Parametric min-cuts analysis in a network

AbstractThe all pairs minimum cuts problem in a capacitated undirected network is well known. Gomory and Hu showed that the all pairs minimum cuts are revealed by a min-cut tree that can be obtained by solving exactly (n−1) maximum flow problems, where n is the number of nodes in the network.In this paper we consider first the problem of finding parametric min-cuts for a specified pair of nodes when the capacity of an arc i is given by min{bi,λ}, where λ is the parameter, ranging from 0 to ∞. Next we seek the parametric min-cuts for all pairs of nodes, and achieve this by constructing min-cut trees for at most 2m different values of λ, where m is the number of edges in the network

Parametric approaches to fractional programs: Analytical and empirical study

Fractional programming is used to model problems where the objective function is a ratio of functions. A parametric modeling approach provides effective technique for obtaining optimal solutions of these fractional programming problems. Although many heuristic algorithms have been proposed and assessed relative to each other, there are limited theoretical studies on the number of steps to obtain the solution. In this dissertation, I focus on the linear fractional combinatorial optimization problem, a special case of fractional programming where all functions in the objective function and constraints are linear and all variables are binary that model certain combinatorial structures. Two parametric algorithms are considered and the efficiency of the algorithms is investigated both theoretically and computationally. I develop the complexity bounds for these algorithms, and show that they can solve the linear fractional combinatorial optimization problem in polynomial time. In the computational study, the algorithms are used to solve fractional knapsack problem, fractional facility location problem, and fractional transportation problem by comparison to other algorithms (e.g., Newton\u27s method). The relative practical performance measured by the number of function calls demonstrates that the proposed algorithms are fast and robust for solving the linear fractional programs with discrete variables

On Fork-Join Queues and Maximum Ratio Cliques

This dissertation consists of two parts. The first part delves into the problem of response time estimation in fork-join queueing networks. These systems have been seen in literature for more than thirty years. The estimation of the mean response time in these systems has been found to be notoriously hard for most forms of these queueing systems. In this work, simple expressions for the mean response time are proposed as conjectures. Extensive experiments demonstrate the remarkable accuracy of these conjectures. Algorithms for the estimation of response time using these conjectures are proposed. For many of the networks studied in this dissertation, no approximations are known in literature for estimation of their response time. Therefore, the contribution of this dissertation in this direction marks significant progress in the analysis of fork-join queues. The second part of this dissertation introduces a fractional version of the classical maximum weight clique problem, the maximum ratio clique problem, which is to find a maximal clique that has the largest ratio of benefit and cost weights associated with the cliques vertices. This problem is formulated to model networks in which the vertices have a benefit as well as a cost associated with them. The maximum ratio clique problem finds applications in a wide range of areas including social networks, stock market graphs and wind farm location. NP-completeness of the decision version of the problem is established, and three solution methods are proposed. The results of numerical experiments with standard graph instances, as well as with real-life instances arising in finance and energy systems, are reported