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23
Minimum cost capacity installation for multicommodity network flows
 MATHEMATICAL PROGRAMMING
, 1998
"... Consider a directed graph G = (V; A), and a set of traffic demands to be shipped between pairs of nodes in V. Capacity has to be installed on the edges of this graph (in integer multiples of a base unit) so that traffic can be routed. In this paper we consider the problem of minimum cost installatio ..."
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Cited by 49 (12 self)
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Consider a directed graph G = (V; A), and a set of traffic demands to be shipped between pairs of nodes in V. Capacity has to be installed on the edges of this graph (in integer multiples of a base unit) so that traffic can be routed. In this paper we consider the problem of minimum cost installation of capacity on the arcs to ensure that the required demands can be shipped simultaneously between node pairs. We study two different approaches for solving problems of this type. The first one is based on the idea of metric inequalities (see Onaga and Kakusho[1971]), and uses a formulation with only jAj variables. The second uses an aggregated multicommodity flow formulation and has jV j \Delta jAj variables. We first describe two classes of strong valid inequalities and use them to obtain a complete polyhedral description of the associated polyhedron for the complete graph on 3 nodes. Next we explain our solution methods for both of the approaches in detail and present computational results. Our computational experience shows that the two formulations are comparable and yield effective algorithms for solving reallife problems.
Polyhedral approaches to machine scheduling
, 1996
"... We provide a review and synthesis of polyhedral approaches to machine scheduling problems. The choice of decision variables is the prime determinant of various formulations for such problems. Constraints, such as facet inducing inequalities for corresponding polyhedra, are often needed, in addition ..."
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Cited by 35 (8 self)
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We provide a review and synthesis of polyhedral approaches to machine scheduling problems. The choice of decision variables is the prime determinant of various formulations for such problems. Constraints, such as facet inducing inequalities for corresponding polyhedra, are often needed, in addition to those just required for the validity of the initial formulation, in order to obtain useful lower bounds and structural insights. We review formulations based on time–indexed variables; on linear ordering, start time and completion time variables; on assignment and positional date variables; and on traveling salesman variables. We point out relationship between various models, and provide a number of new results, as well as simplified new proofs of known results. In particular, we emphasize the important role that supermodular polyhedra and greedy algorithms play in many formulations and we analyze the strength of the lower and upper bounds obtained from different formulations and relaxations. We discuss separation algorithms for several classes of inequalities, and their potential applicability in generating cutting planes for the practical solution of such scheduling problems. We also review some recent results on approximation algorithms based on some of these formulations.
Maximum Planar Subgraphs and Nice Embeddings: Practical Layout Tools
 ALGORITHMICA
, 1996
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Bounds on the Chvátal Rank of Polytopes in the 0/1Cube
"... Gomory's and Chvatal's cuttingplane procedure proves recursively the validity of linear inequalities for the integer hull of a given polyhedron. The number of rounds needed to obtain all valid inequalities is known as the Chvatal rank of the polyhedron. It is wellknown that the Chvatal rank can be ..."
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Cited by 28 (1 self)
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Gomory's and Chvatal's cuttingplane procedure proves recursively the validity of linear inequalities for the integer hull of a given polyhedron. The number of rounds needed to obtain all valid inequalities is known as the Chvatal rank of the polyhedron. It is wellknown that the Chvatal rank can be arbitrarily large, even if the polyhedron is bounded, if it is of dimension 2, and if its integer hull is a 0/1polytope. We prove that the Chvatal rank of polyhedra featured in common relaxations of many combinatorial optimization problems is rather small; in fact, the rank of any polytope contained in the ndimensional 0/1cube is at most 3n² lg n. This improves upon a recent result of Bockmayr et al. [6] who obtained an upper bound of O(n³ lg n). Moreover, we refine this result by showing that the rank of any polytope in the 0/1cube that is defined by inequalities with small coe#cients is O(n). The latter observation explains why for most cutting planes derived in polyhedral st...
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.
Practical Problem Solving with Cutting Plane Algorithms in Combinatorial Optimization
, 1994
"... Cutting plane algorithms have turned out to be practically successful computational tools in combinatorial optimization, in particular, when they are embedded in a branch and bound framework. Implementations of such "branch and cut" algorithms are rather complicated in comparison to many purely comb ..."
