NDLI: A Faster, Better Approximation Algorithm for the Minimum Latency Problem

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Williamson, David P. Archer, Aaron Levin, Asaf
Copyright Year	2008
Abstract	We give a 7.18-approximation algorithm for the minimum latency problem that uses only $O(n \log n)$ calls to the prize-collecting Steiner tree (PCST) subroutine of Goemans and Williamson. This improves the previous best algorithms in both performance guarantee and running time. A previous algorithm of Goemans and Kleinberg for the minimum latency problem requires an approximation algorithm for the k-minimum spanning tree (k-MST) problem which is called as a black box for each value of k. Their algorithm can achieve an approximation factor of 10.77 while making $O(n (n+\log C) \log n)$ PCST calls, a factor of 8.98 using $O(n^3(n+\log C) \log n)$ PCST calls, or a factor of $7.18+\epsilon$ using $n^{O(1/\epsilon)}\log C$ PCST calls, via the k-MST algorithms of Garg, Arya and Ramesh, and Arora and Karakostas, respectively. Here n denotes the number of nodes in the instance, and C is the largest edge cost in the input. In all cases, the running time is dominated by the PCST calls. Since the PCST subroutine can be implemented to run in $O(n^2)$ time, the overall running time of our algorithm is $O(n^3 \log n)$. We also give a faster randomized version of our algorithm that achieves the same approximation guarantee in expectation, but uses only $O(\log^2 n)$ PCST calls, and derandomize it to obtain a deterministic algorithm with factor $7.18+\epsilon$, using $O(\frac{1}{\epsilon} \log^2 n)$ PCST calls. The basic idea for our improvement is that we do not treat the k-MST algorithm as a black box. This allows us to take advantage of some special situations in which the PCST subroutine delivers a 2-approximate k-MST. We are able to obtain the same approximation ratio that would be given by Goemans and Kleinberg if we had access to 2-approximate k-MSTs for all values of k, even though we have them only for some values of k that we are not able to specify in advance. We also extend our algorithm to a weighted version of the minimum latency problem.
Starting Page	1472
Ending Page	1498
Page Count	27
File Format	PDF
ISSN	00975397
DOI	10.1137/07068151X
e-ISSN	10957111
Journal	SIAM Journal on Computing (SMJCAT)
Issue Number	5
Volume Number	37
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2008-01-30
Access Restriction	Subscribed
Subject Keyword	minimum latency problem Lagrangian relaxation Approximation methods and heuristics approximation algorithm Approximation algorithms Combinatorial optimization traveling repairman prize-collecting Steiner tree
Content Type	Text
Resource Type	Article
Subject	Mathematics Computer Science

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