Identifying and locating-dominating codes in (random) geometric networks

International audienceWe model a problem about networks built from wireless devices using identifying and locating-dominating codes in unit disk graphs. It is known that minimising the size of an identifying code is NP-complete even for bipartite graphs. First, we improve this result by showing that the problem remains NP-complete for bipartite planar unit disk graphs. Then, we address the question of the existence of an identifying code for random unit disk graphs. We derive the probability that there exists an identifying code as a function of the radius of the disks and we find that for all interesting ranges of r this probability is bounded away from one. The results obtained are in sharp contrast with those concerning random graphs in the Erdos-Renyi model. Another well-studied class of codes are locating-dominating codes, which are less demanding than identifying codes. A locating-dominating code always exists, but minimising its size is still NP-complete in general. We extend this result to our setting by showing that this question remains NP-complete for arbitrary planar unit disk graphs. Finally, we study the minimum size of such a code in random unit disk graphs, and we prove that with probability tending to one, it is of size (n/r)^{2/3+o(1)} if r ≤ √2/2 − ε is chosen such that nr^2 → ∞ and of size n^{1+o(1)} if nr^2 ≪ ln n

Mixed-Weight Open Locating-Dominating Sets

The detection and location of issues in a network is a common problem encompassing a wide variety of research areas. Location-detection problems have been studied for wireless sensor networks and environmental monitoring, microprocessor fault detection, public utility contamination, and finding intruders in buildings. Modeling these systems as a graph, we want to find the smallest subset of nodes that, when sensors are placed at those locations, can detect and locate any anomalies that arise. One type of set that solves this problem is the open locating-dominating set (OLD-set), a set of nodes that forms a unique and nonempty neighborhood with every node in the graph. For this work, we begin with a study of OLD-sets in circulant graphs. Circulant graphs are a group of regular cyclic graphs that are often used in massively parallel systems. We prove the optimal OLD-set size for two circulant graphs using two proof techniques: the discharging method and Hall\u27s Theorem. Next we introduce the mixed-weight open locating-dominating set (mixed-weight OLD-set), an extension of the OLD-set. The mixed-weight OLD-set allows nodes in the graph to have different weights, representing systems that use sensors of varying strengths. This is a novel approach to the study of location-detection problems. We show that the decision problem for the minimum mixed-weight OLD-set, for any weights up to positive integer d, is NP-complete. We find the size of mixed-weight OLD-sets in paths and cycles for weights 1 and 2. We consider mixed-weight OLD-sets in random graphs by providing probabilistic bounds on the size of the mixed-weight OLD-set and use simulation to reinforce the theoretical results. Finally, we build and study an integer linear program to solve for mixed-weight OLD-sets and use greedy algorithms to generate mixed-weight OLD-set estimates in random geometric graphs. We also extend our results for mixed-weight OLD-sets in random graphs to random geometric graphs by estimating the probabilistic upper bound for the size of the set

Bounds for identifying codes in terms of degree parameters

An identifying code is a subset of vertices of a graph such that each vertex is uniquely determined by its neighbourhood within the identifying code. If \M(G) denotes the minimum size of an identifying code of a graph $G$, it was conjectured by F. Foucaud, R. Klasing, A. Kosowski and A. Raspaud that there exists a constant $c$ such that if a connected graph $G$ with $n$ vertices and maximum degree $d$ admits an identifying code, then \M(G)\leq n-\tfrac{n}{d}+c. We use probabilistic tools to show that for any $d\geq 3$, \M(G)\leq n-\tfrac{n}{\Theta(d)} holds for a large class of graphs containing, among others, all regular graphs and all graphs of bounded clique number. This settles the conjecture (up to constants) for these classes of graphs. In the general case, we prove \M(G)\leq n-\tfrac{n}{\Theta(d^{3})}. In a second part, we prove that in any graph $G$ of minimum degree $\delta$ and girth at least 5, \M(G)\leq(1+o_\delta(1))\tfrac{3\log\delta}{2\delta}n. Using the former result, we give sharp estimates for the size of the minimum identifying code of random $d$-regular graphs, which is about $\tfrac{\log d}{d}n$

Locating and Identifying Codes in Circulant Networks

A set S of vertices of a graph G is a dominating set of G if every vertex u of G is either in S or it has a neighbour in S. In other words, S is dominating if the sets S\cap N[u] where u \in V(G) and N[u] denotes the closed neighbourhood of u in G, are all nonempty. A set S \subseteq V(G) is called a locating code in G, if the sets S \cap N[u] where u \in V(G) \setminus S are all nonempty and distinct. A set S \subseteq V(G) is called an identifying code in G, if the sets S\cap N[u] where u\in V(G) are all nonempty and distinct. We study locating and identifying codes in the circulant networks C_n(1,3). For an integer n>6, the graph C_n(1,3) has vertex set Z_n and edges xy where x,y \in Z_n and |x-y| \in {1,3}. We prove that a smallest locating code in C_n(1,3) has size \lceil n/3 \rceil + c, where c \in {0,1}, and a smallest identifying code in C_n(1,3) has size \lceil 4n/11 \rceil + c', where c' \in {0,1}

