Reconfiguring Directed Trees in a Digraph

Ito, Takehiro; Iwamasa, Yuni; Kobayashi, Yasuaki; Nakahata, Yu; Otachi, Yota; Wasa, Kiyotaka

doi:10.1007/978-3-030-89543-3_29

Cited by 3 publications

(2 citation statements)

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“…A transformation step in the centralized setting is defined as follows: two spanning trees T and T of a graph G are reachable in one step iff there exists two edges e ∈ T and e ∈ T such that T = (T \ e) ∪ e . In the centralized setting, any spanning tree can be reconfigured into any other spanning tree in polynomial time [7] and finding a shortest reconfiguration sequence between two directed spanning trees is polynomial-time solvable [8]. Therefore, more constrained versions of the problem have been studied.…”

Section: Related Workmentioning

confidence: 99%

Brief Announcement: Distributed Reconfiguration of Spanning Trees

Gupta

Kumar

Pai

2022

Lecture Notes in Computer Science

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In a reconfiguration problem, given a problem and two feasible solutions of the problem, the task is to find a sequence of transformations to reach from one solution to the other such that every intermediate state is also a feasible solution to the problem. In this paper, we study the distributed spanning tree reconfiguration problem and we define a new reconfiguration step, called k-simultaneous add and delete, in which every node is allowed to add at most k edges and delete at most k edges such that multiple nodes do not add or delete the same edge.We first observe that, if the two input spanning trees are rooted, then we can do the reconfiguration using a single 1-simultaneous add and delete step in one round in the CONGEST model. Therefore, we focus our attention towards unrooted spanning trees and show that transforming an unrooted spanning tree into another using a single 1-simultaneous add and delete step requires Ω(n) rounds in the LOCAL model. We additionally show that transforming an unrooted spanning tree into another using a single 2-simultaneous add and delete step can be done in O(log n) rounds in the CONGEST model.

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Section: Related Workmentioning

confidence: 99%

Brief Announcement: Distributed Reconfiguration of Spanning Trees

Gupta

Kumar

Pai

2022

Lecture Notes in Computer Science

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“…There are several studies for reconfiguring orientations of a graph, such as strong orientations [15,14,20], acyclic orientations [14], nondeterministic constraint logic [17], and α-orientations [1]. Very recently, Ito et al [21] studied reconfiguration of several subgraphs in directed graphs, such as directed trees, directed paths, and directed acyclic subgraphs. Although these reconfiguration problems are defined on directed graphs, the reconfiguration rules are symmetric, meaning that if one solution X can be obtained from another solution Y by applying some reconfiguration rule, Y can be obtained from X by the same rule as well.…”

Section: Introductionmentioning

confidence: 99%

Independent set reconfiguration on directed graphs

Ito¹,

Iwamasa²,

Kobayashi³

et al. 2022

Preprint

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Directed Token Sliding asks, given a directed graph and two sets of pairwise nonadjacent vertices, whether one can reach from one set to the other by repeatedly applying a local operation that exchanges a vertex in the current set with one of its out-neighbors, while keeping the nonadjacency. It can be seen as a reconfiguration process where a token is placed on each vertex in the current set, and the local operation slides a token along an arc respecting its direction. Previously, such a problem was extensively studied on undirected graphs, where the edges have no directions and thus the local operation is symmetric. Directed Token Sliding is a generalization of its undirected variant since an undirected edge can be simulated by two arcs of opposite directions.In this paper, we initiate the algorithmic study of Directed Token Sliding. We first observe that the problem is PSPACE-complete even if we forbid parallel arcs in opposite directions and that the problem on directed acyclic graphs is NP-complete and W[1]-hard parameterized by the size of the sets in consideration. We then show our main result: a lineartime algorithm for the problem on directed graphs whose underlying undirected graphs are trees, which are called polytrees. Such a result is also known for the undirected variant of the problem on trees [Demaine et al. TCS 2015], but the techniques used here are quite different because of the asymmetric nature of the directed problem. We present a characterization of yes-instances based on the existence of a certain set of directed paths, and then derive simple equivalent conditions from it by some observations, which admits an efficient algorithm. For the polytree case, we also present a quadratic-time algorithm that outputs, if the input is a yes-instance, one of the shortest reconfiguration sequences.

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