All-Norms and All-L_p-Norms Approximation Algorithms
نویسندگان
چکیده
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 Lp-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 Lp norms. A natural problem in this framework is the Lp Set Cover problem, which generalizes SET COVER and MIN-SUM SET COVER. We show that the greedy algorithm 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 submodular set cover, and prove some results for the so-called pipelined set cover problem. We then go on to examine approximation algorithms in the “all-norms” and the “all-Lp-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.
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تاریخ انتشار 2008