Identifying codes in vertex-transitive graphs and strongly regular graphs

We consider the problem of computing identifying codes of graphs and its fractional relaxation. The ratio between the size of optimal integer and fractional solutions is between 1 and 2ln(vertical bar V vertical bar) + 1 where V is the set of vertices of the graph. We focus on vertex-transitive graphs for which we can compute the exact fractional solution. There are known examples of vertex-transitive graphs that reach both bounds. We exhibit infinite families of vertex-transitive graphs with integer and fractional identifying codes of order vertical bar V vertical bar(alpha) with alpha is an element of{1/4, 1/3, 2/5}These families are generalized quadrangles (strongly regular graphs based on finite geometries). They also provide examples for metric dimension of graphs

Identification, location-domination and metric dimension on interval and permutation graphs. II. Algorithms and complexity

We consider the problems of finding optimal identifying codes, (open) locating-dominating sets and resolving sets (denoted Identifying Code, (Open) Open Locating-Dominating Set and Metric Dimension) of an interval or a permutation graph. In these problems, one asks to distinguish all vertices of a graph by a subset of the vertices, using either the neighbourhood within the solution set or the distances to the solution vertices. Using a general reduction for this class of problems, we prove that the decision problems associated to these four notions are NP-complete, even for interval graphs of diameter 2 and permutation graphs of diameter 2. While Identifying Code and (Open) Locating-Dominating Set are trivially fixed-parameter-tractable when parameterized by solution size, it is known that in the same setting Metric Dimension is W[2]-hard. We show that for interval graphs, this parameterization of Metric Dimension is fixed-parameter-tractable

Optimization Approaches for Open-Locating Dominating Sets

An Open Locating-Dominating Set (OLD set) is a subset of vertices in a graph such that every vertex in the graph has a neighbor in the OLD set and every vertex has a unique set of neighbors in the OLD set. This can also represent where sensors, capable of detecting an event occurrence at an adjacent vertex, could be placed such that one could always identify the location of an event by the specific vertices that indicated an event occurred in their neighborhood. By the open neighborhood construct, which differentiates OLD sets from identifying codes, a vertex is not able to report if it is the location of the event. This construct provides a robustness over identifying codes and opens new applications such as disease carrier and dark actor identification in networks. This work explores various aspects of OLD sets, beginning with an Integer Linear Program for quickly identifying the optimal OLD set on a graph. As many graphs do not admit OLD sets, or there may be times when the total size of the set is limited by an external factor, a concept called maximum covering OLD sets is developed and explored. The coverage radius of the sensors is then expanded in a presentation of Mixed-Weight OLD sets where sensors can cover more than just adjacent vertices. Finally, an application is presented to optimally monitor criminal and terrorist networks using OLD sets and related concepts to identify the optimal set of surveillance targets

Optimisation problems in wireless sensor networks : Local algorithms and local graphs

This thesis studies optimisation problems related to modern large-scale distributed systems, such as wireless sensor networks and wireless ad-hoc networks. The concrete tasks that we use as motivating examples are the following: (i) maximising the lifetime of a battery-powered wireless sensor network, (ii) maximising the capacity of a wireless communication network, and (iii) minimising the number of sensors in a surveillance application. A sensor node consumes energy both when it is transmitting or forwarding data, and when it is performing measurements. Hence task (i), lifetime maximisation, can be approached from two different perspectives. First, we can seek for optimal data flows that make the most out of the energy resources available in the network; such optimisation problems are examples of so-called max-min linear programs. Second, we can conserve energy by putting redundant sensors into sleep mode; we arrive at the sleep scheduling problem, in which the objective is to find an optimal schedule that determines when each sensor node is asleep and when it is awake. In a wireless network simultaneous radio transmissions may interfere with each other. Task (ii), capacity maximisation, therefore gives rise to another scheduling problem, the activity scheduling problem, in which the objective is to find a minimum-length conflict-free schedule that satisfies the data transmission requirements of all wireless communication links. Task (iii), minimising the number of sensors, is related to the classical graph problem of finding a minimum dominating set. However, if we are not only interested in detecting an intruder but also locating the intruder, it is not sufficient to solve the dominating set problem; formulations such as minimum-size identifying codes and locating–dominating codes are more appropriate. This thesis presents approximation algorithms for each of these optimisation problems, i.e., for max-min linear programs, sleep scheduling, activity scheduling, identifying codes, and locating–dominating codes. Two complementary approaches are taken. The main focus is on local algorithms, which are constant-time distributed algorithms. The contributions include local approximation algorithms for max-min linear programs, sleep scheduling, and activity scheduling. In the case of max-min linear programs, tight upper and lower bounds are proved for the best possible approximation ratio that can be achieved by any local algorithm. The second approach is the study of centralised polynomial-time algorithms in local graphs – these are geometric graphs whose structure exhibits spatial locality. Among other contributions, it is shown that while identifying codes and locating–dominating codes are hard to approximate in general graphs, they admit a polynomial-time approximation scheme in local graphs