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.
Robustness
dynamic network
temporal graphs
edge removal
connectivity
footprint
packing/covering problems
maximal independent set
maximal matching
minimum dominating set
perfect matching
NP-hardness
Theory of computation~Design and analysis of algorithms
Mathematics of computing~Discrete mathematics
7:1-7:15
Regular Paper
ANR SKYDATA (ANR-22-CE25-0008-02) ANR GrR (ANR-18-CE40-0032).
We thank Nicolas El Maalouly for fruitful discussions about perfect matchings and Dennis Olivetti for pointing out reference [Summer, 1979].
Swan
Dubois
Swan Dubois
Sorbonne Université, CNRS, LIP6, DELYS, Paris, France
https://orcid.org/0000-0003-2320-6178
Laurent
Feuilloley
Laurent Feuilloley
Univ Lyon, CNRS, INSA Lyon, UCBL, LIRIS, UMR5205, Villeurbanne, France
https://orcid.org/0000-0002-3994-0898
Franck
Petit
Franck Petit
Sorbonne Université, CNRS, LIP6, DELYS, Paris, France
https://orcid.org/0000-0002-0948-7842
Mikaël
Rabie
Mikaël Rabie
Université Paris Cité, CNRS, IRIF, Paris, France
10.4230/LIPIcs.SAND.2023.7
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Swan Dubois, Laurent Feuilloley, Franck Petit, and Mikaël Rabie
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