Prabha Sharma

A network simplex algorithm with O(n) consecutive degenerate pivots

In this paper, we suggest a new pivot rule for the primal simplex algorithm for the minimum cost flow problem, known as the network simplex algorithm. Due to degeneracy, cycling may occur in the network simplex algorithm. The cycling can be prevented by maintaining strongly feasible bases proposed by Cunningham (Math. Programming 11 (1976) 105; Math. Oper. Res. 4 (1979) 196); however, if we do not impose any restrictions on the entering variables, the algorithm can still perform an exponentially long sequence of degenerate pivots. This phenomenon is known as stalling. Researchers have suggested several pivot rules with the following bounds on the number of consecutive degenerate pivots: m,n^2,k(k+1)/2, where n is the number of nodes in the network, m is the number of arcs in the network, and k is the number of degenerate arcs in the basis. (Observe that k=

Algorithms for the optimum communication spanning tree problem

Optimum Communication Spanning Tree Problem is a special case of the Network Design Problem. In this problem given a graph, a set of requirements rij and a set of distances dij for all pair of nodes (i,j), the cost of communication for a pair of nodes (i,j), with respect to a spanning tree T is defined as rij times the length of the unique path in T, that connects nodes i and j. Total cost of communication for a spanning tree is the sum of costs for all pairs of nodes of G. The problem is to construct a spanning tree for which the total cost of communication is the smallest among all the spanning trees of G. The problem is known to be NP-hard. Hu (1974) solved two special cases of the problem in polynomial time. In this paper, using Hu’s result the first algorithm begins with a cut-tree by keeping all dij equal to the smallest dij. For arcs (i,j) which are part of this cut-tree the corresponding dij value is increased to obtain a near optimal communication spanning tree in pseudo-polynomial time.In case the distances dij satisfy a generalised triangle inequality the second algorithm in the paper constructs a near optimum tree in polynomial time by parametrising on the rij.

Constrained optimum communication trees and sensitivity analysis

Consider n cities with specified communication requirements between all pairs of cities. An optimum communication tree has the property that among all the spanning trees connecting the n cities, the sum of its costs of communication for the

Permutation Polyhedra and Minimisation of the Variance of Completion Times on a Single Machine

n(n - 1)/2

A new pivot selection rule for the network simplex algorithm

pairs of cities is minimum. The cost of communication between a pair of nodes, with respect to a spanning tree, is the product of the communication requirement and the length of the path between the two cities. We construct constrained optimum communication trees when (i) certain specified cities are required to be the outer nodes of the communication tree, and when (ii) it is required that certain pairs of cities be connected directly in the communication tree. We further analyse changes in the structure of an optimum communication tree when the communication requirement between a pair of cities is subject to changes. We show that for the whole range

A combinatorial arc tolerance analysis for network flow problems

[0,\infty )

An extension of the edge covering problem

, of the communication requirement, there exist at most

An efficient local search scheme for minimizing mean absolute deviation of completion times

(n - 1)

A decomposition algorithm for linear relaxation of the weighted r-covering problem

optimum communication trees...

A local search algorithm: minimizing makespan of deteriorating jobs with relaxed agreeable weights

We consider the problem of minimising variance of completion times when n-jobs are to be processed on a single machine. This problem is known as the CTV problem. The problem has been shown to be difficult. In this paper we consider the polytope Pn whose vertices are in one-to-one correspondence with the n! permutations of the processing times [p1, p2, ..., pn]. Thus each vertex of Pn represents a sequence in which the n-jobs can be processed. We define a V-shaped local optimal solution to the CTV problem to be the V-shaped sequence of jobs corresponding to which the variance of completion times is smaller than for all the sequences adjacent to it on Pn. We show that this local solution dominates V-shaped feasible solutions of the order of 2n−3 where 2n−2 is the total number of V-shaped feasible solutions.Using adjacency structure an Pn, we develop a heuristic for the CTV problem which has a running time of low polynomial order, which is exact for n ≤ 10, and whose domination number is Ω(2n−3). In the end we mention two other classes of scheduling problems for which also ADJACENT provides solutions with the same domination number as for the CTV problem.