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44
On the Solution of Traveling Salesman Problems
 DOC. MATH. J. DMV
, 1998
"... Following the theoretical studies of J.B. Robinson and H.W. Kuhn in the late 1940s and the early 1950s, G.B. Dantzig, R. Fulkerson, and S.M. Johnson demonstrated in 1954 that large instances of the TSP could be solved by linear programming. Their approach remains the only known tool for solving TS ..."
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Cited by 163 (7 self)
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Following the theoretical studies of J.B. Robinson and H.W. Kuhn in the late 1940s and the early 1950s, G.B. Dantzig, R. Fulkerson, and S.M. Johnson demonstrated in 1954 that large instances of the TSP could be solved by linear programming. Their approach remains the only known tool for solving TSP instances with more than several hundred cities; over the years, it has evolved further through the work of M. Grötschel , S. Hong , M. Jünger , P. Miliotis , D. Naddef , M. Padberg
Approximating Minimum Bounded Degree Spanning Trees to within One of Optimal
 In Proc. of ACM Symposium on Theory of computing (STOC
, 2007
"... ABSTRACT In the MINIMUM BOUNDED DEGREE SPANNING TREE problem,we are given an undirected graph with a degree upper bound Bv oneach vertex v, and the task is to find a spanning tree of minimumcost which satisfies all the degree bounds. Let OPT be the costof an optimal solution to this problem. In this ..."
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Cited by 46 (6 self)
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ABSTRACT In the MINIMUM BOUNDED DEGREE SPANNING TREE problem,we are given an undirected graph with a degree upper bound Bv oneach vertex v, and the task is to find a spanning tree of minimumcost which satisfies all the degree bounds. Let OPT be the costof an optimal solution to this problem. In this paper, we present a polynomial time algorithm which returns a spanning tree T ofcost at most OPT and dT (v) ^ Bv + 1 for all v, where dT (v)denotes the degree of v in T. This generalizes a result of Furerand Raghavachari [8] to weighted graphs, and settles a 15yearold conjecture of Goemans [10] affirmatively. The algorithm generalizes when each vertex v has a degree lower bound Av and a degreeupper bound Bv, and returns a spanning tree with cost at most OPTand Av \Gamma 1 ^ dT (v) ^ Bv + 1 for all v. This is essentially thebest possible. The main technique used is an extension of the iterative rounding method introduced by Jain [12] for the design ofapproximation algorithms.
Implementing the DantzigFulkersonJohnson Algorithm for Large Traveling Salesman Problems
, 2003
"... Dantzig, Fulkerson, and Johnson (1954) introduced the cuttingplane method as a means of attacking the traveling salesman problem; this method has been applied to broad classes of problems in combinatorial optimization and integer programming. In this paper we discuss an implementation of Dantzig et ..."
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Cited by 36 (6 self)
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Dantzig, Fulkerson, and Johnson (1954) introduced the cuttingplane method as a means of attacking the traveling salesman problem; this method has been applied to broad classes of problems in combinatorial optimization and integer programming. In this paper we discuss an implementation of Dantzig et al.'s method that is suitable for TSP instances having 1,000,000 or more cities. Our aim is to use the study of the TSP as a step towards understanding the applicability and limits of the general cuttingplane method in largescale applications.
Minimum Bounded Degree Spanning Trees
, 2006
"... We consider the minimum cost spanning tree problem under the restriction that all degrees must be at most a given value k. We show that we can efficiently find a spanning tree of maximum degree at most k + 2 whose cost is at most the cost of the optimum spanning tree of maximum degree at most k. Thi ..."
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Cited by 31 (0 self)
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We consider the minimum cost spanning tree problem under the restriction that all degrees must be at most a given value k. We show that we can efficiently find a spanning tree of maximum degree at most k + 2 whose cost is at most the cost of the optimum spanning tree of maximum degree at most k. This is almost best possible. The approach uses a sequence of simple algebraic, polyhedral and combinatorial arguments. It illustrates many techniques and ideas in combinatorial optimization as it involves polyhedral characterizations, uncrossing, matroid intersection, and graph orientations (or packing of spanning trees). The result generalizes to the setting where every vertex has both upper and lower bounds and gives then a spanning tree which violates the bounds by at most two units and whose cost is at most the cost of the optimum tree. It also gives a better understanding of the subtour relaxation for both the symmetric and asymmetric traveling salesman problems. The generalization to ledgeconnected subgraphs is briefly discussed.
