Finding Paths Between Graph Colourings: PSPACE-Completeness and Superpolynomial Distances

Bonsma, Paul; Cereceda, Luis

doi:10.1007/978-3-540-74456-6_65

Cited by 89 publications

(195 citation statements)

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“…• Token Sliding (TS rule) [6,7,10,11,19,23]: This rule corresponds to sliding token, that is, we can slide a single token only along an edge of a graph.…”

Section: Related and Known Resultsmentioning

confidence: 99%

“…2 for example.) However, ISReconf is PSPACE-complete under any of the three reconfiguration rules for planar graphs [6,10,11], for perfect graphs [19], and for bounded bandwidth graphs [23]. The PSPACE-hardness implies that, unless NP = PSPACE, there exists an instance of sliding token which requires a super-polynomial number of tokenslides even in a minimum-length reconfiguration sequence.…”

Section: Related and Known Resultsmentioning

confidence: 99%

“…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%

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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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“…• Token Sliding (TS rule) [6,7,10,11,19,23]: This rule corresponds to sliding token, that is, we can slide a single token only along an edge of a graph.…”

Section: Related and Known Resultsmentioning

confidence: 99%

Section: Related and Known Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Polynomial-Time Algorithm for Sliding Tokens on Trees

Demaine¹,

Demaine²,

Fox-Epstein

et al. 2014

Algorithms and Computation

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“…The problem of deciding whether two 3-colourings of a graph G are in the same component of R 3 (G) was shown to be solvable in time O(n 2 ) in [12]; it was also proved that the diameter of any component of R 3 (G) is O(n 2 ). In contrast, in [5] the analagous problem for k-colourings, k ≥ 4, was shown to be PSPACE-complete, and examples of reconfiguration graphs with components of superpolynomial diameter were given. In [2], reconfiguration graphs of k-colourings of chordal graphs were shown to be connected with diameter O(n 2 ) whenever k is more than the size of the largest clique (and an infinite class of chordal graphs was described whose reconfiguration graphs have diameter Ω(n 2 )).…”

Section: Introductionmentioning

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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“…2 is an optimal solution for A 0 = {5, 6} and A t = {6, 11}; its objective value is 8. Reconfiguration problems have been studied extensively in recent literature, such as SAT reconfiguration [7,18,19], independent set reconfiguration [9,10,12,16], shortest path reconfiguration [3,15], vertex-coloring reconfiguration [2,4], list edge-coloring reconfiguration [13,14], etc. However, reconfiguration problems for subset sum have not been studied yet.…”

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confidence: 99%

Approximability of the Subset Sum Reconfiguration Problem

Ito

Demaine²

2011

Lecture Notes in Computer Science

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The subset sum problem is a well-known NP-complete problem in which we wish to find a packing (subset) of items (integers) into a knapsack with capacity so that the sum of the integers in the packing is at most the capacity of the knapsack and at least a given integer threshold. In this paper, we study the problem of reconfiguring one packing into another packing by moving only one item at a time, while at all times maintaining the feasibility of packings. First we show that this decision problem is strongly NP-hard, and is PSPACE-complete if we are given a conflict graph for the set of items in which each vertex corresponds to an item and each edge represents a pair of items that are not allowed to be packed together into the knapsack. We then study an optimization version of the problem: we wish to maximize the minimum sum among all packings in a reconfiguration. We show that this maximization problem admits a polynomial-time approximation scheme (PTAS), while the problem is APX-hard if we are given a conflict graph.

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Finding Paths Between Graph Colourings: PSPACE-Completeness and Superpolynomial Distances

Cited by 89 publications

References 5 publications

Polynomial-Time Algorithm for Sliding Tokens on Trees

Polynomial-Time Algorithm for Sliding Tokens on Trees

A Reconfigurations Analogue of Brooks’ Theorem

Approximability of the Subset Sum Reconfiguration Problem

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