We present a number of breakthroughs for coordinated motion planning, in which the objective is to reconfigure a swarm of labeled convex objects by a combination of parallel, continuous, collision-free translations into a given target arrangement. Problems of this type can be traced back to the classic work of Schwartz and Sharir (1983), who gave a method for deciding the existence of a coordinated motion for a set of disks between obstacles; their approach is polynomial in the complexity of the obstacles, but exponential in the number of disks. Other previous work has largely focused on sequential schedules, in which one robot moves at a time.We provide constant-factor approximation algorithms for minimizing the execution time of a coordinated, parallel motion plan for a swarm of robots in the absence of obstacles, provided some amount of separability.Our algorithm achieves constant stretch factor: If all robots are at most d units from their respective starting positions, the total duration of the overall schedule is O(d). Extensions include unlabeled robots and different classes of robots. We also prove that finding a plan with minimal execution time is NP-hard, even for a grid arrangement without any stationary obstacles. On the other hand, we show that for densely packed disks that cannot be well separated, a stretch factor Ω(N 1/4 ) may be required. On the positive side, we establish a stretch factor of O(N 1/2 ) even in this case.

A conflict-free k-coloring of a graph assigns one of k different colors to some of the vertices such that, for every vertex v, there is a color that is assigned to exactly one vertex among v and v's neighbors. Such colorings have applications in wireless networking, robotics, and geometry, and are well-studied in graph theory. Here we study the natural problem of the conflict-free chromatic number χ CF (G) (the smallest k for which conflict-free k-colorings exist). We provide results both for closed neighborhoods N [v], for which a vertex v is a member of its neighborhood, and for open neighborhoods N (v), for which vertex v is not a member of its neighborhood.For closed neighborhoods, we prove the conflict-free variant of the famous Hadwiger Conjecture: If an arbitrary graph G does not contain K k+1 as a minor, then χ CF (G) ≤ k. For planar graphs, we obtain a tight worst-case bound: three colors are sometimes necessary and always sufficient. In addition, we give a complete characterization of the algorithmic/computational complexity of conflict-free coloring. It is NP-complete to decide whether a planar graph has a conflict-free coloring with one color, while for outerplanar graphs, this can be decided in polynomial time. Furthermore, it is NP-complete to decide whether a planar graph has a conflict-free coloring with two colors, while for outerplanar graphs, two colors always suffice. For the bicriteria problem of minimizing the number of colored vertices subject to a given bound k on the number of colors, we give a full algorithmic characterization in terms of complexity and approximation for outerplanar and planar graphs.For open neighborhoods, we show that every planar bipartite graph has a conflict-free coloring with at most four colors; on the other hand, we prove that for k ∈ {1, 2, 3}, it is NP-complete to decide whether a planar bipartite graph has a conflict-free k-coloring. Moreover, we establish that any general planar graph has a conflict-free coloring with at most eight colors.2. It is NP-complete to decide whether a planar graph has a conflict-free coloring with one color.For outerplanar graphs, this question can be decided in polynomial time.

A conflict-free k-coloring of a graph assigns one of k different colors to some of the vertices such that, for every vertex v, there is a color that is assigned to exactly one vertex among v and v's neighbors. Such colorings have applications in wireless networking, robotics, and geometry, and are well-studied in graph theory. Here we study the natural problem of the conflict-free chromatic number χ CF (G) (the smallest k for which conflict-free k-colorings exist), with a focus on planar graphs.For general graphs, we prove the conflict-free variant of the famous Hadwiger Conjecture: If G does not contain K k+1 as a minor, then χ CF (G) ≤ k. For planar graphs, we obtain a tight worst-case bound: three colors are sometimes necessary and always sufficient. In addition, we give a complete characterization of the algorithmic/computational complexity of conflict-free coloring. It is NP-complete to decide whether a planar graph has a conflict-free coloring with one color, while for outerplanar graphs, this can be decided in polynomial time. Furthermore, it is NP-complete to decide whether a planar graph has a conflict-free coloring with two colors, while for outerplanar graphs, two colors always suffice. For the bicriteria problem of minimizing the number of colored vertices subject to a given bound k on the number of colors, we give a full algorithmic characterization in terms of complexity and approximation for outerplanar and planar graphs.

We present algorithmic results for the parallel assembly of many micro-scale objects in two and three dimensions from tiny particles, which has been proposed in the context of programmable matter and self-assembly for building high-yield micro-factories. The underlying model has particles moving under the influence of uniform external forces until they hit an obstacle; particles can bond when being forced together with another appropriate particle.Due to the physical and geometric constraints, not all shapes can be built in this manner; this gives rise to the Tilt Assembly Problem (TAP) of deciding constructibility. For simply-connected polyominoes P in 2D consisting of N unit-squares ("tiles"), we prove that TAP can be decided in O(N log N ) time. For the optimization variant MaxTAP (in which the objective is to construct a subshape of maximum possible size), we show polyAPX-hardness: unless P=NP, MaxTAP cannot be approximated within a factor of Ω(N 1 3 ); for tree-shaped structures, we give an O(N 1 2 )-approximation algorithm. For the efficiency of the assembly process itself, we show that any constructible shape allows pipelined assembly, which produces copies of P in O(1) amortized time, i.e., N copies of P in O(N ) time steps. These considerations can be extended to three-dimensional objects: For the class of polycubes P we prove that it is NP-hard to decide whether it is possible to construct a path between two points of P ; it is also NP-hard to decide constructibility of a polycube P . Moreover, it is expAPX-hard to maximize a path from a given start point.

We contribute results for a set of fundamental problems in the context of programmable matter by presenting algorithmic methods for evaluating and manipulating a collective of particles by a finite automaton that can neither store significant amounts of data, nor perform complex computations, and is limited to a handful of possible physical operations. We provide a toolbox for carrying out fundamental tasks on a given arrangement of particles, using the arrangement itself as a storage device, similar to a higher-dimensional Turing machine with geometric properties. Specific results include time- and space-efficient procedures for bounding, counting, copying, reflecting, rotating or scaling a complex given shape.