We present a self-stabilizing algorithm for the unison problem which is efficient in time, workload, and space in a weak model. Precisely, our algorithm is defined in the atomic-state model and works in anonymous asynchronous connected networks in which even local ports are unlabeled. It makes no assumption on the daemon and thus stabilizes under the weakest one: the distributed unfair daemon. In an n-node network of diameter D and assuming the knowledge B >= 2D + 2, our algorithm only requires Theta(log(B)) bits per node and is fully polynomial as it stabilizes in at most 2D + 2 rounds and O(min(n(2)B, n(3))) moves. In particular, it is the first self-stabilizing unison for arbitrary asynchronous anonymous networks achieving an asymptotically optimal stabilization time in rounds using a bounded memory at each node. Furthermore, we show that our solution can be used to efficiently simulate synchronous self-stabilizing algorithms in asynchronous environments. For example, this simulation allows us to design a new state-of-the-art algorithm solving both the leader election and the BFS (Breadth-First Search) spanning tree construction in any identified connected network which, to the best of our knowledge, beats all existing solutions in the literature.

Distributed systems are ubiquitous, and their distributed nature make them particularly vulnerable to faults. Being able to automatically recover from these faults is of utmost importance, and self-stabilization is a general and lightweight approach to tackle this problem. However, fully asynchronous self-stabilizing algorithms (FASS) are notoriously difficult to design and prove. It thus makes sense to create and prove a transformer that turns synchronous algorithms into FASSes. The Rollback Compiler of Awerbuch and Varghese (FOCS 1991) is such a transformer. The problem is that although it produces FASSes that are fast (time being evaluated in rounds), we prove that their energy requirement (measured through the number of state changes) can be exponential in the number n of nodes. Actually, regardless of the problem, the literature only contains a few FASSes which are asymptotically optimal time-wise. Moreover, several of these algorithms have been shown to require an exponential amount of energy, and no such algorithms are known to be energy-efficient. In this paper, we introduce the first transformer that turns any terminating synchronous algorithm into a fully asynchronous self-stabilizing algorithm with essentially the same time complexity and with a low energy requirement (polynomial in n ). However, as for the rollback compiler, this comes at the cost of a (reasonable) memory increase. Our approach is compatible with most models, ranging from the LOCAL model (a powerful synchronous fault-free model), down to models such as the Stone Age model, in which a node cannot even know how many neighbors it has. In particular, we can transform extremely fast algorithms such as the classical Θ{ log*n )-time ring coloring algorithm by Cole and Vishkin (FOCS 1986) into fast and energy-efficient FASSes. We also provide the best FASSes so far for many classical distributed problems such as leader election and spanning tree constructions ( e.g. , BFS, shortest-path).

This paper deals with the trade-off between time, workload, and versatility in self-stabilization, a general and lightweight fault-tolerant concept in distributed computing.In this context, we propose a transformer that provides an asynchronous silent self-stabilizing version Trans(AlgI) of any terminating synchronous algorithm AlgI. The transformed algorithm Trans(AlgI) works under the distributed unfair daemon and is efficient both in moves and rounds.Our transformer allows to easily obtain fully-polynomial silent self-stabilizing solutions that are also asymptotically optimal in rounds.We illustrate the efficiency and versatility of our transformer with several efficient (i.e., fully-polynomial) silent self-stabilizing instances solving major distributed computing problems, namely vertex coloring, Breadth-First Search (BFS) spanning tree construction, k-clustering, and leader election.

We present a self-stabilizing algorithm for the (asynchronous) unison problem which achieves an efficient trade-off between time, workload, and space in a weak model. Precisely, our algorithm is defined in the atomic-state model and works in anonymous networks in which even local ports are unlabeled. It makes no assumption on the daemon and thus stabilizes under the weakest one: the distributed unfair daemon. In a $n$-node network of diameter $D$ and assuming a period $B \geq 2D+2$, our algorithm only requires $O(\log B)$ bits per node to achieve full polynomiality as it stabilizes in at most $2D-2$ rounds and $O(\min(n^2B, n^3))$ moves. In particular and to the best of our knowledge, it is the first self-stabilizing unison for arbitrary anonymous networks achieving an asymptotically optimal stabilization time in rounds using a bounded memory at each node. Finally, we show that our solution allows to efficiently simulate synchronous self-stabilizing algorithms in an asynchronous environment. This provides a new state-of-the-art algorithm solving both the leader election and the spanning tree construction problem in any identified connected network which, to the best of our knowledge, beat all existing solutions of the literature.

