The problem of finding the correct asymptotic rate of approximation by polynomial loops in dependence of the smoothness of the elements of a loop group seems not well-understood in general. For matrix Lie groups such as SU(N), it can be viewed as a problem of nonlinearly constrained trigonometric approximation. Motivated by applications to optical FIR filter design and control, we present some ...

We provide polynomial-time approximately optimal Bayesian mechanisms for makespan min-imization on unrelated machines as well as for max-min fair allocations of indivisible goods, withapproximation factors of 2 and min{m− k + 1, Õ(√k)} respectively, matching the approxima-tion ratios of best known polynomial-time algorithms (for max-min fairness, the latter claimis true ...

We consider a variant of the generalized assignment problem (GAP) where the amount of space used in each bin is restricted to be either zero (if the bin is not opened) or above a given lower bound (aminimum quantity). We provide several complexity results for different versions of the problem and give polynomial time exact algorithms and approximation algorithms for restricted cases. For the mo...

We give a deterministic polynomial-time 2O(r)-approximation algorithm for the number of bases given matroid rank r and common any two matroids r. To best our knowledge, this is first nontrivial approximation that works arbitrary matroids. Based on lower bound Azar, Broder, Frieze, almost possible result assuming oracle access to independent sets matroid. There are main ingredients in result. Fo...

We prove that some multivariate linear tensor product problems are tractable in the worst case setting if they are defined as tensor products of univariate problems with logarithmically increasing smoothness. This is demonstrated for the approximation problem defined over Korobov spaces and for the approximation problem of certain diagonal operators. For these two problems we show necessary and...

We consider the problem of maximizing a monotone submodular function under a knapsack constraint. We show that, for any fixed ǫ > 0, there exists a polynomial-time algorithm with an approximation ratio 1 − c/e − ǫ, where c ∈ [0, 1] is the (total) curvature of the input function. This approximation ratio is tight up to ǫ for any c ∈ [0, 1]. To the best of our knowledge, this is the first result ...

Low rank approximation is a fundamental computational primitive widely used in data analysis. In many applications the dataset that the algorithm operates on may contain sensitive information about contributing individuals (e.g. user/movie ratings in the Netflix challenge), motivating the need to design low rank approximation algorithms that preserve privacy of individual entries of the input m...