Approximation of a Maximum-Submodular-Coverage problem involving spectral functions, with application to Experimental Design

Please always quote using this URN: urn:nbn:de:0297-zib-14956
  • We study a family of combinatorial optimization problems defined by a parameter $p\in[0,1]$, which involves spectral functions applied to positive semidefinite matrices, and has some application in the theory of optimal experimental design. This family of problems tends to a generalization of the classical maximum coverage problem as $p$ goes to $0$, and to a trivial instance of the knapsack problem as $p$ goes to $1$. In this article, we establish a matrix inequality which shows that the objective function is submodular for all $p\in[0,1]$, from which it follows that the greedy approach, which has often been used for this problem, always gives a design within $1-1/e$ of the optimum. We next study the design found by rounding the solution of the continuous relaxed problem, an approach which has been applied by several authors. We prove an inequality which generalizes a classical result from the theory of optimal designs, and allows us to give a rounding procedure with an approximation factor which tends to $1$ as $p$ goes to $1$.

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Author:Guillaume Sagnol
Document Type:ZIB-Report
Volume:151
Issue:1--2
First Page:258
Last Page:276
Tag:Kiefer's p-criterion; Maximum Coverage; Optimal Experimental Design; Polynomial Time Approximability
MSC-Classification:90-XX OPERATIONS RESEARCH, MATHEMATICAL PROGRAMMING
Date of first Publication:2012/03/27
Series (Serial Number):ZIB-Report (11-53)
ISSN:1438-0064
Published in:Appeared in: Discrete Applied Mathematics
DOI:https://doi.org/10.1016/j.dam.2012.07.016