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Pankaj K. Agarwal, Dan Halperin, Micha Sharir, Alex Steiger

Let $\mathcal{W} \subset \mathbb{R}^2$ be a planar polygonal environment (i.e., a polygon potentially with holes) with a total of $n$ vertices, and let $A,B$ be two robots, each modeled as an axis-aligned unit square, that can translate inside $\mathcal{W}$. Given source and target placements $s_A,t_A,s_B,t_B \in \mathcal{W}$ of $A$ and $B$, respectively, the goal is to compute a \emph{collision-free motion plan} $\mathbf{\pi}^*$, i.e., a motion plan that continuously moves $A$ from $s_A$ to $t_A$ and $B$ from $s_B$ to $t_B$ so that $A$ and $B$ remain inside $\mathcal{W}$ and do not collide with each other during the motion. Furthermore, if such a plan exists, then we wish to return a plan that minimizes the sum of the lengths of the paths traversed by the robots, $\left|\mathbf{\pi}^*\right|$. Given $\mathcal{W}, s_A,t_A,s_B,t_B$ and a parameter $\varepsilon > 0$, we present an $n^2\varepsilon^{-O(1)} \log n$-time $(1+\varepsilon)$-approximation algorithm for this problem. We are not aware of any polynomial time algorithm for this problem, nor do we know whether the problem is NP-Hard. Our result is the first polynomial-time $(1+\varepsilon)$-approximation algorithm for an optimal motion planning problem involving two robots moving in a polygonal environment.

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Mikkel Abrahamsen, Tzvika Geft, Dan Halperin, Barak Ugav

We study a fundamental NP-hard motion coordination problem for multi-robot/multi-agent systems: We are given a graph $G$ and set of agents, where each agent has a given directed path in $G$. Each agent is initially located on the first vertex of its path. At each time step an agent can move to the next vertex on its path, provided that the vertex is not occupied by another agent. The goal is to find a sequence of such moves along the given paths so that each reaches its target, or to report that no such sequence exists. The problem models guidepath-based transport systems, which is a pertinent abstraction for traffic in a variety of contemporary applications, ranging from train networks or Automated Guided Vehicles (AGVs) in factories, through computer game animations, to qubit transport in quantum computing. It also arises as a sub-problem in the more general multi-robot motion-planning problem. We provide a fine-grained tractability analysis of the problem by considering new assumptions and identifying minimal values of key parameters for which the problem remains NP-hard. Our analysis identifies a critical parameter called vertex multiplicity (VM), defined as the maximum number of paths passing through the same vertex. We show that a prevalent variant of the problem, which is equivalent to Sequential Resource Allocation (concerning deadlock prevention for concurrent processes), is NP-hard even when VM is 3. On the positive side, for VM $\le$ 2 we give an efficient algorithm that iteratively resolves cycles of blocking relations among agents. We also present a variant that is NP-hard when the VM is 2 even when $G$ is a 2D grid and each path lies in a single grid row or column. By studying highly distilled yet NP-hard variants, we deepen the understanding of what makes the problem intractable and thereby guide the search for efficient solutions under practical assumptions.

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Pankaj K. Agarwal, Tzvika Geft, Dan Halperin, Erin Taylor

We study the problem of motion planning for a collection of $n$ labeled unit disc robots in a polygonal environment. We assume that the robots have \emph{revolving areas} around their start and final positions: that each start and each final is contained in a radius $2$ disc lying in the free space, not necessarily concentric with the start or final position, which is free from other start or final positions. This assumption allows a \emph{weakly-monotone} motion plan, in which robots move according to an ordering as follows: during the turn of a robot $R$ in the ordering, it moves fully from its start to final position, while other robots do not leave their revolving areas. As $R$ passes through a revolving area, a robot $R'$ that is inside this area may move within the revolving area to avoid a collision. Notwithstanding the existence of a motion plan, we show that minimizing the total traveled distance in this setting, specifically even when the motion plan is restricted to be weakly-monotone, is APX-hard, ruling out any polynomial-time $(1+\epsilon)$-approximation algorithm. On the positive side, we present the first constant-factor approximation algorithm for computing a feasible weakly-monotone motion plan. The total distance traveled by the robots is within an $O(1)$ factor of that of the optimal motion plan, which need not be weakly monotone. Our algorithm extends to an online setting in which the polygonal environment is fixed but the initial and final positions of robots are specified in an online manner. Finally, we observe that the overhead in the overall cost that we add while editing the paths to avoid robot-robot collision can vary significantly depending on the ordering we chose. Finding the best ordering in this respect is known to be NP-hard, and we provide a polynomial time $O(\log n \log \log n)$-approximation algorithm for this problem.

