# The Parameterized Complexity of Motion Planning for Snake-Like Robots

## TL;DR

This paper investigates the parameterized complexity of a snake-like robot motion planning problem, proving it is fixed-parameter tractable and exploring kernelization and treewidth reduction techniques, filling a gap in motion planning complexity analysis.

## Contribution

It introduces the first fixed-parameter tractability result for snake-like robot motion planning and develops novel color-coding and graph sparsification methods.

## Key findings

- Problem is fixed-parameter tractable (FPT) with respect to snake size
- Color-coding sparsifies the configuration graph to size $k^{O(k)}|I|^{O(1)}$
- Problem unlikely to admit a polynomial kernel on grid graphs

## Abstract

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 [Biasi and Ophelders, 2016]. Nevertheless, we prove that even on general graphs, the problem is solvable in time $k^{\mathcal{O}(k)}|I|^{\mathcal{O}(1)}$ 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 $k^{\mathcal{O}(k)}|I|^{\mathcal{O}(1)}$ rather than $|I|^{\mathcal{O}(k)}$. 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.