We study unlabeled multi-robot motion planning for unit-disk robots in a polygonal environment. Although the problem is hard in general, polynomial-time solutions exist under appropriate separation assumptions on start and target positions. Solovey et al. (RSS'15) provide a near-optimal solution assuming that start/target positions must have pairwise distance at least 4, and at least √5≈2.236 from obstacles. This raises the question of whether polynomial-time algorithms can be obtained in even more densely packed environments. In this paper we present a generalized algorithm that achieve different trade-offs on the robots-separation and obstacles-separation bounds, all significantly improving upon the state of the art. Specifically, we obtain polynomial-time constant-approximation algorithms to minimize the total path length when (i) the robots-separation is 2 2/3 and the obstacles-separation is 1 2/3, or (ii) the robots-separation is ≈3.291 and the obstacles-separation ≈1.354. Additionally, we introduce a different strategy yielding a polynomial-time solution when the robots-separation is only 2, and the obstacles-separation is 3. Finally, we show that without any robots-separation assumption, obstacles-separation of at least 1.5 may be necessary for a solution to exist.

Motivated by applications for robust guarding, we consider a variant of the multiple-watchmen problem that ensures that every point within a polygon P is seen from more than one direction: we search for two routes W₁,W₂, such that every point p ∈ P is contained in a segment w₁w₂ ⊆ P such that w₁ ∈ W₁ and w₂ ∈ W₂. We call such routes segment watchman routes. We show that finding the two routes that are optimal with respect to the min-max criterion is weakly NP-hard even in simple polygons, and that finding the routes that are optimal with respect to the min-sum criterion is NP-hard in polygons with holes. Moreover, we present sufficient conditions for routes to be segment watchman routes, and provide a polynomial-time 2-approximation under both the min-max criterion and the min-sum criterion, both in simple polygons. Finally, we show how to generalize our results for k watchmen.

The minimum convex cover problem seeks to cover a polygon \(P\) with the fewest convex polygons that lie within \(P\). This problem is \(\exists\mathbb{R}\)-complete, and the best previously known algorithm, due to Eidenbenz and Widmayer (2001), achieves an \(O(\log n)\)-approximation in \(O(n^{29} \log n)\) time, where \(n\) is the complexity of \(P\).

Abstract We study the Art Gallery Problem under k-hop visibility in polyominoes. In this visibility model, two unit squares of a polyomino can see each other if and only if the shortest path between the respective vertices in the dual graph of the polyomino has length at most k. In this paper, we show that the VC dimension of this problem is 3 in simple polyominoes, and 4 in polyominoes with holes. Furthermore, we provide a reduction from Planar Monotone 3Sat, thereby showing that the problem is -complete even in thin polyominoes (i.e., polyominoes that do not a contain a $$2\times 2$$ 2 × 2 block of cells). Complementarily, we present a linear-time 4-approximation algorithm for simple 2-thin polyominoes (which do not contain a $$3\times 3$$ 3 × 3 block of cells) for all $$k\in {\mathbb {N}}$$ k ∈ N .

We propose precise notions of what it means to guard a domain "robustly", under a variety of models. While approximation algorithms for minimizing the number of (precise) point guards in a polygon is a notoriously challenging area of investigation, we show that imposing various degrees of robustness on the notion of visibility coverage leads to a more tractable (and realistic) problem for which we can provide approximation algorithms with constant factor guarantees.

The classical and extensively-studied Fréchet distance between two curves is defined as an inf max, where the infimum is over all traversals of the curves, and the maximum is over all concurrent positions of the two agents. In this article we investigate a "flipped" Fréchet measure defined by a sup min - the supremum is over all traversals of the curves, and the minimum is over all concurrent positions of the two agents. This measure produces a notion of "social distance" between two curves (or general domains), where agents traverse curves while trying to stay as far apart as possible. We first study the flipped Fréchet measure between two polygonal curves in one and two dimensions, providing conditional lower bounds and matching algorithms. We then consider this measure on polygons, where it denotes the minimum distance that two agents can maintain while restricted to travel in or on the boundary of the same polygon. We investigate several variants of the problem in this setting, for some of which we provide linear time algorithms. Finally, we consider this measure on graphs. We draw connections between our proposed flipped Fréchet measure and existing related work in computational geometry, hoping that our new measure may spawn investigations akin to those performed for the Fréchet distance, and into further interesting problems that arise.

