Topic
Minimum degree spanning tree
About: Minimum degree spanning tree is a research topic. Over the lifetime, 1433 publications have been published within this topic receiving 34888 citations.
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TL;DR: It is shown that maxπwπ = C* precisely when a certain well-known linear program has an optimal solution in integers.
Abstract: This paper explores new approaches to the symmetric traveling-salesman problem in which 1-trees, which are a slight variant of spanning trees, play an essential role. A 1-tree is a tree together with an additional vertex connected to the tree by two edges. We observe that i a tour is precisely a 1-tree in which each vertex has degree 2, ii a minimum 1-tree is easy to compute, and iii the transformation on "intercity distances" cij â Cij + πi + πj leaves the traveling-salesman problem invariant but changes the minimum 1-tree. Using these observations, we define an infinite family of lower bounds wπ on C*, the cost of an optimum tour. We show that maxπwπ = C* precisely when a certain well-known linear program has an optimal solution in integers. We give a column-generation method and an ascent method for computing maxπwπ, and construct a branch-and-bound method in which the lower bounds wπ control the search for an optimum tour.
1,448 citations
TL;DR: An efficient iterative method for approximating this bound closely from below is presented, and a branch-and-bound procedure based upon these considerations has easily produced proven optimum solutions to all traveling-salesman problems presented to it.
Abstract: The relationship between the symmetric traveling-salesman problem and the minimum spanning tree problem yields a sharp lower bound on the cost of an optimum tour. An efficient iterative method for approximating this bound closely from below is presented. A branch-and-bound procedure based upon these considerations has easily produced proven optimum solutions to all traveling-salesman problems presented to it, ranging in size up to sixty-four cities. The bounds used are so sharp that the search trees are minuscule compared to those normally encountered in combinatorial problems of this type.
1,041 citations
854 citations
TL;DR: This paper develops the reverse search technique in a general framework and shows its broader applications to various problems in operations research, combinatorics, and geometry, and proposes new algorithms for listing.
808 citations
TL;DR: By employing a subroutine that solves the post office problem, it is shown that, for fixed k $\geq$ 3, such a minimum spanning tree can be found in time O($n^{2-a(k)} {(log n)}^{1-a (k)}$), where a(k) = $2^{-(k+1)}$.
Abstract: The problem of finding a minimum spanning tree connecting n points in a k-dimensional space is discussed under three common distance metrics -- Euclidean, rectilinear, and $L_\infty$. By employing a subroutine that solves the post office problem, we show that, for fixed k $\geq$ 3, such a minimum spanning tree can be found in time O($n^{2-a(k)} {(log n)}^{1-a(k)}$), where a(k) = $2^{-(k+1)}$. The bound can be improved to O(${(n log n)}^{1.8}$) for points in the 3-dimensional Euclidean space. We also obtain o($n^2$) algorithms for finding a farthest pair in a set of n points and for other related problems.
683 citations