Solving Multiconstraint Assignment Problems Using Learning Automata

This paper considers the
NP
-hard problem of object assignment with respect to multiple constraints: assigning a set of elements (or objects) into mutually exclusive
classes
(or groups), where the elements which are ldquosimilarrdquo to each other are hopefully located in the same class. The literature reports solutions in which the similarity constraint consists of a
single
index that is inappropriate for the type of multiconstraint problems considered here and where the constraints could simultaneously be
contradictory
. Such a scenario is illustrated with the
static mapping problem
, which consists of distributing the processes of a parallel application onto a set of computing nodes. This is a classical and yet very important problem within the areas of parallel computing, grid computing, and
cloud
computing. We have developed four learning-automata (LA)-based algorithms to solve this problem: First, a fixed-structure stochastic automata algorithm is presented, where the processes try to form pairs to go onto the same node. This algorithm solves the problem, although it requires some centralized coordination. As it is desirable to avoid centralized control, we subsequently present
three
different variable-structure stochastic automata (VSSA) algorithms, which have superior partitioning properties in certain settings, although they forfeit some of the scalability features of the fixed-structure algorithm. All three VSSA algorithms model the processes as automata having first the hosting nodes as possible actions; second, the processes as possible actions; and, third, attempting to estimate the process communication digraph prior to probabilistically mapping the processes. This paper, which, we believe, comprehensively reports the pioneering LA solutions to this problem, unequivocally demonstrates that LA can play an important role in solving complex combinatorial and integer optimization problems.