A proof-labeling scheme (PLS) for a boolean predicate $\Pi$ on labeled graphs is a mechanism used for certifying the legality with respect to $\Pi$ of global network states in a distributed manner. In a PLS, a certificate is assigned to each processing node of the network, and the nodes are in charge of checking that the collection of certificates forms a global proof that the system is in a correct state, by exchanging the certificates once, between neighbors only. The main measure of complexity is the size of the certificates. Many PLSs have been designed for certifying specific predicates, including cycle-freeness, minimum-weight spanning tree, planarity, etc. In 2021, a breakthrough has been obtained, as a meta-theorem stating that a large set of properties have compact PLSs in a large class of networks. Namely, for every $\mathrm{MSO}_2$ property $\Pi$ on labeled graphs, there exists a PLS for $\Pi$ with $O(\log n)$-bit certificates for all graphs of bounded tree-depth. This result has been extended to the larger class of graphs with bounded {tree-width}, using certificates on $O(\log^2 n)$ bits. We extend this result even further, to the larger class of graphs with bounded clique-width, which, as opposed to the other two aforementioned classes, includes dense graphs. We show that, for every $\mathrm{MSO}_1$ property $\Pi$ on labeled graphs, there exists a PLS for $\Pi$ with $O(\log^2 n)$ bit certificates for all graphs of bounded clique-width.

Let s1, t1,. . . sk, tk be vertices in a graph G embedded on a surface \sigma of genus g. A vertex v of G is "redundant" if there exist k vertex disjoint paths linking si and ti (1 \lequal i \lequal k) in G if and only if such paths also exist in G - v. Robertson and Seymour proved in Graph Minors VII that if v is "far" from the vertices si and tj and v is surrounded in a planar part of \sigma by l(g, k) disjoint cycles, then v is redundant. Unfortunately, their proof of the existence of l(g, k) is not constructive. In this paper, we give an explicit single exponential bound in g and k.

We give a simple proof of the "tree-width duality theorem" of Seymour and Thomas that the tree-width of a finite graph is exactly one less than the largest order of its brambles.

In Graph minors III, Robertson and Seymour write: ''It seems that the tree-width of a planar graph and the tree-width of its geometric dual are approximately equal - indeed, we have convinced ourselves that they differ by at most one.'' They never gave a proof of this. In this paper, we prove a generalisation of this statement to embedding of hypergraphs on general surfaces, and we prove that our bound is tight.

In a recent paper, Amini et al. introduced a general framework to prove duality theorems between tree-decompositions and their dual combinatorial object. They unify all known ad hoc proofs in one duality theorem based on submodular partition functions. This general theorem remains however a bit technical and relies on this particular submodularity property. Instead of partition functions, we propose here a simple combinatorial property of a set of partitions which also gives these duality results. Our approach is both simpler, and a little bit more general.

Adapting the method introduced in Graph Minors X, we propose a new proof of the duality between the bramble number of a graph and its tree-width. Our approach is based on a new definition of submodularity on partition functions which naturally extends the usual one on set functions. The proof does not rely on Menger’s theorem, and thus generalises the original one. It thus provides a dual for matroid tree-width. One can also derive all known dual notions of other classical width-parameters from it.

Given an arbitrary graph G and a number k, it is well-known by a result of Seymour and Thomas [22] that G has treewidth strictly larger than k if and only if it has a bramble of order k + 2. Brambles are used in combinatorics as certificates proving that the treewidth of a graph is large. From an algorithmic point of view there are several algorithms computing tree-decompositions of G of width at most k, if such decompositions exist and the running time is polynomial for constant k. Nevertheless, when the treewidth of the input graph is larger than k, to our knowledge there is no algorithm constructing a bramble of order k + 2. We give here such an algorithm, running in 𝒪(n^k+4) time. For classes of graphs with polynomial number of minimal separators, we define a notion of compact brambles and show how to compute compact brambles of order k + 2 in polynomial time, not depending on k.

