A Spectral Approach to Network Design and Experimental Design

Abstract

Over the last decade, the spectral sparsification technique has become a powerful tool in designing fast graph algorithms for various problems with numerous applications. In this thesis, we extend this spectral approach, and show that it is also very powerful in designing approximation algorithms for classical network design and experimental design problems.

The central piece in this thesis is a problem called spectral rounding, which is inspired by spectral sparsification and studied in an earlier work on experimental design. In this problem, we are given vectors $\vv_1, \ldots, \vv_m$ each with a non-negative cost, and a fractional solution $\vx \in [0,1]^m$. The task is to find an integral solution $\vz \in \{0,1\}^m$ such that the spectrum of the integral solution is similar to the one of the fractional solution, i.e.~$\sum_i \vz(i) \cdot \vv_i \vv_i^\top \approx \sum_i \vx(i) \cdot \vv_i \vv_i^\top$, and the integral cost is approximately equal to the fractional cost.

We observe that the spectral rounding problem underlies a large family of network design and experimental design problems. With this perspective, we bring new insights into these well-studied problems. For network design, we show that the spectral rounding technique provides a novel and general approach to significantly extend the scope of problems that can be solved efficiently. For experimental design, we show that the spectral rounding technique provides a unified and elegant framework that matches and improves all known existing algorithmic results.

There are two key techniques that we will use in this thesis. The first one is regret minimization, which is well-known to the online optimization community and has been used for spectral sparsification. We use it to control the spectrum of the integral solution in the spectral rounding problem. The second key technique is concentration inequalities for analyzing adaptive random sampling processes, which enable us to satisfy spectral and linear constraints simultaneously with high probability.