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  1. Semi-definite programming is a powerful tool in the design and analysis of approximation algorithms for combinatorial optimization problems. In particular, the random hyperplane rounding method of Goemans and Williamson [23] has been extensively studied for more than two decades, resulting in various extensions to the original technique and beautiful algorithms for a wide range of applications. Despite the fact that this approach yields tight approximation guarantees for some problems, e.g., Max-Cut, for many others, e.g., Max-SAT and Max-DiCut, the tight approximation ratio is still unknown. One of the main reasons for this is the fact that very few techniques for rounding semi-definite relaxations are known. In this work, we present a new general and simple method for rounding semi-definite programs, based on Brownian motion. Our approach is inspired by recent results in algorithmic discrepancy theory. We develop and present tools for analyzing our new rounding algorithms, utilizing mathematical machinery from the theory of Brownian motion, complex analysis, and partial differential equations. Focusing on constraint satisfaction problems, we apply our method to several classical problems, including Max-Cut, Max-2SAT, and Max-DiCut, and derive new algorithms that are competitive with the best known results. To illustrate the versatility and general applicability of ourmore »approach, we give new approximation algorithms for the Max-Cut problem with side constraints that crucially utilizes measure concentration results for the Sticky Brownian Motion, a feature missing from hyperplane rounding and its generalizations.« less
  2. We study the A-optimal design problem where we are given vectors υ1, …, υn ∊ ℝd, an integer k ≥ d, and the goal is to select a set S of k vectors that minimizes the trace of (∑i∊Svivi⊺)−1. Traditionally, the problem is an instance of optimal design of experiments in statistics [35] where each vector corresponds to a linear measurement of an unknown vector and the goal is to pick k of them that minimize the average variance of the error in the maximum likelihood estimate of the vector being measured. The problem also finds applications in sensor placement in wireless networks [22], sparse least squares regression [8], feature selection for k-means clustering [9], and matrix approximation [13, 14, 5]. In this paper, we introduce proportional volume sampling to obtain improved approximation algorithms for A-optimal design. Given a matrix, proportional volume sampling involves picking a set of columns S of size k with probability proportional to µ(S) times det(∑i∊Svivi⊺) for some measure µ. Our main result is to show the approximability of the A-optimal design problem can be reduced to approximate independence properties of the measure µ. We appeal to hardcore distributions as candidate distributions µ that allow usmore »to obtain improved approximation algorithms for the A-optimal design. Our results include a d-approximation when k = d, an (1 + ∊)-approximation when and -approximation when repetitions of vectors are allowed in the solution. We also consider generalization of the problem for k ≤ d and obtain a k-approximation. We also show that the proportional volume sampling algorithm gives approximation algorithms for other optimal design objectives (such as D-optimal design [36] and generalized ratio objective [27]) matching or improving previous best known results. Interestingly, we show that a similar guarantee cannot be obtained for the E-optimal design problem. We also show that the A-optimal design problem is NP-hard to approximate within a fixed constant when k = d.« less