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From: jse@netcom.com (Jamieson Science Jse Engineer)
Subject: Re: Algorithm for Held and Karp lower bound on the TSP
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Date: Thu, 20 Jul 1995 14:41:39 GMT
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David Neto (neto@eecg.toronto.edu) wrote:
: In article <NEWTNews.2008.806112556.martin@netcomsv.netcom.com.jse.slip.netcom.com>,
:  <martin@as.arizona.edu> wrote:
: >
: >Hello,
: >
: >I've read the two papers from Held and Karp (1970 and 1971) on the lower
: >bound for the TSP.  Does anybody know if there exists code, or a simple
: >algorithm (as in the paper without the proofs, etc.) for their branch and
: >bound method (1971 paper) on finding a lower bound?
: >
: >Martin Schlapfer
: >martin@as.arizona.edu

: If all you want to do is find a lower bound, then you don't need to do
: any branch and bound at all.  However, if you want to find an optimal
: tour, then you do.

Its my understanding that Held and Karp present an algorithm which
produces a tight lower bound for an instance of the TSP.  They use minimum
1-trees to do this.  Obviously finding a minimum 1-tree *is* a lower
bound, however it is not tight.  Held and Karp (1971) introduce a real
vector pi of length n = |V|, of additional weights added to the cost of
entering a vertex (edge c(ij) = c(ij)+pi(i)+pi(j)).  Since the TSP enters
and exits each vertex exactly once, this does not change the optimal tour. 
However, it may change the minimum 1-tree. They then iterate (I assume on
pi) to produce a tight lower bound.  What I'm looking for is someone who
may have this algorithm coded, or possibly a paper that describes this
more explicitly and without the proofs.  It seems that the Held and Karp
lower bound is used for comparing how well an approximation algorithms do
(cost wise) and is still the standard. 
 
Thanks for the following refs.:
Martin.
martin@as.arizona.edu

: Here are a few more up to date works to look into:

: @incollection{ BalasToth1985,
: 	author = "E.~Balas and P.~Toth",
: 	title = "Branch and bound methods",
: 	booktitle = "The Traveling Salesman Problem",
: 	publisher = "John Wiley \& Sons Ltd.",
: 	chapter = 10,
: 	year = 1985,
: 	editor = {E.~L.~Lawler and J.~K.~Lenstra and A.~H.~G.~Rinnooy Kan 
: 		and D.~B.~Shmoys},
: 	pages = "361--401"
: }

: @article { VolgenantJonker1982,
: 	fullauthor = "Ton~Volgenant and Roy~Jonker",
: 	author = "T.~Volgenant and R.~Jonker",
: 	title = {A branch and bound algorithm for the symmetric traveling 
: 		salesman problem based on the 1-tree relaxation},
: 	journal = "European Journal of Operational Research",
: 	year = 1982,
: 	volume = 9,
: 	pages = "83--89"
: }

: @article { VolgenantJonker1983,
: 	fullauthor = "Ton~Volgenant and Roy~Jonker",
: 	author = "T.~Volgenant and R.~Jonker",
: 	title = {The symmetric traveling salesman problem and edge exchanges
: 		in minimal 1-trees},
: 	journal = "European Journal of Operational Research",
: 	year = 1983,
: 	volume = 12,
: 	pages = "394--403"
: }
: 	
: @article { VolgenantJonker1990,
: 	fullauthor = "Ton~Volgenant and Roy~Jonker",
: 	author = "T.~Volgenant and R.~Jonker",
: 	title = {Fictitious upper bounds in an algorithm for the symmetric
: 		traveling salesman problem},
: 	journal = "Computers and Operations Research",
: 	volume = 17,
: 	number = 1,
: 	year = 1990,
: 	pages = "113--117",
: 	datereviewed = "95/2/24"
: }

: I have found an error in VolgenantJonker1983 (in their procedure to
: find required branches), but you can ignore that portion of the paper.

: An alternative to the above is chapter 10 in the following
: book:

: @book { Reinelt1994,
: 	fullauthor = "Gerhard Reinelt",
: 	author = "G.~Reinelt",
: 	title = {The traveling salesman: Computational solutions for {TSP}
: 		applications},
: 	publisher = "Springer {V}erlag",
: 	year = 1994,
: 	isbn = "0-387-58334-3",
: 	note = "LNCS 840"
: }

: I hope this helps.

: -- David Neto
: neto@eecg.utoronto.ca
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