Fault-Detection in Networks

To find broken links in networks we use the cut-set space. Information on which nodes can talk, or not, to which other nodes allows reduction of the problem to that of decoding the cut-set code of a graph. Special classes of such codes are known to have polynomial-time decoding algorithms. We present a simple algorithm to achieve the reduction and apply it in two examples

Locating-dominating sets and identifying codes in graphs of girth at least 5

Locating-dominating sets and identifying codes are two closely related notions in the area of separating systems. Roughly speaking, they consist in a dominating set of a graph such that every vertex is uniquely identified by its neighbourhood within the dominating set. In this paper, we study the size of a smallest locating-dominating set or identifying code for graphs of girth at least 5 and of given minimum degree. We use the technique of vertex-disjoint paths to provide upper bounds on the minimum size of such sets, and construct graphs who come close to meet these bounds.Comment: 20 pages, 9 figure

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

Metric Dimension Parameterized by Feedback Vertex Set and Other Structural Parameters

For a graph $G$, a subset $S \subseteq V(G)$ is called a \emph{resolving set}if for any two vertices $u,v \in V(G)$, there exists a vertex $w \in S$ suchthat $d(w,u) \neq d(w,v)$. The {\sc Metric Dimension} problem takes as input agraph $G$ and a positive integer $k$, and asks whether there exists a resolvingset of size at most $k$. This problem was introduced in the 1970s and is knownto be NP-hard~[GT~61 in Garey and Johnson's book]. In the realm ofparameterized complexity, Hartung and Nichterlein~[CCC~2013] proved that theproblem is W[2]-hard when parameterized by the natural parameter $k$. They alsoobserved that it is FPT when parameterized by the vertex cover number and askedabout its complexity under \emph{smaller} parameters, in particular thefeedback vertex set number. We answer this question by proving that {\sc MetricDimension} is W[1]-hard when parameterized by the feedback vertex set number.This also improves the result of Bonnet and Purohit~[IPEC 2019] which statesthat the problem is W[1]-hard parameterized by the treewidth. Regarding theparameterization by the vertex cover number, we prove that {\sc MetricDimension} does not admit a polynomial kernel under this parameterizationunless $NP\subseteq coNP/poly$. We observe that a similar result holds when theparameter is the distance to clique. On the positive side, we show that {\scMetric Dimension} is FPT when parameterized by either the distance to clusteror the distance to co-cluster, both of which are smaller parameters than thevertex cover number.<br

Bounds and extremal graphs for total dominating identifying codes

An identifying code $C$ of a graph $G$ is a dominating set of $G$ such that any two distinct vertices of $G$ have distinct closed neighbourhoods within $C$. The smallest size of an identifying code of $G$ is denoted $\gamma^{\text{ID}}(G)$. When every vertex of $G$ also has a neighbour in $C$, it is said to be a total dominating identifying code of $G$, and the smallest size of a total dominating identifying code of $G$ is denoted by $\gamma_t^{\text{ID}}(G)$. Extending similar characterizations for identifying codes from the literature, we characterize those graphs $G$ of order $n$ with $\gamma_t^{\text{ID}}(G)=n$ (the only such connected graph is $P_3$) and $\gamma_t^{\text{ID}}(G)=n-1$ (such graphs either satisfy $\gamma^{\text{ID}}(G)=n-1$ or are built from certain such graphs by adding a set of universal vertices, to each of which a private leaf is attached). Then, using bounds from the literature, we remark that any (open and closed) twin-free tree of order $n$ has a total dominating identifying code of size at most $\frac{3n}{4}$. This bound is tight, and we characterize the trees reaching it. Moreover, by a new proof, we show that this bound actually holds for the larger class of all twin-free graphs of girth at least 5. The cycle $C_8$ also attains this bound. We also provide a generalized bound for all graphs of girth at least 5 (possibly with twins). Finally, we relate $\gamma_t^{\text{ID}}(G)$ to the related parameter $\gamma^{\text{ID}}(G)$ as well as the location-domination number of $G$ and its variants, providing bounds that are either tight or almost tight

Locating-dominating sets and identifying codes in Graphs of Girth at least 5

Locating-dominating sets and identifying codes are two closely related notions in the area of separating systems. Roughly speaking, they consist in a dominating set of a graph such that every vertex is uniquely identified by its neighbourhood within the dominating set. In this paper, we study the size of a smallest locating-dominating set or identifying code for graphs of girth at least 5 and of given minimum degree. We use the technique of vertex-disjoint paths to provide upper bounds on the minimum size of such sets, and construct graphs who come close to meeting these bounds.Award-winningPostprint (author’s final draft

Fault-tolerant Stochastic Distributed Systems

The present doctoral thesis discusses the design of fault-tolerant distributed systems, placing emphasis in addressing the case where the actions of the nodes or their interactions are stochastic. The main objective is to detect and identify faults to improve the resilience of distributed systems to crash-type faults, as well as detecting the presence of malicious nodes in pursuit of exploiting the network. The proposed analysis considers malicious agents and computational solutions to detect faults. Crash-type faults, where the affected component ceases to perform its task, are tackled in this thesis by introducing stochastic decisions in deterministic distributed algorithms. Prime importance is placed on providing guarantees and rates of convergence for the steady-state solution. The scenarios of a social network (state-dependent example) and consensus (time- dependent example) are addressed, proving convergence. The proposed algorithms are capable of dealing with packet drops, delays, medium access competition, and, in particular, nodes failing and/or losing network connectivity. The concept of Set-Valued Observers (SVOs) is used as a tool to detect faults in a worst-case scenario, i.e., when a malicious agent can select the most unfavorable sequence of communi- cations and inject a signal of arbitrary magnitude. For other types of faults, it is introduced the concept of Stochastic Set-Valued Observers (SSVOs) which produce a confidence set where the state is known to belong with at least a pre-specified probability. It is shown how, for an algorithm of consensus, it is possible to exploit the structure of the problem to reduce the computational complexity of the solution. The main result allows discarding interactions in the model that do not contribute to the produced estimates. The main drawback of using classical SVOs for fault detection is their computational burden. By resorting to a left-coprime factorization for Linear Parameter-Varying (LPV) systems, it is shown how to reduce the computational complexity. By appropriately selecting the factorization, it is possible to consider detectable systems (i.e., unobservable systems where the unobservable component is stable). Such a result plays a key role in the domain of Cyber-Physical Systems (CPSs). These techniques are complemented with Event- and Self-triggered sampling strategies that enable fewer sensor updates. Moreover, the same triggering mechanisms can be used to make decisions of when to run the SVO routine or resort to over-approximations that temporarily compromise accuracy to gain in performance but maintaining the convergence characteristics of the set-valued estimates. A less stringent requirement for network resources that is vital to guarantee the applicability of SVO-based fault detection in the domain of Networked Control Systems (NCSs)