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All-Norms and All-L_p-Norms Approximation Algorithms

Authors Daniel Golovin, Anupam Gupta, Amit Kumar, Kanat Tangwongsan



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Daniel Golovin
Anupam Gupta
Amit Kumar
Kanat Tangwongsan

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Daniel Golovin, Anupam Gupta, Amit Kumar, and Kanat Tangwongsan. All-Norms and All-L_p-Norms Approximation Algorithms. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 2, pp. 199-210, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2008)
https://doi.org/10.4230/LIPIcs.FSTTCS.2008.1753

Abstract

In many optimization problems, a solution can be viewed as ascribing a ``cost\'\' to each client, and the goal is to optimize some aggregation of the per-client costs. We often optimize some $L_p$-norm (or some other symmetric convex function or norm) of the vector of costs---though different applications may suggest different norms to use. Ideally, we could obtain a solution that optimizes several norms simultaneously. In this paper, we examine approximation algorithms that simultaneously perform well on all norms, or on all $L_p$ norms. A natural problem in this framework is the $L_p$ Set Cover problem, which generalizes \textsc{Set Cover} and \textsc{Min-Sum Set Cover}. We show that the greedy algorithm \emph{simultaneously gives a $(p + \ln p + O(1))$-approximation for all $p$, and show that this approximation ratio is optimal up to constants} under reasonable complexity-theoretic assumptions. We additionally show how to use our analysis techniques to give similar results for the more general \emph{submodular set cover}, and prove some results for the so-called \emph{pipelined set cover} problem. We then go on to examine approximation algorithms in the ``all-norms\'\' and the ``all-$L_p$-norms\'\' frameworks more broadly, and present algorithms and structural results for other problems such as $k$-facility-location, TSP, and average flow-time minimization, extending and unifying previously known results.
Keywords
  • Approximation algorithms
  • set-cover problems
  • combinatorial optimization
  • sampling minkowski norms

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