D-Optimal Data Fusion: Exact and Approximation Algorithms

Abstract

We study the D-optimal Data Fusion (DDF) problem, which aims to select new data points, given an existing Fisher information matrix, so as to maximize the logarithm of the determinant of the overall Fisher information matrix. We show that the DDF problem is NP-hard and has no constant-factor polynomial-time approximation algorithm unless P = NP. Therefore, to solve the DDF problem effectively, we propose two convex integer-programming formulations and investigate their corresponding complementary and Lagrangian-dual problems. Leveraging the concavity of the objective functions in the two proposed convex integer-programming formulations, we design an exact algorithm, aimed at solving the DDF problem to optimality. We further derive a family of submodular valid inequalities and optimality cuts, which can significantly enhance the algorithm performance. We also develop scalable randomized-sampling and local-search algorithms with provable performance guarantees. Finally, we test our algorithms using real-world data on the new phasor-measurement-units placement problem for modern power grids, considering the existing conventional sensors. Our numerical study demonstrates the efficiency of our exact algorithm and the scalability and high-quality outputs of our approximation algorithms. History: Accepted by Andrea Lodi, Area Editor for Design & Analysis of Algorithms—Discrete. Funding: Y. Li and W. Xie were supported in part by Division of Civil, Mechanical and Manufacturing Innovation [Grant 2046414] and Division of Computing and Communication Foundations [Grant 2246417]. J. Lee was supported in part by Air Force Office of Scientific Research [Grants FA9550-19-1-0175 and FA9550-22-1-0172]. M. Fampa was supported in part by Conselho Nacional de Desenvolvimento Científico e Tecnológico [Grants 305444/2019-0 and 434683/2018-3]. F. Qiu and R. Yao were supported in part by the U.S. Department of Energy Advanced Grid Modeling Program under [Grant DE-OE0000875]. Supplemental Material: The e-companion is available at https://doi.org/10.1287/ijoc.2022.0235 .