The complexity of snake and undirected NCL variants

Biasi, Marzio De; Ophelders, Tim

doi:10.1016/j.tcs.2017.10.031

Cited by 6 publications

(11 citation statements)

References 13 publications

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“…We now give a procedure (Algorithm 1) to construct a new graph, called a (k − 1)-sparse configuration graph of SG, based on a (V (G), 3k − 2)-permuter and Lemma 3. 3. We first give an outline of the procedure.…”

Section: Fpt Algorithm On General Graphsmentioning

confidence: 99%

“…Our proof is based on the exhibition of a cross-composition (see Definition 2.9). We remind that our cross-composition is inspired by the proof of NP-hardness of Snake Game on grid graphs by Biasi and Ophelders [3]. As the source problem Π for the cross-composition, we select the Hamiltonian Cycle problem on grid graphs, and the target problem is Snake Game on grid graphs.…”

Section: Non-existence Of a Polynomial Kernel On Grid Graphsmentioning

confidence: 99%

“…Lemma 4. 3. If G, n, init i , fin is a Yes-instance of Snake Game for some 1 ≤ i ≤ t, then G i is a Yes-instance of the Hamiltonian Cycle problem.…”

Section: Proofmentioning

confidence: 99%

“…Unlike most other video games, the popularity of Snake has hardly decreased despite its age-indeed, new versions of Snake still appear to this day. We study the parameterized complexity of a variant of Snake that was introduced by Biasi and Ophelders [3], which models real-world problems of motion planning for agents of a "snake-like" shape. 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.…”

Section: Introductionmentioning

confidence: 99%

“…A snake-like shape and serpentine locomotion offer immediate advantages for such purposes; the restricted area of mobility also makes the requirement of the robot to avoid intersecting itself and other obstacles a highly non-trivial issue that is mandatory to take into account. Moreover, the Snake Game problem is a natural 1 abstraction to model a wide-variety of other scenarios, which range 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 . Biasi and Ophelders [3] proved that the Snake Game problem is PSPACE-complete even on grid graphs (see Section 2). Additionally, they considered the version aligned with the video game, where "food" items are located on vertices.…”

Section: Introductionmentioning

confidence: 99%

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The Parameterized Complexity of Motion Planning for Snake-Like Robots

Gupta

Sa'ar

Zehavi

2019

Proceedings of the Twenty-Eighth International Joint Conference on Artificial Intelligence

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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 PSPACEcomplete [Biasi and Ophelders, 2016]. Nevertheless, we prove that even on general graphs, the problem is solvable in time k O(k) |I| O(1) where k is the size of the snake, and |I| is the input size. 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 have size k O(k) |I| O(1) rather than |I| O(k) . 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. * Ben-Gurion University of the Negev, Israel. siddhart@post.bgu.ac.il † Ben-Gurion University of the Negev, Israel. saag@bgu.ac.il ‡ Ben-Gurion University of the Negev, Israel. meiravze@bgu.ac.il

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Section: Fpt Algorithm On General Graphsmentioning

confidence: 99%

Section: Non-existence Of a Polynomial Kernel On Grid Graphsmentioning

confidence: 99%

“…Lemma 4. 3. If G, n, init i , fin is a Yes-instance of Snake Game for some 1 ≤ i ≤ t, then G i is a Yes-instance of the Hamiltonian Cycle problem.…”

Section: Proofmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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