# Distributed Lagrangian Relaxation Protocol for the Over-constrained Generalized Mutual Assignment Problem

## Abstract

*The Generalized Mutual Assignment Problem* (GMAP) is a distributed combinatorial optimization problem in which, with no centralized control, multiple agents search for an optimal assignment of goods that satisfies their individual knapsack constraints. Previously, in the GMAP protocol, problem instances were assumed to be feasible, meaning that the total capacities of the agents were large enough to assign the goods. However, this assumption may not be realistic in some situations. In this paper, we present two methods for dealing with such “over-constrained” GMAP instances. First, we introduce a *disposal agent* who has an unlimited capacity and is in charge of the unassigned goods. With this method, we can use any off-the-shelf GMAP protocol since the disposal agent can make the instances feasible. Second, we formulate the GMAP instances as an Integer Programming (IP) problem, in which the assignment constraints are described with inequalities. With this method, we need to devise a new protocol for such a formulation. We experimentally compared these two methods on the variants of *Generalized Assignment Problem* (GAP) benchmark instances. Our results indicate that the first method finds a solution faster for fewer over-constrained instances, and the second finds a better solution faster for more over-constrained instances.

## Keywords

generalized mutual assignment problem distributed optimization Lagrangian relaxation## Preview

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