Self-stabilizing algorithms for Connected Vertex Cover and Clique decomposition problems

In many wireless networks, there is no fixed physical backbone nor centralized network management. The nodes of such a network have to self-organize in order to maintain a virtual backbone used to route messages. Moreover, any node of the network can be a priori at the origin of a malicious attack. Thus, in one hand the backbone must be fault-tolerant and in other hand it can be useful to monitor all network communications to identify an attack as soon as possible. We are interested in the minimum \emph{Connected Vertex Cover} problem, a generalization of the classical minimum Vertex Cover problem, which allows to obtain a connected backbone. Recently, Delbot et al.~\cite{DelbotLP13} proposed a new centralized algorithm with a constant approximation ratio of $2$ for this problem. In this paper, we propose a distributed and self-stabilizing version of their algorithm with the same approximation guarantee. To the best knowledge of the authors, it is the first distributed and fault-tolerant algorithm for this problem. The approach followed to solve the considered problem is based on the construction of a connected minimal clique partition. Therefore, we also design the first distributed self-stabilizing algorithm for this problem, which is of independent interest

Lattice Linear Problems vs Algorithms

Modelling problems using predicates that induce a partial order among global states was introduced as a way to permit asynchronous execution in multiprocessor systems. A key property of such problems is that the predicate induces one lattice in the state space which guarantees that the execution is correct even if nodes execute with old information about their neighbours. Thus, a compiler that is aware of this property can ignore data dependencies and allow the application to continue its execution with the available data rather than waiting for the most recent one. Unfortunately, many interesting problems do not exhibit lattice linearity. This issue was alleviated with the introduction of eventually lattice linear algorithms. Such algorithms induce a partial order in a subset of the state space even though the problem cannot be defined by a predicate under which the states form a partial order. This paper focuses on analyzing and differentiating between lattice linear problems and algorithms. It also introduces a new class of algorithms called (fully) lattice linear algorithms. A characteristic of these algorithms is that the entire reachable state space is partitioned into one or more lattices and the initial state locks into one of these lattices. Thus, under a few additional constraints, the initial state can uniquely determine the final state. For demonstration, we present lattice linear self-stabilizing algorithms for minimal dominating set and graph colouring problems, and a parallel processing 2-approximation algorithm for vertex cover. The algorithm for minimal dominating set converges in n moves, and that for graph colouring converges in n+2m moves. The algorithm for vertex cover is the first lattice linear approximation algorithm for an NP-Hard problem; it converges in n moves. Some part is cut due to 1920 character limit. Please see the pdf for full abstract.Comment: arXiv admin note: text overlap with arXiv:2209.1470

Polynomial Silent Self-Stabilizing p-Star Decomposition

We present a silent self-stabilizing distributed algorithm computing a maximal p-star decomposition of the underlying communication network. Under the unfair distributed scheduler, the most general scheduler model, the algorithm converges in at most 12∆m + O(m + n) moves, where m is the number of edges, n is the number of nodes, and ∆ is the maximum node degree. Regarding the move complexity, our algorithm outperforms the previously known best algorithm by a factor of ∆. While the round complexity for the previous algorithm was unknown, we show a 5 [n/(p+1)] + 5 bound for our algorithm

Towards an Isomorphism Dichotomy for Hereditary Graph Classes

In this paper we resolve the complexity of the isomorphism problem on all but finitely many of the graph classes characterized by two forbidden induced subgraphs. To this end we develop new techniques applicable for the structural and algorithmic analysis of graphs. First, we develop a methodology to show isomorphism completeness of the isomorphism problem on graph classes by providing a general framework unifying various reduction techniques. Second, we generalize the concept of the modular decomposition to colored graphs, allowing for non-standard decompositions. We show that, given a suitable decomposition functor, the graph isomorphism problem reduces to checking isomorphism of colored prime graphs. Third, we extend the techniques of bounded color valence and hypergraph isomorphism on hypergraphs of bounded color size as follows. We say a colored graph has generalized color valence at most k if, after removing all vertices in color classes of size at most k, for each color class C every vertex has at most k neighbors in C or at most k non-neighbors in C. We show that isomorphism of graphs of bounded generalized color valence can be solved in polynomial time.Comment: 37 pages, 4 figure

