What Would Edmonds Do? Augmenting Paths and Witnesses for DegreeBounded MSTs
 Kamalika Chaudhuri,
 Satish Rao,
 Samantha Riesenfeld,
 Kunal Talwar
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Abstract
Given a graph and degree upper bounds on vertices, the BDMST problem requires us to find a minimum cost spanning tree respecting the given degree bounds. This problem generalizes the Travelling Salesman Path Problem (TSPP), even in unweighted graphs, and so we expect that it is necessary to relax the degree constraints to get efficient algorithms. Könemann and Ravi (Proceedings of the Thirty Second Annual ACM Symposium on Theory of Computing, pp. 537–546, 2000; Proceedings of the ThirtyFifth ACM Symposium on Theory of Computing, pp. 389–395, 2003) give bicriteria approximation algorithms for the problem using local search techniques of Fischer (Technical Report 14853, Cornell University, 1993). Their algorithms find solutions which make a tradeoff of the approximation factor for the cost of the resulting tree against the factor by which degree constraints are violated. In particular, they give an algorithm which, for a graph with a spanning tree of cost C and degree B, and for parameters b,w>1, produces a tree whose cost is at most wC and whose degree is at most $\frac{w}{w1}bB+\log_{b}n.$
A primary contribution of Könemann and Ravi is to use a Lagrangean relaxation to formally relate the BDMST problem to what we call the MDMST problem, which is the problem of finding an MST of minimum degree in a graph. In their solution to the MDMST problem, they make central use of a localsearch approximation algorithm of Fischer.
In this paper, we give the first approximation algorithms for the BDMST problem—both our algorithms find trees of optimal cost. We achieve this improvement using a primaldual cost bounding methodology from Edmonds’ weighted matching algorithms which was not previously used in this context. In order to follow Edmonds’ approach, we develop algorithms for a variant of the MDMST problem in which there are degree lower bound requirements. This variant may be of independent interest; in particular, our results extend to a generalized version of the BDMST problem in which both upper and lower degree bounds are given.
First we give a polynomialtime algorithm that finds a tree of optimal cost and with maximum degree at most $\frac{b}{2b}B+O(\log_{b}n)$ for any b∈(1,2). We also give a quasipolynomialtime approximation algorithm which produces a tree of optimal cost C and maximum degree at most B+O(log n/log log n). That is, the error is additive as well as restricted just to the degree. This further improvement in degree is obtained by using augmentingpath techniques that search over a larger solution space than Fischer’s localsearch algorithm.
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 Title
 What Would Edmonds Do? Augmenting Paths and Witnesses for DegreeBounded MSTs
 Journal

Algorithmica
Volume 55, Issue 1 , pp 157189
 Cover Date
 20090901
 DOI
 10.1007/s0045300791155
 Print ISSN
 01784617
 Online ISSN
 14320541
 Publisher
 SpringerVerlag
 Additional Links
 Topics
 Keywords

 Approximation algorithms
 Combinatorial optimization
 Minimum spanning trees
 Matching
 Industry Sectors
 Authors

 Kamalika Chaudhuri ^{(1)}
 Satish Rao ^{(1)}
 Samantha Riesenfeld ^{(1)}
 Kunal Talwar ^{(2)}
 Author Affiliations

 1. University of California, Berkeley, USA
 2. Microsoft Research, Mountain View, CA, USA