Given a curve P with points in ℝ d in a streaming fashion, and parameters ɛ > 0 and k , we construct a distance oracle that uses \(O(\frac{1}{\varepsilon })^{kd}\log \varepsilon ^{-1}\) space, and given a query curve Q with k points in ℝ d returns in \(\tilde{O}(kd)\) time a 1+ɛ approximation of the discrete Fréchet distance between Q and P . In addition, we construct simplifications in the streaming model, oracle for distance queries to a sub-curve (in the static setting), and introduce the zoom-in problem. Our algorithms work in any dimension d , and therefore we generalize some useful tools and algorithms for curves under the discrete Fréchet distance to work efficiently in high dimensions.

Consider a set of voters V, represented by a multiset in a metric space (X,d). The voters have to reach a decision - a point in X. A choice p∈ X is called a β-plurality point for V, if for any other choice q∈ X it holds that |{v∈ V ∣ β⋅ d(p,v)≤ d(q,v)}| ≥|V|/2 . In other words, at least half of the voters ``prefer'' over q, when an extra factor of β is taken in favor of p. For β=1, this is equivalent to Condorcet winner, which rarely exists. The concept of β-plurality was suggested by Aronov, de Berg, Gudmundsson, and Horton [SoCG 2020] as a relaxation of the Condorcet criterion. Denote by β*(X,d) the value sup{ β ∣ every finite multiset V in X admits a β-plurality point}}. The parameter β* determines the amount of relaxation required in order to reach a stable decision. Aronov et al. showed that for the Euclidean plane β*(ℝ2,\|⋅\|2)=√3/2 , and more generally, for d-dimensional Euclidean space, 1/√d ≤ β*(ℝd,\|⋅\|2)≤√3/2 . In this paper, we show that 0.557≤ β*(ℝd,\|⋅\|2) for any dimension d (notice that 1/√d

Given a curve P with points in ℝd in a streaming fashion, and parameters ∊ > 0 and k, we construct a distance oracle that uses space, and given a query curve Q with k points in ℝd, returns in O(kd) time a 1 + ∊ approximation of the discrete Fréchet distance between Q and P. In addition, we construct simplifications in the streaming model, oracle for distance queries to a sub-curve (in the static setting), and introduce the zoom-in problem. Our algorithms work in any dimension d, and therefore we generalize some useful tools and algorithms for curves under the discrete Fréchet distance to work efficiently in high dimensions.

In the (1+ε,r)-approximate near-neighbor problem for curves (ANNC) under some similarity measure δ, the goal is to construct a data structure for a given set 𝒞 of curves that supports approximate near-neighbor queries: Given a query curve Q, if there exists a curve C ∈ 𝒞 such that δ(Q,C)≤ r, then return a curve C' ∈ 𝒞 with δ(Q,C') ≤ (1+ε)r. There exists an efficient reduction from the (1+ε)-approximate nearest-neighbor problem to ANNC, where in the former problem the answer to a query is a curve C ∈ 𝒞 with δ(Q,C) ≤ (1+ε)⋅δ(Q,C^*), where C^* is the curve of 𝒞 most similar to Q. Given a set 𝒞 of n curves, each consisting of m points in d dimensions, we construct a data structure for ANNC that uses n⋅ O(1/ε)^{md} storage space and has O(md) query time (for a query curve of length m), where the similarity measure between two curves is their discrete Fréchet or dynamic time warping distance. Our method is simple to implement, deterministic, and results in an exponential improvement in both query time and storage space compared to all previous bounds. Further, we also consider the asymmetric version of ANNC, where the length of the query curves is k ≪ m, and obtain essentially the same storage and query bounds as above, except that m is replaced by k. Finally, we apply our method to a version of approximate range counting for curves and achieve similar bounds.