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Cited by 20 (5 self)
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Cutting plane algorithms have turned out to be practically successful computational tools in combinatorial optimization, in particular, when they are embedded in a branch and bound framework. Implementations of such "branch and cut" algorithms are rather complicated in comparison to many purely combinatorial algorithms. The purpose of this article is to give an introduction to cutting plane algorithms from an implementor's point of view. Special emphasis is given to control and data structures used in practically successful implementations of branch and cut algorithms. We also address the issue of parallelization. Finally, we point out that in important applications branch and cut algorithms are not only able to produce optimal solutions but also approximations to the optimum with certified good quality in moderate computation times. We close with an overview of successful practical applications in the literature.
The 2hop spanning tree problem
 OPERATIONS RESEARCH LETTERS
, 1998
"... Given a graph G with a specied root node r. A spanning tree in G where each node has distance at most 2 from r is called a 2hop spanning tree. For given edge weights the 2hop spanning tree problem is to nd a minimum weight 2hop spanning tree. The problem is NPhard and has some interesting ap ..."
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Cited by 12 (0 self)
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Given a graph G with a specied root node r. A spanning tree in G where each node has distance at most 2 from r is called a 2hop spanning tree. For given edge weights the 2hop spanning tree problem is to nd a minimum weight 2hop spanning tree. The problem is NPhard and has some interesting applications. We study a polytope associated with a directed model of the problem give a completeness result for wheels and a vertex description of a linear relaxation. Some classes of valid inequalities for the convex hull of incidence vectors of 2hop spanning trees are derived by projection techniques.
Finding low cost TSP and 2matching solutions using certain halfinteger subtour vertices
, 1998
"... Consider the traveling salesman problem (TSP) defined on the complete graph, where the edge costs satisfy the triangle inequality. Let TOUR denote the optimal solution value for the TSP. Two well known relaxations of the TSP are the subtour elimination problem and the 2matching problem. If we let S ..."
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Cited by 6 (3 self)
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Consider the traveling salesman problem (TSP) defined on the complete graph, where the edge costs satisfy the triangle inequality. Let TOUR denote the optimal solution value for the TSP. Two well known relaxations of the TSP are the subtour elimination problem and the 2matching problem. If we let SUBT and 2M represent the optimal solution values for these two relaxations, then it has been conjectured that TOUR/SUBT ≤ 4/3, and that 2M/SUBT ≤ 10/9. In this paper we exploit the structure of certain 1/2integer solutions for the subtour elimination problem in order to obtain low cost TSP and 2matching solutions. In particular, we show that for cost functions for which the optimal subtour elimination solutions fall into our classes, the above two conjectures are true. We also discuss how this class of subtour elimination solutions is important for the potential resolution of the 4/3 conjecture above. Our proofs are constructive and could be implemented in polynomial time, and thus for such cost functions provide a 4/3 (or better) approximation algorithm for the TSP. Key words: traveling salesman problem, subtour elimination problem, 2matching, approximation algorithm. 1
Two EdgeDisjoint HopConstrained Paths and Polyhedra
 SIAM J. Discrete Math
, 2002
"... Given a graph G with distinguished nodes s and t, a cost on each edge of G, and a xed integer L 2, the Two edgedisjoint Hopconstrained Paths Problem (THPP for short) is to nd a minimum cost subgraph such that between s and t there exist at least two edgedisjoint paths of length at most L. In thi ..."
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Cited by 5 (1 self)
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Given a graph G with distinguished nodes s and t, a cost on each edge of G, and a xed integer L 2, the Two edgedisjoint Hopconstrained Paths Problem (THPP for short) is to nd a minimum cost subgraph such that between s and t there exist at least two edgedisjoint paths of length at most L. In this paper, we consider that problem from a polyhedral point of view. We give an integer programming formulation for the problem and discuss the associated polytope, P (G; L), when L = 2; 3. In particular, we show in this case that the linear relaxation of P (G; L), Q(G;L), given by the trivial, the stcut, and the socalled Lpathcut inequalities, is integral. As a consequence, we obtain a polynomial time cutting plane algorithm for the problem when L = 2; 3. We also give necessary and sucient conditions for these inequalities to dene facets of P (G; L) for L 2. We nally investigate the dominant of P (G; L) and give a complete description of this polyhedron for L 2, when P (G; L) = Q(G;L). Key words. Survivable network, edgedisjoint paths, hopconstraints, polyhedron, facet AMS subject classications. 90B10, 90C27, 90C57 1