The Steiner tree polytope and related polyhedra
, 1994
"... We consider the vertexweighted version of the undirected Steiner tree problem. In this problem, a cost is incurred both for the vertices and the edges present in the Steiner tree. We completely describe the associated polytope by linear inequalities when the underlying graph is seriesparallel. For ..."
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Cited by 27 (1 self)
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We consider the vertexweighted version of the undirected Steiner tree problem. In this problem, a cost is incurred both for the vertices and the edges present in the Steiner tree. We completely describe the associated polytope by linear inequalities when the underlying graph is seriesparallel. For general graphs, this formulation can be interpreted as a (partial) extended formulation for the Steiner tree problem. By projecting this formulation, we obtain some very large classes of facetdefining valid inequalities for the Steiner tree polytope.
TSP cuts which do not conform to the template paradigm
 IN COMPUTATIONAL COMBINATORIAL OPTIMIZATION
, 2001
"... The first computer implementation of the DantzigFulkersonJohnson cuttingplane method for solving the traveling salesman problem, written by Martin, used subtour inequalities as well as cutting planes of Gomory’s type. The practice of looking for and using cuts that match prescribed templates in c ..."
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Cited by 25 (1 self)
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The first computer implementation of the DantzigFulkersonJohnson cuttingplane method for solving the traveling salesman problem, written by Martin, used subtour inequalities as well as cutting planes of Gomory’s type. The practice of looking for and using cuts that match prescribed templates in conjunction with Gomory cuts was continued in computer codes of Miliotis, Land, and Fleischmann. Grötschel, Padberg, and Hong advocated a different policy, where the template paradigm is the only source of cuts; furthermore, they argued for drawing the templates exclusively from the set of linear inequalities that induce facets of the TSP polytope. These policies were adopted in the work of Crowder and Padberg, in the work of Grötschel and Holland, and in the work of Padberg and Rinaldi; their computer codes produced the most impressive computational TSP successes of the nineteen eighties. Eventually, the template paradigm became the standard frame of reference for cutting planes in the TSP. The purpose of this paper is to describe a technique
WorstCase Comparison of Valid Inequalities for the TSP
 MATH. PROG
, 1995
"... We consider most of the known classes of valid inequalities for the graphical travelling salesman polyhedron and compute the worstcase improvement resulting from their addition to the subtour polyhedron. For example, we show that the comb inequalities cannot improve the subtour bound by a factor gr ..."
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Cited by 25 (1 self)
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We consider most of the known classes of valid inequalities for the graphical travelling salesman polyhedron and compute the worstcase improvement resulting from their addition to the subtour polyhedron. For example, we show that the comb inequalities cannot improve the subtour bound by a factor greater than 10/9. The corresponding factor for the class of clique tree inequalities is 8/7, while it is 4/3 for the path configuration inequalities.
The Circuit Polytope: Facets
, 1994
"... Given an undirected graph G = (V; E) and a cost vector c 2 IR E , the weighted girth problem is to find a circuit in G having minimum total cost. This problem is in general NPhard since the traveling salesman problem can be reduced to it. A promising approach to hard combinatorial optimization p ..."
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Cited by 15 (1 self)
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Given an undirected graph G = (V; E) and a cost vector c 2 IR E , the weighted girth problem is to find a circuit in G having minimum total cost. This problem is in general NPhard since the traveling salesman problem can be reduced to it. A promising approach to hard combinatorial optimization problems is given by the socalled cutting plane methods. These involve linear programming techniques based on a partial description of the convex hull of the incidence vectors of possible solutions. We consider the weighted girth problem in the case where G is the complete graph K n and study the facial structure of the circuit polytope P n C and some related polyhedra. In the appendix we give complete characterizations of P n C for n up to 6.
Design of Survivable Networks: A survey
 In Networks
, 2005
"... For the past few decades, combinatorial optimization techniques have been shown to be powerful tools for formulating and solving optimization problems arising from practical situations. In particular, many network design problems have been formulated as combinatorial optimization problems. With the ..."
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Cited by 14 (0 self)
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For the past few decades, combinatorial optimization techniques have been shown to be powerful tools for formulating and solving optimization problems arising from practical situations. In particular, many network design problems have been formulated as combinatorial optimization problems. With the advances of optical technologies and the explosive growth of the Internet, telecommunication networks have seen an important evolution and therefore, designing survivable networks has become a major objective for telecommunication operators. Over the past years, a big amount of research has then been done for devising efficient methods for survivable network models, and particularly cutting plane based algorithms. In this paper, we attempt to survey some of these models and the optimization methods used for solving them.