Self-stabilizing k-clustering in mobile ad hoc networks

In this thesis, two silent self-stabilizing asynchronous distributed algorithms are given for constructing a k-clustering of a connected network of processes. These are the first self-stabilizing solutions to this problem. One algorithm, FLOOD, takes O( k) time and uses O(k log n) space per process, while the second algorithm, BFS-MIS-CLSTR, takes O(n) time and uses O(log n) space; where n is the size of the network. Processes have unique IDs, and there is no designated leader. BFS-MIS-CLSTR solves three problems; it elects a leader and constructs a BFS tree for the network, constructs a minimal independent set, and finally a k-clustering. Finding a minimal k-clustering is known to be NP -hard. If the network is a unit disk graph in a plane, BFS-MIS-CLSTR is within a factor of O(7.2552k) of choosing the minimal number of clusters; A lower bound is given, showing that any comparison-based algorithm for the k-clustering problem that takes o( diam) rounds has very bad worst case performance; Keywords: BFS tree construction, K-clustering, leader election, MIS construction, self-stabilization, unit disk graph

Optimal Dynamic Distributed MIS

Finding a maximal independent set (MIS) in a graph is a cornerstone task in distributed computing. The local nature of an MIS allows for fast solutions in a static distributed setting, which are logarithmic in the number of nodes or in their degrees. The result trivially applies for the dynamic distributed model, in which edges or nodes may be inserted or deleted. In this paper, we take a different approach which exploits locality to the extreme, and show how to update an MIS in a dynamic distributed setting, either \emph{synchronous} or \emph{asynchronous}, with only \emph{a single adjustment} and in a single round, in expectation. These strong guarantees hold for the \emph{complete fully dynamic} setting: Insertions and deletions, of edges as well as nodes, gracefully and abruptly. This strongly separates the static and dynamic distributed models, as super-constant lower bounds exist for computing an MIS in the former. Our results are obtained by a novel analysis of the surprisingly simple solution of carefully simulating the greedy \emph{sequential} MIS algorithm with a random ordering of the nodes. As such, our algorithm has a direct application as a $3$-approximation algorithm for correlation clustering. This adds to the important toolbox of distributed graph decompositions, which are widely used as crucial building blocks in distributed computing. Finally, our algorithm enjoys a useful \emph{history-independence} property, meaning the output is independent of the history of topology changes that constructed that graph. This means the output cannot be chosen, or even biased, by the adversary in case its goal is to prevent us from optimizing some objective function.Comment: 19 pages including appendix and reference

Linear-Time Algorithms for Edge-Based Problems

There is a dearth of algorithms that deal with edge-based problems in trees, specifically algorithms for edge sets that satisfy a particular parameter. The goal of this thesis is to create a methodology for designing algorithms for these edge-based problems. We will present a variant of the Wimer method [Wimer et al. 1985] [Wimer 1987] that can handle edge properties. We call this variant the Wimer edge variant. The thesis is divided into three sections, the first being a chapter devoted to defining and discussing the Wimer edge variant in depth, showing how to develop an algorithm using this variant, and an example of this process, including a run of an algorithm developed using this method. The second section involves algorithms developed using the Wimer edge variant. We will provide algorithms for a variety of edge parameters, including four different matching parameters (connected, disconnected, induced and 2-matching), three different domination parameters (edge, total edge and edge-vertex) and two covering parameters (edge cover and edge cover irredundance). Each of these algorithms are discussed in detail and run in linear time. The third section involves an attempt to characterize the Wimer edge variant. We show how the variant can be applied to three classes of graphs: weighted trees, unicyclic graphs and generalized series-parallel graphs. For each of these classes, we detail what adaptations are required (if any) and design an algorithm, including showing a run on an example graph. The fourth chapter is devoted to a discussion of what qualities a parameter has to have in order to be likely to have a solution using the Wimer edge variant. Also in this chapter we discuss classes of graphs that can utilize the Wimer edge variant. Other topics discussed in this thesis include a literature review, and a discussion of future work. There are plenty of options for future work on this topic, which hopefully this thesis can inspire. The intent of this thesis is to provide the foundation for future algorithms and other work in this area