A constructive algorithm for max–min paths problems on energy networks

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Abstract

Max–min paths problems on energy networks are the main center of interest of this article. They typically arose as an auxiliary problem within the study of a special class of discrete min–max control models and within so-called cyclic games. These two classes generalize the well-known combinatorial problem of the shortest and the longest paths in a weighted directed graph. A constructive algorithm for determining the tree of max–min paths in these special networks is proposed. Furthermore, we apply it as a new approach to the solution of special zero value cyclic games. Such a class is not too restrictive. Furthermore we refer to more general models which are very close to real-world examples.

Section snippets

Introduction and problem formulation

In this paper we consider the max–min paths problem on energy networks, which generalizes the classical combinatorial problems of the shortest and the longest paths in weighted directed graphs [2]. This max–min paths problem arose as an auxiliary problem within the study of a game theoretic version of the discrete control problem from [8], [9], [12], [13] and within the examination of optimal stationary strategies of players in cyclic games [3], [5], [12], [16].

This approach extends the results

An algorithm for solving the problem on acyclic networks

The formulated problem for acyclic networks has been studied in [8].

Let G=(V,E) be a finite directed graph without directed cycles and a given sink vertex vf. The partition V=VAVB (VAVB=) of the vertex set of G is given and the cost function c:ER on the edges is defined. We consider the dynamic c-game on G with a given starting position vV.

It is easy to observe that for fixed strategies of the players sASA and sBSB the subgraph Gs=(V,Es) has the structure of a directed tree with sink

The main results for the problem on an arbitrary network

First of all we give an example which demonstrates that equality (1) may fail to hold. In Fig. 1 a certain network is given with starting position v0=1 and final position vf=4, where the positions of the first player are represented by circles and the positions of the second player are represented by squares; values of the cost functions are given on the edges.

It is easy to observe thatmaxsASAminsBSBF1(sA,sB)=2,minsBSBmaxsASAF1(sA,sB)=3.The following theorem gives conditions for the

An application of the algorithm for solving zero value cyclic games

In this section we show that the zero value ergodic cycle game can be regarded as a max–min paths problem and therefore the proposed algorithm can be used for determining the optimal strategies of the players in such special cyclic games.

At first we summarize the formulation of cyclic games and some necessary preliminary results.

Summary

The approach presented here extends the results of a cooperative treatment of the technology emission means (TEM) model which was introduced to model and simulate investment strategies within energy markets. The TEM model and its mathematical structure are related to those general control problems. In [7] suitable transformation principles are developed which lead to an algorithmic determination of Pareto optima. Furthermore it is obvious that the mathematical assumptions are not too

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