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Bounded geometries, fractals, and low-distortion embeddings
This work considers both general doubling metrics, as well as more restricted families such as those arising from trees, from graphs excluding a fixed minor, and from snowflaked metrics, which contains many families of metrics that occur in applied settings.
Polylogarithmic inapproximability
It is shown that for every fixed ε>0, the GROUP-STEINER-TREE problem admits no efficient log2-ε k approximation, where k denotes the number of groups (or, alternatively, the input size), unless NP has quasi polynomial Las-Vegas algorithms.
Navigating nets: simple algorithms for proximity search
This work presents a simple deterministic data structure for maintaining a set S of points in a general metric space, while supporting proximity search and updates to S (insertions and deletions) and is essentially optimal in a certain model of distance computation.
Estimating the sortedness of a data stream
It is conjecture that any deterministic (1 + ε) approximation algorithm for LIS requires Ω (√n) space, and a lower bound of Ω(n) is proved for a restricted yet natural class of deterministic algorithms.
Approximating edit distance efficiently
Algorithms are developed that solve gap versions of the edit distance problem: given two strings of length n with the promise that their edit distance is either at most k or greater than /spl lscr/, decide which of the two holds and develop an n/sup 3/7/-approximation quasilinear time algorithm.
On the Hardness of Approximating Multicut and Sparsest-Cut
We show that the MULTICUT, SPARSEST-CUT, and MIN-2CNF/spl equiv/DELETION problems are NP-hard to approximate within every constant factor, assuming the unique games conjecture of Khot [STOC, 2002]. A
Partitioning graphs into balanced components
This work considers the k-balanced partitioning problem, where the goal is to partition the vertices of an input graph G into k equally sized components, while minimizing the total weight of the edges connecting different components, and presents a (bi-criteria) approximation algorithm achieving an approximation of O(log n log k), which matches or improves over previous algorithms for all relevant values of k.
Improved lower bounds for embeddings into L1
It is shown that for infinitely many values of n, there are metric spaces of negative type that require a distortion of Ω(log log <i>n</i>) for such an embedding, implying the same lower bound on the integrality gap of a well-known SDP relaxation for SPARSEST-CUT.
Sketching Cuts in Graphs and Hypergraphs
This work shows that every r-uniform hypergraph admits a (1+ ε)-cut-sparsifier (i.e., a weighted subhypergraph that approximately preserves all the cuts) with O(ε-2n(r+log n)) edges, and makes first steps towards sketching general CSPs (Constraint Satisfaction Problems).
On Sketching Quadratic Forms
The results provide the first separation between the sketch size needed for the "for all" and "for each" guarantees for Laplacian matrices.