We give an overview of the 2021 Computational Geometry Challenge, which targeted the problem of optimally coordinating a set of robots by computing a family of collision-free trajectories for a set set S of n pixel-shaped objects from a given start configuration into a desired target configuration.
We provide a tight result for a fundamental problem arising from packing squares into a circular container: The critical density of packing squares into a disk is $delta=frac{8}{5pi}approx 0.509$. This implies that any set of (not necessarily equal)
squares of total area $A leq frac{8}{5}$ can always be packed into a disk with radius 1; in contrast, for any $varepsilon>0$ there are sets of squares of total area $frac{8}{5}+varepsilon$ that cannot be packed, even if squares may be rotated. This settles the last (and arguably, most elusive) case of packing circular or square objects into a circular or square container: The critical densities for squares in a square $left(frac{1}{2}right)$, circles in a square $left(frac{pi}{(3+2sqrt{2})}approx 0.539right)$ and circles in a circle $left(frac{1}{2}right)$ have already been established, making use of recursive subdivisions of a square container into pieces bounded by straight lines, or the ability to use recursive arguments based on similarity of objects and container; neither of these approaches can be applied when packing squares into a circular container. Our proof uses a careful manual analysis, complemented by a computer-assisted part that is based on interval arithmetic. Beyond the basic mathematical importance, our result is also useful as a blackbox lemma for the analysis of recursive packing algorithms. At the same time, our approach showcases the power of a general framework for computer-assisted proofs, based on interval arithmetic.
We give an overview of the 2020 Computational Geometry Challenge, which targeted the problem of partitioning the convex hull of a given planar point set P into the smallest number of convex faces, such that no point of P is contained in the interior of a face.
We provide the solution for a fundamental problem of geometric optimization by giving a complete characterization of worst-case optimal disk coverings of rectangles: For any $lambdageq 1$, the critical covering area $A^*(lambda)$ is the minimum value
for which any set of disks with total area at least $A^*(lambda)$ can cover a rectangle of dimensions $lambdatimes 1$. We show that there is a threshold value $lambda_2 = sqrt{sqrt{7}/2 - 1/4} approx 1.035797ldots$, such that for $lambda<lambda_2$ the critical covering area $A^*(lambda)$ is $A^*(lambda)=3pileft(frac{lambda^2}{16} +frac{5}{32} + frac{9}{256lambda^2}right)$, and for $lambdageq lambda_2$, the critical area is $A^*(lambda)=pi(lambda^2+2)/4$; these values are tight. For the special case $lambda=1$, i.e., for covering a unit square, the critical covering area is $frac{195pi}{256}approx 2.39301ldots$. The proof uses a careful combination of manual and automatic analysis, demonstrating the power of the employed interval arithmetic technique.
We study emph{parallel} online algorithms: For some fixed integer $k$, a collective of $k$ parallel processes that perform online decisions on the same sequence of events forms a $k$-emph{copy algorithm}. For any given time and input sequence, th
e overall performance is determined by the best of the $k$ individual total results. Problems of this type have been considered for online makespan minimization; they are also related to optimization with emph{advice} on future events, i.e., a number of bits available in advance. We develop textsc{Predictive Harmonic}$_3$ (PH3), a relatively simple family of $k$-copy algorithms for the online Bin Packing Problem, whose joint competitive factor converges to 1.5 for increasing $k$. In particular, we show that $k=6$ suffices to guarantee a factor of $1.5714$ for PH3, which is better than $1.57829$, the performance of the best known 1-copy algorithm textsc{Advanced Harmonic}, while $k=11$ suffices to achieve a factor of $1.5406$, beating the known lower bound of $1.54278$ for a single online algorithm. In the context of online optimization with advice, our approach implies that 4 bits suffice to achieve a factor better than this bound of $1.54278$, which is considerably less than the previous bound of 15 bits.
We provide a tight result for a fundamental problem arising from packing disks into a circular container: The critical density of packing disks in a disk is 0.5. This implies that any set of (not necessarily equal) disks of total area $deltaleq 1/2$
can always be packed into a disk of area 1; on the other hand, for any $varepsilon>0$ there are sets of disks of area $1/2+varepsilon$ that cannot be packed. The proof uses a careful manual analysis, complemented by a minor automatic part that is based on interval arithmetic. Beyond the basic mathematical importance, our result is also useful as a blackbox lemma for the analysis of recursive packing algorithms.
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. Pr
oblems 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 {em sequential} schedules, in which one robot moves at a time. We provide constant-factor approximation algorithms for minimizing the execution time of a coordinated, {em parallel} motion plan for a swarm of robots in the absence of obstacles, provided some amount of separability. Our algorithm achieves {em 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 $Omega(N^{1/4})$ may be required. On the positive side, we establish a stretch factor of $O(N^{1/2})$ even in this case.
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-factor
ies. 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 $O(Nlog N)$ 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 $Omega(N^{frac{1}{3}})$; for tree-shaped structures, we give an $O(N^{frac{1}{2}})$-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.
A conflict-free $k$-coloring of a graph $G=(V,E)$ 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 a
pplications in wireless networking, robotics, and geometry, and are well studied in graph theory. Here we study the conflict-free coloring of geometric intersection graphs. We demonstrate that the intersection graph of $n$ geometric objects without fatness properties and size restrictions may have conflict-free chromatic number in $Omega(log n/loglog n)$ and in $Omega(sqrt{log n})$ for disks or squares of different sizes; it is known for general graphs that the worst case is in $Theta(log^2 n)$. For unit-disk intersection graphs, we prove that it is NP-complete to decide the existence of a conflict-free coloring with one color; we also show that six colors always suffice, using an algorithm that colors unit disk graphs of restricted height with two colors. We conjecture that four colors are sufficient, which we prove for unit squares instead of unit disks. For interval graphs, we establish a tight worst-case bound of two.