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Dan Halperin, Steven M. LaValle, Barak Ugav

Given a polygon $W$, a depth sensor placed at point $p=(x,y)$ inside $W$ and oriented in direction $\theta$ measures the distance $d=h(x,y,\theta)$ between $p$ and the closest point on the boundary of $W$ along a ray emanating from $p$ in direction $\theta$. We study the following problem: Give a polygon $W$, possibly with holes, with $n$ vertices, preprocess it such that given a query real value $d\geq 0$, one can efficiently compute the preimage $h^{-1}(d)$, namely determine all the possible poses (positions and orientations) of a depth sensor placed in $W$ that would yield the reading $d$. We employ a decomposition of $W\times S^1$, which is an extension of the celebrated trapezoidal decomposition, and which we call rotational trapezoidal decomposition and present an efficient data structure, which computes the preimage in an output-sensitive fashion relative to this decomposition: if $k$ cells of the decomposition contribute to the final result, we will report them in $O(k+1)$ time, after $O(n^2\log n)$ preprocessing time and using $O(n^2)$ storage space. We also analyze the shape of the projection of the preimage onto the polygon $W$; this projection describes the portion of $W$ where the sensor could have been placed. Furthermore, we obtain analogous results for the more useful case (narrowing down the set of possible poses), where the sensor performs two depth measurement from the same point $p$, one in direction $\theta$ and the other in direction $\theta+\pi$. While localizations problems in robotics are often carried out by exploring the full visibility polygon of a sensor placed at a fixed point of the environment, the approach that we propose here opens the door to sufficing with only few depth measurements, which is advantageous as it allows for usage of inexpensive sensors and could also lead to savings in storage and communication costs.

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Bahareh Banyassady, Mark de Berg, Karl Bringmann, Kevin Buchin, Henning Fernau, Dan Halperin, Irina Kostitsyna, Yoshio Okamoto, Stijn Slot

We consider the unlabeled motion-planning problem of $m$ unit-disc robots moving in a simple polygonal workspace of $n$ edges. The goal is to find a motion plan that moves the robots to a given set of $m$ target positions. For the unlabeled variant, it does not matter which robot reaches which target position as long as all target positions are occupied in the end. If the workspace has narrow passages such that the robots cannot fit through them, then the free configuration space, representing all possible unobstructed positions of the robots, will consist of multiple connected components. Even if in each component of the free space the number of targets matches the number of start positions, the motion-planning problem does not always have a solution when the robots and their targets are positioned very densely. In this paper, we prove tight bounds on how much separation between start and target positions is necessary to always guarantee a solution. Moreover, we describe an algorithm that always finds a solution in time $O(n \log n + mn + m^2)$ if the separation bounds are met. Specifically, we prove that the following separation is sufficient: any two start positions are at least distance $4$ apart, any two target positions are at least distance $4$ apart, and any pair of a start and a target positions is at least distance $3$ apart. We further show that when the free space consists of a single connected component, the separation between start and target positions is not necessary.

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Tzvika Geft, Dan Halperin

We study the computational complexity of multi-agent path finding (MAPF). Given a graph $G$ and a set of agents, each having a start and target vertex, the goal is to find collision-free paths minimizing the total distance traveled. To better understand the source of difficulty of the problem, we aim to study the simplest and least constrained graph class for which it remains hard. To this end, we restrict $G$ to be a 2D grid, which is a ubiquitous abstraction, as it conveniently allows for modeling well-structured environments (e.g., warehouses). Previous hardness results considered highly constrained 2D grids having only one vertex unoccupied by an agent, while the most restricted hardness result that allowed multiple empty vertices was for (non-grid) planar graphs. We therefore refine previous results by simultaneously considering both 2D grids and multiple empty vertices. We show that even in this case distance-optimal MAPF remains NP-hard, which settles an open problem posed by Banfi et al. (2017). We present a reduction directly from 3-SAT using simple gadgets, making our proof arguably more informative than previous work in terms of potential progress towards positive results. Furthermore, our reduction is the first linear one for the case where $G$ is planar, appearing nearly four decades after the first related result. This allows us to go a step further and exploit the Exponential Time Hypothesis (ETH) to obtain an exponential lower bound for the running time of the problem. Finally, as a stepping stone towards our main results, we prove the NP-hardness of the monotone case, in which agents move one by one with no intermediate stops.

