Distributed Reconfiguration of Maximal Independent Sets

by   Keren Censor-Hillel, et al.

Consider the following problem: given a graph and two maximal independent sets (MIS), is there a valid sequence of independent sets starting from the first one and ending in the second, in which a single node is inserted to or removed from the set at each step? While this would be trivial without any restrictions by simply removing all the nodes and then inserting the required ones, this problem, called the MIS reconfiguration problem, has been studied in the centralized setting with the caveat that intermediate sets in the sequence (schedule) must be at least of a certain size. In this paper, we investigate a distributed MIS reconfiguration problem, in which nodes can be inserted or removed from the sets concurrently. Each node of the graph is aware of its membership in the initial and final independent sets, and the nodes communicate with their neighbors in order to produce a reconfiguration schedule. The schedule is restricted by forbidding two neighbors to change their membership status at the same step. Here, we do not impose a lower bound on the size of the intermediate independent sets, as this would be hard to coordinate in a non-centralized fashion. However, we do want the independent sets to be non-trivial. We show that obtaining an actual MIS (and even a 3-dominating set) in each intermediate step is impossible. However, we provide efficient solutions when the intermediate sets are only required to be independent and 4-dominating. We prove that a constant length schedule can be found in O(MIS+R32) rounds, where MIS is the complexity of finding an MIS on a worst-case graph and R32 is the complexity of finding a (3,2)-ruling set. For bounded degree graphs, this is O(^*n) rounds and we show that it is necessary. On the other extreme, we show that with a constant number of rounds we can find a linear length schedule.


page 1

page 2

page 3

page 4


Lower bounds for maximal matchings and maximal independent sets

There are distributed graph algorithms for finding maximal matchings and...

Distributed Distance-r Dominating Set on Sparse High-Girth Graphs

The dominating set problem and its generalization, the distance-r domina...

Tight Complexity Bounds for Counting Generalized Dominating Sets in Bounded-Treewidth Graphs Part II: Hardness Results

For a well-studied family of domination-type problems, in bounded-treewi...

Improved Distributed Lower Bounds for MIS and Bounded (Out-)Degree Dominating Sets in Trees

Recently, Balliu, Brandt, and Olivetti [FOCS '20] showed the first ω(log...

Tight Complexity Bounds for Counting Generalized Dominating Sets in Bounded-Treewidth Graphs

We investigate how efficiently a well-studied family of domination-type ...

Online matching in lossless expanders

Bauwens and Zimand [BZ 2019] have shown that lossless expanders have an ...

Distributed Recoloring

Given two colorings of a graph, we consider the following problem: can w...

Please sign up or login with your details

Forgot password? Click here to reset