A LEARNING AUTOMATA-BASED ALGORITHM TO THE STOCHASTIC MIN-DEGREE CONSTRAINED MINIMUM SPANNING TREE PROBLEM

Min-degree constrained minimum spanning tree (md-MST) problem is an NP-hard combinatorial optimization problem seeking for the minimum weight spanning tree in which the vertices are either of degree one (leaf) or at least degree d ≥ 2. md-MST problem is new to the literature and very few studies have been conducted on this problem in deterministic graph. md-MST problem has several appealing real-world applications. Though in realistic applications the graph conditions and parameters are stochastic and vary with time, to the best of our knowledge no work has been done on solving md-MST problem in stochastic graph. This paper proposes a decentralized learning automata-based algorithm for finding a near optimal solution to the md-MST problem in stochastic graph. In this work, it is assumed that the weight associated with the graph edge is random variable with a priori unknown probability distribution. This assumption makes the md-MST problem incredibly harder to solve. The proposed algorithm exploits an intelligent sampling technique avoiding the unnecessary samples by focusing on the edges of the min-degree spanning tree with the minimum expected weight. On the basis of the Martingale theorem, the convergence of the proposed algorithm to the optimal solution is theoretically proven. Extensive simulation experiments are performed on the stochastic graph instances to show the performance of the proposed algorithm. The obtained results are compared with those of the standard sampling method in terms of the sampling rate and solution optimality. Simulation experiments show that the proposed method outperforms the standard sampling method.

A Note on the ExponentialG-Martingale

We get the exponentialG-martingale theorem with the Kazamaki condition and tell a distinct difference between the Kazamaki’s and Novikov’s criteria with an example.

On the weak convergence of U-statistic processes, and of the empirical process

1. Summary and introductionIn (5) a weak convergence result for U-statistics was obtained as a special case of a reverse martingale theorem; in (7) Miller and Sen obtained another such result for U-statistics by a direct argument. As they stand these results are not very closely connected, since one is concerned with U-statistics Uk for k ≥ n, while the other deals with Uk for k ≤ n, but if one instead thinks of k as unrestricted and transforms the random functions Xn which enter into one of these results into new functions Yn by setting Yn(t) = tXn(t−1) one finds that the Yn are (aside from variations in interpolated values) just the functions with which the other result is concerned. As the limiting Wiener process W is well-known to have the property that tW(t−1) is another Wiener process it is not too surprising that both results should hold, and part of the purpose of this paper is to provide a general framework within which the relationship between these results will become clear. A second purpose is to illustrate the simplification that the martingale property brings to weak convergence studies; this is shown both in the U-statistic example and in a new proof of the convergence of the empirical process.