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Tzvika Geft, Dan Halperin

In multi-robot motion planning (MRMP) the aim is to plan the motion of several robots operating in a common workspace, while avoiding collisions with obstacles or with fellow robots. The main contribution of this paper is a simple construction that serves as a lower bound for the computational cost of a variety of prevalent MRMP problems. In particular we show that optimal decoupling of multi-robot motion -- decoupling being a standard approach to practically addressing MRMP -- is NP-hard. The basic problem for which we present our construction is monotone MRMP, a restricted and natural MRMP variant, where robots move one by one to their targets with no intermediate stops. Observing the hardness of these restricted versions of MRMP is significant as it guides the search for efficient solutions towards techniques that can cope with intractable problems. Furthermore, our construction highlights structural properties of difficult instances, such as the need of robots to pass through many start and target positions of other robots. These insights can lead to useful problem relaxations resulting in efficient solutions and to suitable engineering of workspaces.

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Dror Dayan, Kiril Solovey, Marco Pavone, Dan Halperin

An underlying structure in several sampling-based methods for continuous multi-robot motion planning (MRMP) is the tensor roadmap (TR), which emerges from combining multiple PRM graphs constructed for the individual robots via a tensor product. We study the conditions under which the TR encodes a near-optimal solution for MRMP---satisfying these conditions implies near optimality for a variety of popular planners, including dRRT*, and the discrete methods M* and CBS when applied to the continuous domain. We develop the first finite-sample analysis of this kind, which specifies the number of samples, their deterministic distribution, and magnitude of the connection radii that should be used by each individual PRM graph, to guarantee near-optimality using the TR. This significantly improves upon a previous asymptotic analysis, wherein the number of samples tends to infinity, and supports guaranteed high-quality solutions in practice, within bounded running time. To achieve our new result, we first develop a sampling scheme, which we call the staggered grid, for finite-sample motion planning for individual robots, which requires significantly less samples than previous work. We then extend it to the much more involved MRMP setting which requires to account for interactions among multiple robots. Finally, we report on a few experiments that serve as a verification of our theoretical findings and raise interesting questions for further investigation.

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Pankaj K. Agarwal, Boris Aronov, Tzvika Geft, Dan Halperin

Assembly planning, which is a fundamental problem in robotics and automation, aims to design a sequence of motions that will bring the separate constituent parts of a product into their final placement in the product. It is convenient to study assembly planning in reverse order, where the following key problem, assembly partitioning, arises: Given a set of parts in their final placement in a product, partition them into two sets, each regarded as a rigid body, which we call a subassembly, such that these two subassemblies can be moved sufficiently far away from each other, without colliding with one another. The basic assembly planning problem is further complicated by practical consideration such as how to hold the parts in a subassembly together. Therefore, a desired property of a valid assembly partition is that each of the two subassemblies will be connected. We show that even an utterly simple case of the connected-assembly-partitioning problem is hard: Given a connected set $A$ of unit squares in the plane, each forming a distinct cell of the uniform integer grid, find a subset $S\subset A$ such that $S$ can be rigidly translated to infinity along a prescribed direction without colliding with $A\setminus S$, and both subassemblies $S$ and $A\setminus S$ are each connected. We show that this problem is NP-Complete, and by that settle an open problem posed by Wilson et al. (1995) a quarter of a century ago. We complement the hardness result with two positive results for the aforementioned problem variant of grid squares. First, we show that it is fixed parameter tractable and give an $O(2^k n^2)$-time algorithm, where $n=|A|$ and $k=|S|$. Second, we describe a special case of this variant where a connected partition can always be found in linear time. Each of the positive results sheds further light on the special geometric structure of the problem at hand.

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Tzvika Geft, Aviv Tamar, Ken Goldberg, Dan Halperin

To compute robust 2D assembly plans, we present an approach that combines geometric planning with a deep neural network. We train the network using the Box2D physics simulator with added stochastic noise to yield robustness scores--the success probabilities of planned assembly motions. As running a simulation for every assembly motion is impractical, we train a convolutional neural network to map assembly operations, given as an image pair of the subassemblies before and after they are mated, to a robustness score. The neural network prediction is used within a planner to quickly prune out motions that are not robust. We demonstrate this approach on two-handed planar assemblies, where the motions are one-step translations. Results suggest that the neural network can learn robustness to plan robust sequences an order of magnitude faster than physics simulation.

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