The complexity of rerouting shortest paths

Bonsma, Paul

doi:10.1016/j.tcs.2013.09.012

Cited by 63 publications

(74 citation statements)

References 18 publications

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“…We distinguish the following two cases: First, we assume that each internal vertex of T is b-tight. Then we pick S = v 0 v 1 and R = v 1 . .…”

Section: T ∩ M Is Internally 1-reconfigurable To T ∩ N and 2 T ∩ Nmentioning

confidence: 99%

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Degree-Constrained Subgraph Reconfiguration is in P

Mühlenthaler

2015

Mathematical Foundations of Computer Science 2015

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The degree-constrained subgraph problem asks for a subgraph of a given graph such that the degree of each vertex is within some specified bounds. We study the following reconfiguration variant of this problem: Given two solutions to a degree-constrained subgraph instance, can we transform one solution into the other by adding and removing individual edges, such that each intermediate subgraph satisfies the degree constraints and contains at least a certain minimum number of edges? This problem is a generalization of the matching reconfiguration problem, which is known to be in P. We show that even in the more general setting the reconfiguration problem is in P.

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“…We distinguish the following two cases: First, we assume that each internal vertex of T is b-tight. Then we pick S = v 0 v 1 and R = v 1 . .…”

Section: T ∩ M Is Internally 1-reconfigurable To T ∩ N and 2 T ∩ Nmentioning

confidence: 99%

“…Then T ∩ M is not internally k-reconfigurable to T ∩ N for any k ≥ 1, since no edges of T ∩ M can be removed and no edges of T ∩ N can be added without violating the degree constraints. If T is b-tight, i.e., 3a is violated, then we choose R = v 1 . .…”

Section: T ∩ M Is Internally 1-reconfigurable To T ∩ N and 2 T ∩ Nmentioning

confidence: 99%

Degree-Constrained Subgraph Reconfiguration is in P

Mühlenthaler

2015

Mathematical Foundations of Computer Science 2015

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“…Reconfiguration graphs have been studied for a number of combinatorial problems; the questions asked are typically (as we have seen for colouring) is the graph connected?, what is the diameter of the graph (or of its connected components)?, how difficult is it to decide whether there is a path between a pair of given solutions? Problems studied include boolean satisfiability [13,21],clique and vertex cover [16], independent set [6,20], list edge colouring [17],shortest path [3,4], and subset sum [15] (see also a recent survey [14]). Recent work has included looking at finding the shortest path in the reconfiguration graph between given solutions [19], and studying the fixed-parameter-tractability of these problems [7,18,23,24].…”

Section: (The Answer Is Always Yes)mentioning

confidence: 99%

A Reconfigurations Analogue of Brooks’ Theorem

Feghali

Johnson

Paulusma

2014

Lecture Notes in Computer Science

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The full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-pro t purposes provided that:• a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.Please consult the full DRO policy for further details. Abstract. Let G be a simple undirected graph on n vertices with maximum degree ∆. Brooks' Theorem states that G has a ∆-colouring unless G is a complete graph, or a cycle with an odd number of vertices. To recolour G is to obtain a new proper colouring by changing the colour of one vertex. We show that from a k-colouring, k > ∆, a ∆-colouring of G can be obtained by a sequence of O(n 2 ) recolourings using only the original k colours unless -G is a complete graph or a cycle with an odd number of vertices, or -k = ∆ + 1, G is ∆-regular and, for each vertex v in G, no two neighbours of v are coloured alike. We use this result to study the reconfiguration graph R k (G) of the kcolourings of G. The vertex set of R k (G) is the set of all possible kcolourings of G and two colourings are adjacent if they differ on exactly one vertex. It is known that -if k ≤ ∆(G), then R k (G) might not be connected and it is possible that its connected components have superpolynomial diameter, -if k ≥ ∆(G) + 2, then R k (G) is connected and has diameter O(n 2 ). We complete this structural classification by settling the missing case:-if k = ∆(G) + 1, then R k (G) consists of isolated vertices and at most one further component which has diameter O(n 2 ). We also describe completely the computational complexity classification of the problem of deciding whether two k-colourings of a graph G of maximum degree ∆ belong to the same component of R k (G) by settling the case k = ∆(G) + 1. The problem is -O(n 2 ) time solvable for k = 3, -PSPACE-complete for 4 ≤ k ≤ ∆(G), -O(n) time solvable for k = ∆(G) + 1, -O(1) time solvable for k ≥ ∆(G) + 2 (the answer is always yes).

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“…The problem arises when we wish to find a step-by-step transformation between two feasible solutions of a problem such that all intermediate results are also feasible and each step abides by a fixed reconfiguration rule (i.e., an adjacency relation defined on feasible solutions of the original problem). This kind of reconfiguration problem has been studied extensively for several well-known problems, including independent set [2,5,7,10,11,13,15,19,[21][22][23], satisfiability [9,20], set cover, clique, matching [13], vertexcoloring [3,6,8,23], list edge-coloring [14,17], list L(2, 1)-labeling [16], subset sum [12], shortest path [4,18], and so on.…”

Section: Introductionmentioning

confidence: 99%

Polynomial-Time Algorithm for Sliding Tokens on Trees

Demaine¹,

Demaine²,

Fox-Epstein

et al. 2014

Algorithms and Computation

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Abstract. Suppose that we are given two independent sets I b and Ir of a graph such that |I b | = |Ir|, and imagine that a token is placed on each vertex in I b . Then, the sliding token problem is to determine whether there exists a sequence of independent sets which transforms I b into Ir so that each independent set in the sequence results from the previous one by sliding exactly one token along an edge in the graph. This problem is known to be PSPACE-complete even for planar graphs, and also for bounded treewidth graphs. In this paper, we thus study the problem restricted to trees, and give the following three results: (1) the decision problem is solvable in linear time; (2) for a yes-instance, we can find in quadratic time an actual sequence of independent sets between I b and Ir whose length (i.e., the number of token-slides) is quadratic; and (3) there exists an infinite family of instances on paths for which any sequence requires quadratic length.

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The complexity of rerouting shortest paths

Cited by 63 publications

References 18 publications

Degree-Constrained Subgraph Reconfiguration is in P

Degree-Constrained Subgraph Reconfiguration is in P

A Reconfigurations Analogue of Brooks’ Theorem

Polynomial-Time Algorithm for Sliding Tokens on Trees

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