Laboratoire d ’ Analyses et Modélisation de Systèmes pour l ’ Aide à la Décision UMR 7243 Juillet 2012

We present a natural wireless sensor network problem, which we model as a probabilistic version of the min dominating set problem. We show that this problem, being a generalization of the classical min dominating set, is NP-hard, even in bipartite graphs. We first study the complexity of probabilistic min dominating set in graphs where min dominating set is polynomial, mainly in trees and paths and then we give some approximation results for it. 1 Wireless sensor networks and probabilistic dominating set Very frequently, in wireless sensor networks [31], one wishes to identify a subset of sensors, called “master” sensors, that will have a particular role in messages transmission, namely, to centralize and process messages sent by the rest of the sensors, called “slave” sensors, in the network. These latter sensors will be only nodes of intermediate messages transmission, while the former ones will be authorized to make several operations on messages received and will be, for this reason, better or fully equipped and preprogrammed. So, the objective for designing such a network is to identify a subset of sensors (the master sensors) such that, every other sensor is linked to some sensor in this set. In other words, one wishes to find a dominating set in the graph of sensors. If we suppose that equipment of master sensors induces some additional cost with respect to that of the slave ones, if this cost is the same for all master sensors, we have a minimum cardinality dominating set problem (min dominating set), while if any master sensor has its own cost, we have a minimum weight dominating set problem. Sensors can be broken down any time but, since the network must be remain operational, once a break down, one must be able to recompute a new set of master sensors very quickly (and in any case as quickly as solution from scratch is not allowed). This is the problem handled in this paper. For simplicity, we deal with master sensors of uniform equipment cost (hopefully, it will be clear later that this assumption is not restrictive for the model) and we suppose that any sensor, can be broken down with some probability qi (so, it remains operational, i.e., present in the Research supported by the French Agency for Research under the program TODO, ANR-09-EMER-010 Author’s current address: D.A.I., Politecnico di Torino, Torino, Italy Institut Universitaire de France 1 ha l-0 08 74 39 6, v er si on 1 18 O ct 2 01 3 network, with probability pi = 1− qi) depending on its construction, proper equipment, age, etc. Informally, the approach we propose, in order to maintain the network operational at any time, is the following: • design an algorithm M that, given a set D of master sensors of the network, if some sensors of D fail, it adapts D to the surviving network (in other words, the new D becomes the new master set of sensors for the surviving part of the network); this algorithm must be as efficient as possible, in order that long idle periods for the network are avoided; • given the network, its sensors’ surviving probabilities pi and M, compute a solution D ∗, called “a priori” solution that, informally, has the basic property to be close, in a sense that will be defined later, to every possible solution obtained by a modification of D∗ when applying M. The problem of determining an optimal a priori master set D∗ is the problem handled in this paper. It is clear that given a network of sensors, identifying master set of them is equivalent to determining a dominating set in the associated graph where vertices are the sensors of the network and, for any linked pair of them, an edge links the corresponding vertices. The min dominating set problem is formally defined as follows. Let G(V,E) be a connected graph defined on a set V of vertices with a set E ⊆ V × V of edges. A vertex-set D is said to be a dominating set of G if, for any v ∈ V \ D, v has at least one neighbor in D. In the min dominating set problem, the objective is to determine a minimum-size dominating set in G. The decision version of min dominating set problem is one of the first 21 NP-complete problems [21] and remains NP-complete even in bipartite graphs, while it is polynomial in trees. v1 v2 v3 v4