New Formulations for an Optimal Connectivity Approach for Mobile Ad-hoc Networks

Abstract

In this paper, we propose new formulations for the optimal connectivity of a tree backbone topology for mobile ad-hoc networks (MANETs). Applications of MANETs include military communications, emergency and disaster recovery, and e-commerce to name a few. Formally, given a graph $G=(V \cup K,E)$ with set of wireless sensor nodes V, a set of connection links E, and a set of K users, the problem is to find a backbone spanning tree network topology with as many leaves as possible in order to maximize capacity at the lowest power costs for the users which are connected to the leaf nodes of the backbone network. For this purpose, we model a MANET by means of disk graphs where each disk represents the Euclidean distance transmission range of a node $v \in V$. We propose an exponential and a compact polynomial formulation for the problem. The exponential model is characterized with constraints from the classical maximum leaf spanning tree polytope [18] whilst the compact formulation is characterized with constraints adapted from the classical minimum dominating tree problem [2]. The latter formulation is further strengthened with selected valid inequalities referred to as generalized sub-tour elimination constraints [9]. Our preliminary numerical results indicate that the compact model with additional valid inequalities allows to solve instances to optimality in significantly short CPU time for transmission distances ranging from 100 to 150 ms.