Combinatorial Algorithms for Optimal Design
Combinatorial Algorithms for Optimal Design
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发表时间：
2019-06
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通讯作者：
V. Madan;Mohit Singh;U. Tantipongpipat;Weijun Xie
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作者：
V. Madan;Mohit Singh;U. Tantipongpipat;Weijun Xie
在最佳设计问题中，我们得到了一组线性实验v 1，vn∈Rd和k≥D，我们的目标是选择一个或多个大小k φ（（pi∈SV I V> i） - 1）当φ（m）=确定性（m）1 /d时，该问题被称为D-最佳设计问题，并且当φ（m）=痕迹（m），被称为a-ofimal设计问题。 1980年；米勒（Nguyen），1994年，阿特金森（Atkinson）等人，2007年。我们弥合了此差距，并证明了D-最佳设计的本地搜索算法和A-Aftimal设计问题的近似值。当KD较大时，D-最佳设计和A-最佳设计问题的贪婪算法保证。
In an optimal design problem, we are given a set of linear experiments v 1 , . . . , v n ∈ R d and k ≥ d , and our goal is to select a set or a multiset S ⊆ [ n ] of size k such that Φ(( P i ∈ S v i v > i ) − 1 ) is minimized. When Φ( M ) = Determinant ( M ) 1 /d , the problem is known as the D-optimal design problem, and when Φ( 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 kd 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 kd is large.