Let A and B be two sets of points in R^d, where |A|=|B|=n and the distance between them is defined by some bipartite measure dist(A, B). We study several problems in which the goal is to translate the set B, so that dist(A, B) is minimized. The main measures that we consider are (i) the diameter in two and three dimensions, that is diam(A,B) = max {d(a,b) | a in A, b in B}, where d(a,b) is the Euclidean distance between a and b, (ii) the uniformity in the plane, that is uni(A,B) = diam(A,B) - d(A,B), where d(A,B)=min{d(a,b) | a in A, b in B}, and (iii) the union width in two and three dimensions, that is union_width(A,B) = width(A cup B). For each of these measures we present efficient algorithms for finding a translation of B that minimizes the distance: For diameter we present near-linear-time algorithms in R^2 and R^3, for uniformity we describe a roughly O(n^{9/4})-time algorithm, and for union width we offer a near-linear-time algorithm in R^2 and a quadratic-time one in R^3.

The Chain Pair Simplification problem (CPS) was posed by Bereg et al. who were motivated by the problem of efficiently computing and visualizing the structural resemblance between a pair of protein backbones. In this problem, given two polygonal chains of lengths n and m, the goal is to simplify both of them simultaneously, so that the lengths of the resulting simplifications as well as the discrete Frechet distance between them are bounded. When the vertices of the simplifications are arbitrary (i.e., not necessarily from the original chains), the problem is called General CPS (GCPS). In this paper we consider for the first time the complexity of GCPS under both the discrete Frechet distance (GCPS-3F) and the Hausdorff distance (GCPS-2H). (In the former version, the quality of the two simplifications is measured by the discrete Fr'echet distance, and in the latter version it is measured by the Hausdorff distance.) We prove that GCPS-3F is polynomially solvable, by presenting an widetilde-O((n+m)^6 min{n,m}) time algorithm for the corresponding minimization problem. We also present an O((n+m)^4) 2-approximation algorithm for the problem. On the other hand, we show that GCPS-2H is NP-complete, and present an approximation algorithm for the problem.

The Fréchet distance is a well-studied similarity measure between curves. The discrete Fréchet distance is an analogous similarity measure, defined for two sequences of m and n points, where the points are usually sampled from input curves. We consider a variant, called the discrete Fréchet distance with shortcuts , which captures the similarity between (sampled) curves in the presence of outliers. When shortcuts are allowed only in one noise-containing curve, we give a randomized algorithm that runs in O (( m + n ) 6/5 + ε ) expected time, for any ε > 0. When shortcuts are allowed in both curves, we give an O (( m 2/3 n 2/3 + m + n )log 3 ( m + n ))-time deterministic algorithm. We also consider the semicontinuous Fréchet distance with one-sided shortcuts, where we have a sequence of m points and a polygonal curve of n edges, and shortcuts are allowed only in the sequence. We show that this problem can be solved in randomized expected time O (( m + n ) 2/3 m 2/3 n 1/3 log ( m + n )). Our techniques are novel and may find further applications. One of the main new technical results is: Given two sets of points A and B in the plane and an interval I , we develop an algorithm that decides whether the number of pairs ( x , y ) ∈ A × B whose distance dist( x , y ) is in I is less than some given threshold L . The running time of this algorithm decreases as L increases. In case there are more than L pairs of points whose distance is in I , we can get a small sample of pairs that contain a pair at approximate median distance (i.e., we can approximately “bisect” I ). We combine this procedure with additional ideas to search, with a small overhead, for the optimal one-sided Fréchet distance with shortcuts, using a very fast decision procedure. We also show how to apply this technique for approximating distance selection (with respect to rank), and a somewhat more involved variant of this technique is used in the solution of the semicontinuous Fréchet distance with one-sided shortcuts. In general, the new technique can be applied to optimization problems for which the decision procedure is very fast but standard techniques like parametric search makes the optimization algorithm substantially slower.

The Fréchet distance is a well studied similarity measure between curves. The discrete Fréchet distance is an analogous similarity measure, defined for two sequences of m and n points, where the points are usually sampled from input curves. We consider a variant, called the discrete Fréchet distance with shortcuts, which captures the similarity between (sampled) curves in the presence of outliers. When shortcuts are allowed only in one noise-containing curve, we give a randomized algorithm that runs in O((m+n)6/5+ϵ) expected time, for any ϵ > 0. When shortcuts are allowed in both curves, we give an O((m2/3n2/3 + m + n) log3(m + n))-time deterministic algorithm.