The Parameterized Complexity of Motion Planning for Snake-Like Robots

Gupta, Siddharth; Sa'ar, Guy; Zehavi, Meirav

doi:10.1613/jair.1.11864

Cited by 6 publications

(4 citation statements)

References 30 publications

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“…Snake is typically played on grid graphs, and it is known to be PSPACE-complete to determine whether the Snake can reach a specific goal state from a given start state on generalized grid graphs [4]. Independently of our work, Gupta et al [7] have found that reconfiguring snakes (paths that can move only unidirectionally) is fixed-parameter tractable in the length of the path, analogously to our Theorem 1. Fig.…”

Section: Related Workmentioning

confidence: 84%

Reconfiguring Undirected Paths

Demaine¹,

Eppstein

Hesterberg

et al. 2019

Lecture Notes in Computer Science

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We consider problems in which a simple path of fixed length, in an undirected graph, is to be shifted from a start position to a goal position by moves that add an edge to either end of the path and remove an edge from the other end. We show that this problem may be solved in linear time in trees, and is fixed-parameter tractable when parameterized either by the cyclomatic number of the input graph or by the length of the path. However, it is PSPACE-complete for paths of unbounded length in graphs of bounded bandwidth.

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Section: Related Workmentioning

confidence: 84%

Reconfiguring Undirected Paths

Demaine¹,

Eppstein

Hesterberg

et al. 2019

Lecture Notes in Computer Science

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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%

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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“…Overall, we believe that more subtle underlying parameters need to be considered. Previous positive results that employ parameterized complexity in discrete motion planning problems [1,17] provide some encouragement.…”

Section: Future Workmentioning

confidence: 95%

Refined Hardness of Distance-Optimal Multi-Agent Path Finding

Geft¹,

Halperin²

2022

Preprint

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We study the computational complexity of multi-agent path finding (MAPF). Given a graph 𝐺 and a set of agents, each having a start and target vertex, the goal is to find collision-free paths minimizing the total distance traveled. To better understand the source of difficulty of the problem, we aim to study the simplest and least constrained graph class for which it remains hard. To this end, we restrict 𝐺 to be a 2D grid, which is a ubiquitous abstraction, as it conveniently allows for modeling well-structured environments (e.g., warehouses). Previous hardness results considered highly constrained 2D grids having only one vertex unoccupied by an agent, while the most restricted hardness result that allowed multiple empty vertices was for (non-grid) planar graphs. We therefore refine previous results by simultaneously considering both 2D grids and multiple empty vertices. We show that even in this case distance-optimal MAPF remains NP-hard, which settles an open problem posed by Banfi et al. [4]. We present a reduction directly from 3-SAT using simple gadgets, making our proof arguably more informative than previous work in terms of potential progress towards positive results. Furthermore, our reduction is the first linear one for the case where 𝐺 is planar, appearing nearly four decades after the first related result. This allows us to go a step further and exploit the Exponential Time Hypothesis (ETH) to obtain an exponential lower bound for the running time of the problem. Finally, as a stepping stone towards our main results, we prove the NP-hardness of the monotone case, in which agents move one by one with no intermediate stops.

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The Parameterized Complexity of Motion Planning for Snake-Like Robots

Cited by 6 publications

References 30 publications

Reconfiguring Undirected Paths

Reconfiguring Undirected Paths

Reconfiguring (non-spanning) arborescences

Refined Hardness of Distance-Optimal Multi-Agent Path Finding

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