of scheduling and power control, where we seek both an assignment of power settings and a partition of the links so that each set satisfies the signal-to-interference-plus-noise (SINR) constraints. We give an algorithm that attains an approximation ratio of $O(\log n \cdot \log\log \Delta)$, where $n$ is the number of links and $\Delta$ is the ratio between the longest and the shortest link length. Under the natural assumption that lengths are represented in binary, this gives the first approximation ratio that is polylogarithmic in the size of the input. The algorithm has the desirable property of using an oblivious power assignment, where the power assigned to a sender depends only on the length of the link. We give evidence that this dependence on $\Delta$ is unavoidable, showing that any reasonably-behaving oblivious power assignment results in a $\Omega(\log\log \Delta)$-approximation. These results hold also for the (weighted) capacity problem of finding a maximum (weighted) subset of links that can be scheduled in a single time slot. In addition, we obtain improved approximation for a bidirectional variant of the scheduling problem, give partial answers to questions about the utility of graphs for modeling physical interference, and generalize the setting from the standard 2-dimensional Euclidean plane to doubling metrics. Finally, we explore the utility of graph models in capturing wireless interference.Comment: Revised full versio

Dynamic Clustering to Minimize the Sum of Radii

In this paper, we study the problem of opening centers to cluster a set of clients in a metric space so as to minimize the sum of the costs of the centers and of the cluster radii, in a dynamic environment where clients arrive and depart, and the solution must be updated efficiently while remaining competitive with respect to the current optimal solution. We call this dynamic sum-of-radii clustering problem. We present a data structure that maintains a solution whose cost is within a constant factor of the cost of an optimal solution in metric spaces with bounded doubling dimension and whose worst-case update time is logarithmic in the parameters of the problem