The Complexity of Bounded Length Graph Recoloring and CSP Reconfiguration

Bonsma, Paul; Mouawad, Amer E.; Nishimura, Nobuya; Raman, Venkatesh

doi:10.1007/978-3-319-13524-3_10

“…In contrast, for k ≥ 4, there are both yes-instances and no-instances for connectivity [31], and there exists a family of graphs and a k ≥ 4 such that for every graph in the family there exist components of diameter superpolynomial in |V(G)| [35]. Moreover, for k ≥ 4, reachability is strongly NP-hard [79] and PSPACE-complete, even for bipartite graphs, planar graphs for 4 ≤ k ≤ 6, and bipartite planar graphs for k = 4 [35].…”

Section: K-coloring Reconfigurationmentioning

confidence: 99%

Introduction to Reconfiguration

Nishimura

¹

2018

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Reconfiguration is concerned with relationships among solutions to a problem instance, where the reconfiguration of one solution to another is a sequence of steps such that each step produces an intermediate feasible solution. The solution space can be represented as a reconfiguration graph, where two vertices representing solutions are adjacent if one can be formed from the other in a single step. Work in the area encompasses both structural questions (Is the reconfiguration graph connected?) and algorithmic ones (How can one find the shortest sequence of steps between two solutions?) This survey discusses techniques, results, and future directions in the area.

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“…In contrast, for k ≥ 4, there are both yes-instances and no-instances for connectivity [31], and there exists a family of graphs and a k ≥ 4 such that for every graph in the family there exist components of diameter superpolynomial in |V(G)| [35]. Moreover, for k ≥ 4, reachability is strongly NP-hard [79] and PSPACE-complete, even for bipartite graphs, planar graphs for 4 ≤ k ≤ 6, and bipartite planar graphs for k = 4 [35].…”

Section: K-coloring Reconfigurationmentioning

confidence: 99%

“…For bounded-degree graphs, reachability can be solved in constant time when ∆(G) ≤ k − 2, linear time when k ≥ 3 and ∆(G) = k − 1, and quadratic time when k = 3 and ∆(G) ≥ 3 [81]. With respect to parameterized complexity, the problem is fixed-parameter tractable when parameterized by k + (where is the length of the reconfiguration sequence) [79,87] but W[1]-hard when parameterized by alone [79]. The problem of finding a path for k = ∆(G) + 1 was shown to be solvable in quadratic time for any G such that ∆(G) ≥ 1 and col(G) = ∆(G) − 1 [80]; by a recent result applying to ∆(G)-regular graphs, the result was extended to hold for any connected graph with ∆(G) ≥ 3 [88].…”

Section: K-coloring Reconfigurationmentioning

confidence: 99%

“…The reachability problem for LIST COLORING RECONFIGURATION has been shown to be PSPACE-complete for graphs of bounded bandwidth [47] as well as for complete split graphs (with modular-width zero) [44], and to be W[1]-hard parameterized by size of minimum vertex cover [75]. Fixed-parameter algorithms have been found when parameterized by k + (where is the length of the reconfiguration sequence) [79,87], parameterized by k and modular-width of input graph (and hence for cographs when parameterized by k) [75], and for shortest transformation, parameterized by k and the size of the minimum vertex cover (and hence for split graphs parameterized by k) [75]. Other variants for which reconfiguration has been studied include LIST EDGE-COLORING [36,40,100], LIST(2,1)-LABELING [37], CIRCULAR COLORING [101,102], ACYCLIC COLORING [103], and EQUITABLE COLORING [103].…”

Section: Variants Of Coloringmentioning

confidence: 99%

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Introduction to Reconfiguration

Nishimura¹

2017

Preprint

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Reconfiguration is concerned with relationships among solutions to a problem instance, where the reconfiguration of one solution to another is a sequence of steps such that each step produces an intermediate feasible solution. The solution space can be represented as a reconfiguration graph, where two vertices representing solutions are adjacent if one can be formed from the other in a single step. Work in the area encompasses both structural questions (Is the reconfiguration graph connected?) and algorithmic ones (How can one find the shortest sequence of steps between two solutions?) This survey discusses techniques, results, and future directions in the area.

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“…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

¹

,

Johnson

²

,

Paulusma

³

2014

Mathematical Foundations of Computer Science 2014

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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 Complexity of Bounded Length Graph Recoloring and CSP Reconfiguration

Cited by 33 publications

References 24 publications

Introduction to Reconfiguration

Introduction to Reconfiguration

Introduction to Reconfiguration

A Reconfigurations Analogue of Brooks’ Theorem

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