In graph searching, a team of searchers is aiming at capturing a fugitive moving in a graph. In the initial variant, called invisible graph searching, the searchers do not know the position of the fugitive until they catch it. In another variant, the searchers know the position of the fugitive, i.e. the fugitive is visible. This latter variant is called visible graph searching. A search strategy that catches any fugitive in such a way that, the part of the graph reachable by the fugitive never grows is called monotone. A priori, monotone strategies may require more searchers than general strategies to catch any fugitive. This is however not the case for visible and invisible graph searching. Two important consequences of the monotonicity of visible and invisible graph searching are: (1) the decision problem corresponding to the computation of the smallest number of searchers required to clear a graph is in NP, and (2) computing optimal search strategies is simplified by taking into account that there exist some that never backtrack. Fomin et al. (2005) introduced an important graph searching variant, called non-deterministic graph searching, that unifies visible and invisible graph searching. In this variant, the fugitive is invisible, and the searchers can query an oracle that knows the current position of the fugitive. The question of the monotonicity of non-deterministic graph searching is however left open. In this paper, we prove that non-deterministic graph searching is monotone. In particular, this result is a unified proof of monotonicity for visible and invisible graph searching. As a consequence, the decision problem corresponding to non-determinisitic graph searching belongs to NP. Moreover, the exact algorithms designed by Fomin et al. do compute optimal non-deterministic search strategies.

We prove that the branch-width of circular-arc graphs can be computed in polynomial time.

Minimal triangulations and potential maximal cliques are the main ingredients for a number of polynomial time algorithms on different graph classes computing the treewidth of a graph. Potential maximal cliques are also the main engine of the fastest so far 𝒪 (1.9601 n )-time exact treewidth algorithm. Based on the recent results of Mazoit, we define the structures that can be regarded as minimal triangulations and potential maximal cliques for branchwidth: efficient triangulations and blocks. We show how blocks can be used to construct an algorithm computing the branchwidth of a graph on n vertices in time (2 + √( 3) ) ^ n · n O(1).

We prove that the branchwidth of a bridgeless graph is equal to the branchwidth of its cycle matroid. Our proof is based on branch-decompositions of hypergraphs. By matroid duality, a direct corollary of this result is that the branchwidth of a bridgeless planar graph is equal to the branchwidth of its planar dual.

We adapt some decision theorems about treewidth to the branchwidth and use this theorems to prove that the branchwidth of circular-arc graphs can be computed in polynomial time.

Dans cette these, nous nous interessons a deux types de decompositions des graphes introduits par Robertson et Seymour: les decompositions arborescentes et les decompositions en branches. A ces decompositions sont associes deux parametres des graphes: la largeur arborescente et la largeur de branches. Nous montrons que ces deux decompositions peuvent etre vues comme issues d'une meme structure combinatoire; les deux parametres mentionne ci-dessus sont egaux aux valeurs minimales de deux parametres de cette structure commune. En poussant plus avant cette analogie, nous montrons comment adapter une technique de calcul de la largeur arborescente au calcul de la largeur de branches. Ceci nous permet de calculer la largeur de branches des graphes de nombre asteroide borne ayant un nombre polynomial de separateurs minimaux et celle des graphes d-trapezoides circulaires. Ce parallele nous permet aussi d'adapter certains resultats structurels sur les decompositions en branches aux decompositions arborescentes. Dans le cas des graphes planaires, nous interpretons ces proprietes a l'aide d'outils topologiques. De cette facon, nous donnons une demonstration simple d'un theoreme de dualite reliant la largeur arborescente d'un graphe planaire et celle de son dual. Ces outils nous permettent aussi d'enumerer de facon efficace les separateurs minimaux des graphes planaires.

Abstract This paper presents a tool to automatically discover the network topology. The goal is to evaluate the performance of concurrent transfers (for example to improve collective communications) and not to discover the physical machines interconnection scheme (for administration purposes). The problems encountered, preliminary algorithms to solve them, as well as theoretical proofs of their validity (under some conditions) are presented. Keywords: Grid computing, simulation, network topology, communication performance prediction R´esum´e

Robertson and Seymour conjectured that the treewidth of a planar graph and the treewidth of its geometric dual differ by at most one. Lapoire solved the conjecture in the affirmative, using algebraic techniques. We give here a much shorter proof of this result.