Simple dynamic algorithms for Maximal Independent Set, Maximum Flow and Maximum Matching

Peer reviewe

All Maximal Independent Sets and Dynamic Dominance for Sparse Graphs

We describe algorithms, based on Avis and Fukuda's reverse search paradigm, for listing all maximal independent sets in a sparse graph in polynomial time and delay per output. For bounded degree graphs, our algorithms take constant time per set generated; for minor-closed graph families, the time is O(n) per set, and for more general sparse graph families we achieve subquadratic time per set. We also describe new data structures for maintaining a dynamic vertex set S in a sparse or minor-closed graph family, and querying the number of vertices not dominated by S; for minor-closed graph families the time per update is constant, while it is sublinear for any sparse graph family. We can also maintain a dynamic vertex set in an arbitrary m-edge graph and test the independence of the maintained set in time O(sqrt m) per update. We use the domination data structures as part of our enumeration algorithms.Comment: 10 page

Unifying discrete and continuous Weyl-Titchmarsh theory via a class of linear Hamiltonian systems on Sturmian time scales

In this study, we are concerned with introducing Weyl-Titchmarsh theory for a class of dynamic linear Hamiltonian nabla systems over a half-line on Sturmian time scales. After developing fundamental properties of solutions and regular spectral problems, we introduce the corresponding maximal and minimal operators for the system. Matrix disks are constructed and proved to be nested and converge to a limiting set. Some precise relationships among the rank of the matrix radius of the limiting set, the number of linearly independent square summable solutions, and the defect indices of the minimal operator are established. Using the above results, a classification of singular dynamic linear Hamiltonian nabla systems is given in terms of the defect indices of the minimal operator, and several equivalent conditions on the cases of limit point and limit circle are obtained, respectively. These results unify and extend certain classic and recent results on the subject in the continuous and discrete cases, respectively, to Sturmian time scales.Comment: 34 page

When Should You Wait Before Updating? - Toward a Robustness Refinement

Consider a dynamic network and a given distributed problem. At any point in time, there might exist several solutions that are equally good with respect to the problem specification, but that are different from an algorithmic perspective, because some could be easier to update than others when the network changes. In other words, one would prefer to have a solution that is more robust to topological changes in the network; and in this direction the best scenario would be that the solution remains correct despite the dynamic of the network. In [Arnaud Casteigts et al., 2020], the authors introduced a very strong robustness criterion: they required that for any removal of edges that maintain the network connected, the solution remains valid. They focus on the maximal independent set problem, and their approach consists in characterizing the graphs in which there exists a robust solution (the existential problem), or even stronger, where any solution is robust (the universal problem). As the robustness criteria is very demanding, few graphs have a robust solution, and even fewer are such that all of their solutions are robust. In this paper, we ask the following question: Can we have robustness for a larger class of networks, if we bound the number k of edge removals allowed? To answer this question, we consider three classic problems: maximal independent set, minimal dominating set and maximal matching. For the universal problem, the answers for the three cases are surprisingly different. For minimal dominating set, the class does not depend on the number of edges removed. For maximal matching, removing only one edge defines a robust class related to perfect matchings, but for all other bounds k, the class is the same as for an arbitrary number of edge removals. Finally, for maximal independent set, there is a strict hierarchy of classes: the class for the bound k is strictly larger than the class for bound k+1. For the robustness notion of [Arnaud Casteigts et al., 2020], no characterization of the class for the existential problem is known, only a polynomial-time recognition algorithm. We show that the situation is even worse for bounded k: even for k = 1, it is NP-hard to decide whether a graph has a robust maximal independent set

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