Translating polygons in the plane
Proceedings on STACS 85 2nd annual symposium on theoretical aspects of computer science
On separating two simple polygons by a single translation
Discrete & Computational Geometry
Gross motion planning—a survey
ACM Computing Surveys (CSUR)
On movable separability and isotheticity
Information Sciences: an International Journal
Computers and Intractability: A Guide to the Theory of NP-Completeness
Computers and Intractability: A Guide to the Theory of NP-Completeness
The Complexity and Decidability of Separation
Proceedings of the 11th Colloquium on Automata, Languages and Programming
Reconfigurations in Graphs and Grids
SIAM Journal on Discrete Mathematics
Computational Geometry: Theory and Applications - Special issue on the Japan conference on discrete and computational geometry 2004
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We revisit two natural reconfiguration models for systems of disjoint objects in the plane: translation and sliding. Consider a set of n pairwise interior-disjoint objects in the plane that need to be brought from a given start (initial) configuration S into a desired goal (target) configuration T, without causing collisions. In the translation model, in one move an object is translated along a fixed direction to another position in the plane. In the sliding model, one move is sliding an object to another location in the plane by means of a continuous rigid motion (that could involve rotations). We obtain various combinatorial and computational results for these two models:(I)For systems of n congruent unlabeled disks in the translation model, Abellanas et al. showed that 2n-1 moves always suffice and @?8n/5@? moves are sometimes necessary for transforming the start configuration into the target configuration. Here we further improve the lower bound to @?5n/3@?-1, and thereby give a partial answer to one of their open problems. (II)We show that the reconfiguration problem with congruent disks in the translation model is NP-hard, in both the labeled and unlabeled variants. This answers another open problem of Abellanas et al. (III)We also show that the reconfiguration problem with congruent disks in the sliding model is NP-hard, in both the labeled and unlabeled variants. (IV)For the reconfiguration with translations of n arbitrary labeled convex bodies in the plane, 2n moves are always sufficient and sometimes necessary.