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37
Dynamic Trees as Search Trees via Euler Tours, Applied to the Network Simplex Algorithm
 Mathematical Programming
, 1997
"... The dynamic tree is an abstract data type that allows the maintenance of a collection of trees subject to joining by adding edges (linking) and splitting by deleting edges (cutting), while at the same time allowing reporting of certain combinations of vertex or edge values. For many applications of ..."
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The dynamic tree is an abstract data type that allows the maintenance of a collection of trees subject to joining by adding edges (linking) and splitting by deleting edges (cutting), while at the same time allowing reporting of certain combinations of vertex or edge values. For many applications of dynamic trees, values must be combined along paths. For other applications, values must be combined over entire trees. For the latter situation, we show that an idea used originally in parallel graph algorithms, to represent trees by Euler tours, leads to a simple implementation with a time of O(log n) per tree operation, where n is the number of tree vertices. We apply this representation to the implementation of two versions of the network simplex algorithm, resulting in a time of O(log n) per pivot, where n is the number of vertices in the problem network.
A linear bound on the diameter of the transportation polytope
"... We prove that the combinatorial diameter of the skeleton of the polytope of feasible solutions of any m × n transportation problem is at most 8(m + n − 2). The transportation problem ( TP) is a classic problem in operations research. The problem was posed for the first time by Hitchcock in 1941 [9] ..."
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Cited by 13 (1 self)
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We prove that the combinatorial diameter of the skeleton of the polytope of feasible solutions of any m × n transportation problem is at most 8(m + n − 2). The transportation problem ( TP) is a classic problem in operations research. The problem was posed for the first time by Hitchcock in 1941 [9] and independently by Koopmans in 1947 [12], and appears in any standard introductory course on operations research. The m × n TP has m supply points and n demand points. Each supply point i holds a quantity ri> 0, and each demand point j wants a quantity cj> 0, with m� ri = n� cj. A solution to the problem can be written as an m × n matrix X, where entries are decision variables xij having value equal to the amount transported from supply point i to demand point j. The set of feasible solutions of TP, the transportation polytope T, is described by n� xij = ri, i = 1, 2,...,m; j=1 m� xij = cj, j = 1, 2,...,n; i=1 xij ≥ 0, i = 1, 2,...,m, j = 1, 2,...n. The 1skeleton ( edge graph) of T is defined as the graph with vertices the vertices of the polytope and edges its 1dimensional faces. The diameter of T, which we denote by diam(T), is the diameter of its 1skeleton. In 1957 W.M. Hirsch stated his famous conjecture (cf. [5]) saying that any ddimensional polytope with n facets has diameter at most n − d. So far the best known bound for arbitrary polytopes is O(n log d+1) [10]. Any polynomial bound is still lacking. Such bounds have been proved for some special classes of polytopes ( for examples, see [14]). Among those are some special classes of transportation polytopes [1, 3] and the polytope of the dual of TP [1].
Random walks on the vertices of transportation polytopes with constant number of sources
 Proc. 14th Ann. ACMSIAM Symp. Disc. Alg. (Baltimore, MD) 330–339, ACM
, 2003
"... We consider the problem of uniformly sampling a vertex of a transportation polytope with m sources and n destinations, where m is a constant. We analyse a natural random walk on the edgevertex graph of the polytope. The analysis makes use of the multicommodity flow technique of Sinclair [30] toget ..."
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Cited by 10 (2 self)
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We consider the problem of uniformly sampling a vertex of a transportation polytope with m sources and n destinations, where m is a constant. We analyse a natural random walk on the edgevertex graph of the polytope. The analysis makes use of the multicommodity flow technique of Sinclair [30] together with ideas developed by Morris and Sinclair [24, 25] for the knapsack problem, and Cryan et al. [3] for contingency tables, to establish that the random walk approaches the uniform distribution in time n O(m2). 1
Relaxed most negative cycle and most positive cut canceling algorithms for minimum cost flow
 Math. of OR
, 2000
"... This paper presents two new scaling algorithms for the minimum cost network flow problem, one a primal cycle canceling algorithm, the other a dual cut canceling algorithm. Both algorithms scale a relaxed optimality parameter, and create a second, inner relaxation. The primal algorithm uses the inne ..."
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Cited by 7 (4 self)
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This paper presents two new scaling algorithms for the minimum cost network flow problem, one a primal cycle canceling algorithm, the other a dual cut canceling algorithm. Both algorithms scale a relaxed optimality parameter, and create a second, inner relaxation. The primal algorithm uses the inner relaxation to cancel a most negative nodedisjoint family of cycles w.r.t. the scaled parameter, the dual algorithm uses it to cancel most positive cuts w.r.t. the scaled parameter. We show that in a network with n nodes and m arcs, both algorithms need to cancel only O(mn) objects per scaling phase. Furthermore, we show how to efficiently implement both algorithms to yield weakly polynomial running times that are as fast as any other cycle or cut canceling algorithms. Our algorithms have potential practical advantages compared to some other canceling algorithms as well. Along the way, we give a comprehensive survey of cycle and cut canceling algorithms for mincost flow. We also clarify the formal duality between cycles and cuts.
A quadratic bound on the diameter of the transportation polytope
 TECHNISCHE UNIVERSITEIT EINDHOVEN, AND CDAM RESEARCH REPORT 200209, LONDON SCHOOL OF ECONOMICS
, 2002
"... We prove that the combinatorial diameter of the skeleton of the polytope of feasible solutions of any m × n transportation problem is less than 1/2 (m + n)². ..."
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Cited by 5 (1 self)
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We prove that the combinatorial diameter of the skeleton of the polytope of feasible solutions of any m × n transportation problem is less than 1/2 (m + n)².
Container Terminals: Scheduling Decisions, their Formulation and Solutions
 SUBMISSION TO JOURNAL OF SCHEDULING
, 2006
"... The growths of containerization and transporting goods in containers have created many problems for ports. In this paper, we systematically survey a literature over problems in container terminals. The operations that take place in container terminals are described and then the problems are classif ..."
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Cited by 5 (0 self)
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The growths of containerization and transporting goods in containers have created many problems for ports. In this paper, we systematically survey a literature over problems in container terminals. The operations that take place in container terminals are described and then the problems are classified into five scheduling decisions. For each of the decisions, on overview of the literature is presented. After that, each of the decisions is formulated as Constraint Satisfaction and Optimisation Problems (CSOPs). The literature also includes solutions, implementation and performance. The solutions are classified and summarized. Two frameworks for the implementation are suggested. To confirm the results and to measure the terminal’s efficiency in each of the decisions, several indices are suggested.
MINIMUM COST NETWORK FLOWS: PROBLEMS, ALGORITHMS, AND SOFTWARE
"... Abstract: We present a wide range of problems concerning minimum cost network flows, and give an overview of the classic linear singlecommodity Minimum Cost Network Flow Problem (MCNFP) and some other closely related problems, either tractable or intractable. We also discuss stateoftheart algori ..."
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Cited by 5 (4 self)
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Abstract: We present a wide range of problems concerning minimum cost network flows, and give an overview of the classic linear singlecommodity Minimum Cost Network Flow Problem (MCNFP) and some other closely related problems, either tractable or intractable. We also discuss stateoftheart algorithmic approaches and recent advances in the solution methods for the MCNFP. Finally, optimization software packages for the MCNFP are presented.
On subdeterminants and the diameter of polyhedra
 In Proceedings of the 28th annual ACM symposium on Computational geometry, SoCG ’12
, 2012
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