Experimental evaluation of algorithmic solutions for generalized network flow models

Copyright @ 2000 Mathematical Programming SocietyThe generalised network flow problem is to maximise the net flow into a specified sink node in a network with gain-loss factors associated with edges. In practice, computation of solutions for instances of this problem is almost always done using general-purpose linear programming codes, but this may change because a number of specialized combinatorial generalised-flow algorithms have been recently proposed. To complement the known theoretical analyses of these algorithms, we develop their implementations and investigate their practical performance. We include in our study different versions of Goldberg, Plotkin, and Tardos's Fat-Path algorithm and Wayne's Push-Related algorithm. We compare the performance of our implementations of these algorithms with implementations of the straightforward highest-gain path-augmentation algorithms. We use various classes of networks, including a type of layered networks which may appear in the multiperiod portfolio revision problem.This work was supported by the EPSRC grant GR/L8146

Experimental evaluation of algorithmic solutions for the maximum generalised network flow problem

Also available as Technical Report No. TR-01-09, Department of Computer Science, King's College London, UK, 2001.The maximum generalised network flow problem is to maximise the net flow into a specified node in a network with capacities and gain-loss factors associated with edges. In practice, input instances of this problem are usually solved using general-purpose linear programming codes, but this may change because a number of specialised combinatoria generalised-flow algorithms have been recently proposed. To complement the knwon theoretical analyses of these algorithms, we develop their implementations and investigate their actual performance. We focus in this study on Goldfarb, Jin and Orlin's excess-scaling algorithm and Tardos and Wayne's push-relabel algorithm. We develop variants of these algorithms to implementations of simple, but non-polynomial, combinatorial algorithms proposed by Onaga and Truemper, and with performance of CPLEX, a commercial general-purpose linear progamming package.This work was supported by the EPSRC grant GR/L8146

Smoothed Analysis of the Minimum-Mean Cycle Canceling Algorithm and the Network Simplex Algorithm

The minimum-cost flow (MCF) problem is a fundamental optimization problem with many applications and seems to be well understood. Over the last half century many algorithms have been developed to solve the MCF problem and these algorithms have varying worst-case bounds on their running time. However, these worst-case bounds are not always a good indication of the algorithms' performance in practice. The Network Simplex (NS) algorithm needs an exponential number of iterations for some instances, but it is considered the best algorithm in practice and performs best in experimental studies. On the other hand, the Minimum-Mean Cycle Canceling (MMCC) algorithm is strongly polynomial, but performs badly in experimental studies. To explain these differences in performance in practice we apply the framework of smoothed analysis. We show an upper bound of $O(mn^2\log(n)\log(\phi))$ for the number of iterations of the MMCC algorithm. Here $n$ is the number of nodes, $m$ is the number of edges, and $\phi$ is a parameter limiting the degree to which the edge costs are perturbed. We also show a lower bound of $\Omega(m\log(\phi))$ for the number of iterations of the MMCC algorithm, which can be strengthened to $\Omega(mn)$ when $\phi=\Theta(n^2)$. For the number of iterations of the NS algorithm we show a smoothed lower bound of $\Omega(m \cdot \min \{ n, \phi \} \cdot \phi)$.Comment: Extended abstract to appear in the proceedings of COCOON 201

Nonzero-sum Stochastic Games

This paper treats of stochastic games. We focus on nonzero-sum games and provide a detailed survey of selected recent results. In Section 1, we consider stochastic Markov games. A correlation of strategies of the players, involving ``public signals'', is described, and a correlated equilibrium theorem proved recently by Nowak and Raghavan for discounted stochastic games with general state space is presented. We also report an extension of this result to a class of undiscounted stochastic games, satisfying some uniform ergodicity condition. Stopping games are related to stochastic Markov games. In Section 2, we describe a version of Dynkin's game related to observation of a Markov process with random assignment mechanism of states to the players. Some recent contributions of the second author in this area are reported. The paper also contains a brief overview of the theory of nonzero-sum stochastic games and stopping games which is very far from being complete