Combinatorial Algorithms for Optimal Design.

In an optimal design problem, we are given a set of linear experiments $v_1,…,v_n\in \mathbb{R}^d$ and $k \geq d$, and our goal is to select a set or a multiset $S \subseteq [n]$ of size $k$ such that $\Phi((\sum_{i \in S} v_i v_i^\top )^{-1})$ is minimized. When $\Phi(M) = Determinant(M)^{1/d}$, the problem is known as the D-optimal design problem, and when $\Phi(M) = Trace(M)$, it is known as the A-optimal design problem. One of the most common heuristics used in practice to solve these problems is the local search heuristic, also known as the Fedorov’s exchange method (Fedorov, 1972). This is due to its simplicity and its empirical performance (Cook and Nachtrheim, 1980; Miller and Nguyen, 1994; Atkinson et al., 2007). However, despite its wide usage no theoretical bound has been proven for this algorithm. In this paper, we bridge this gap and prove approximation guarantees for the local search algorithms for D-optimal design and A-optimal design problems. We show that the local search algorithms are asymptotically optimal when $\frac{k}{d}$ is large. In addition to this, we also prove similar approximation guarantees for the greedy algorithms for D-optimal design and A-optimal design problems when $\frac{k}{d}$ is large.