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URN: urn:nbn:de:0030-drops-17537
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All-Norms and All-L_p-Norms Approximation Algorithms

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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.

BibTeX - Entry

@InProceedings{golovin_et_al:LIPIcs:2008:1753,
author =	{Daniel Golovin and Anupam Gupta and Amit Kumar and Kanat Tangwongsan},
title =	{{All-Norms and All-L_p-Norms Approximation Algorithms}},
booktitle =	{IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science},
pages =	{199--210},
series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN =	{978-3-939897-08-8},
ISSN =	{1868-8969},
year =	{2008},
volume =	{2},
editor =	{Ramesh Hariharan and Madhavan Mukund and V Vinay},
publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},