Reconfiguring Undirected Paths

Demaine, Erik D.; Eppstein, David; Hesterberg, Adam; Jain, Kshitij; Lubiw, Anna; Uehara, Ryuhei; Uno, Yushi

doi:10.1007/978-3-030-24766-9_26

Cited by 12 publications

(12 citation statements)

References 16 publications

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“…3. Simultaneously to our work,Demaine et al (2019) also studied reconfiguration of undirected paths and found that path reconfiguration is fixed-parameter tractable when parameterized by the length of the path.…”

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confidence: 75%

The Parameterized Complexity of Motion Planning for Snake-Like Robots

Gupta

Sa'ar

Zehavi

2020

jair

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We study the parameterized complexity of a variant of the classic video game Snake that models real-world problems of motion planning. Given a snake-like robot with an initial position and a final position in an environment (modeled by a graph), our objective is to determine whether the robot can reach the final position from the initial position without intersecting itself. Naturally, this problem models a wide-variety of scenarios, ranging from the transportation of linked wagons towed by a locomotor at an airport or a supermarket to the movement of a group of agents that travel in an “ant-like” fashion and the construction of trains in amusement parks. Unfortunately, already on grid graphs, this problem is PSPACE-complete. Nevertheless, we prove that even on general graphs, the problem is solvable in FPT time with respect to the size of the snake. In particular, this shows that the problem is fixed-parameter tractable (FPT). Towards this, we show how to employ color-coding to sparsify the configuration graph of the problem to reduce its size significantly. We believe that our approach will find other applications in motion planning. Additionally, we show that the problem is unlikely to admit a polynomial kernel even on grid graphs, but it admits a treewidth-reduction procedure. To the best of our knowledge, the study of the parameterized complexity of motion planning problems (where the intermediate configurations of the motion are of importance) has so far been largely overlooked. Thus, our work is pioneering in this regard.

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confidence: 75%

The Parameterized Complexity of Motion Planning for Snake-Like Robots

Gupta

Sa'ar

Zehavi

2020

jair

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“…It would be also interesting to know whether Directed Path Reconfiguration and Directed Path Sliding are fixed-parameter tractable (FPT) parameterized by the length of input paths. Although the undirected counterparts are known to be FPT [3,4], it would be difficult to apply their techniques directly to our cases.…”

Section: Discussionmentioning

confidence: 99%

“…In particular, they showed that if S(G) consists of all trees in G, every instance of the corresponding reconfiguration problem is a yes-instance unless the two input trees have different numbers of edges. Motivated by applications in motion planning, Biasi and Ophelders [1], Demaine et al [3], and Gupta et al [4] studied some variants of reconfiguring undirected paths and showed that these problems are PSPACE-complete, while they are fixed-parameter tractable (FPT) when parameterized by the length of input paths.…”

Section: Introductionmentioning

confidence: 99%

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Reconfiguring (non-spanning) arborescences

Ito

Iwamasa²,

Kobayashi³

et al. 2021

Preprint

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In this paper, we investigate the computational complexity of subgraph reconfiguration problems in directed graphs. More specifically, we focus on the problem of determining whether, given two directed trees in a digraph, there is a (reconfiguration) sequence of directed trees such that for every pair of two consecutive trees in the sequence, one of them is obtained from the other by removing an arc and then adding another arc. We show that this problem can be solved in polynomial time, whereas the problem is PSPACE-complete when we restrict directed trees in a reconfiguration sequence to form directed paths. We also show that there is a polynomial-time algorithm for finding a shortest reconfiguration sequence between two directed spanning trees.

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“…Several other results on reconfiguration of paths are known in the literature. For example, Demaine et al [7] proved that the reachability problem for shifting paths (“Given two paths in a graph, can one be transformed into the other one by a sequence of shifts?”) is

sans-serifPSPACE

‐complete. For shortest

u, v

‐paths, for which each transformation consists in changing a single vertex, the same result was obtained by Bonsma [5].…”

Section: Introductionmentioning

confidence: 99%

Shifting paths to avoidable ones

Gurvich

Krnc

Milanič

et al. 2021

Journal of Graph Theory

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An extension of an induced path P in a graph G is an induced path P false′ such that deleting the endpoints of P false′ results in P. An induced path in a graph is said to be avoidable if each of its extensions is contained in an induced cycle. In 2019, Beisegel, Chudovsky, Gurvich, Milanič, and Servatius conjectured that every graph that contains an induced k‐vertex path also contains an avoidable induced path of the same length, and proved the result for k = 2. The case k = 1 was known much earlier, due to a work of Ohtsuki, Cheung, and Fujisawa in 1976. The conjecture was proved for all k in 2020 by Bonamy, Defrain, Hatzel, and Thiebaut. In the present paper, using a similar approach, we strengthen their result from a reconfiguration point of view. Namely, we show that in every graph, each induced path can be transformed to an avoidable one by a sequence of shifts, where two induced k‐vertex paths are shifts of each other if their union is an induced path with k + 1 vertices. We also obtain analogous results for not necessarily induced paths and for walks. In contrast, the statement cannot be extended to trails or to isometric paths.

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Reconfiguring Undirected Paths

Cited by 12 publications

References 16 publications

The Parameterized Complexity of Motion Planning for Snake-Like Robots

The Parameterized Complexity of Motion Planning for Snake-Like Robots

Reconfiguring (non-spanning) arborescences

Shifting paths to avoidable ones

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