We study the problems of covering or partitioning a polygon P (possibly with holes) using a minimum number of small pieces, where a small piece is a connected sub-polygon contained in an axis-aligned unit square. For covering, we seek to write P as a union of small pieces, and in partitioning, we furthermore require the pieces to be pairwise interior-disjoint. We show that these problems are in fact equivalent: Optimum covers and partitions have the same number of pieces. For covering, a natural local search algorithm repeatedly attempts to replace k pieces from a candidate cover with k-1 pieces. In two dimensions and for sufficiently large k, we show that when no such swap is possible, the cover is a 1+O(1/√(k))-approximation, hence obtaining the first PTAS for the problem. Prior to our work, the only known algorithm was a 13-approximation that only works for polygons without holes [Abrahamsen and Rasmussen, SODA 2025]. In contrast, in the three dimensional version of the problem, for a polyhedron P of complexity n, we show that it is NP-hard to approximate an optimal cover or partition to within a factor that is logarithmic in n, even if P is simple, i.e., has genus 0 and no holes.

A circle graph is the intersection graph of a set of chords in a circle. A dominating set of a graph G=(V,E) is a subset D⊆ V such that every vertex in V∖ D is adjacent to at least one vertex of D. Computing a minimum dominating set is known to be NP-hard on circle graphs. In this paper, we study the minimum dominating set problem on bipartite circle graphs, namely, circle graphs admitting a chord representation in which the chords can be partitioned into two color classes such that no two chords of the same color intersect. We prove that the problem remains NP-hard for this restricted graph class by a reduction from Planar Monotone 3-SAT. On the positive side, we present a polynomial-time 2-approximation algorithm and develop a polynomial-time approximation scheme (PTAS) based on local search.

We consider the problem of searching for an intruder in a geometric domain by utilizing multiple search robots. The domain is a simply connected orthogonal polygon with edges parallel to the cartesian coordinate axes. Each robot has a limited sensing capability. We study the problem for both static and mobile intruders. It turns out that the problem of finding an intruder is NP-hard, even for a stationary intruder. Given this intractability, we turn our attention towards developing efficient and robust algorithms, namely methods based on space-filling curves, random search, and cooperative random search. Moreover, for each proposed algorithm, we evaluate the trade-off between the number of search robots and the time required for the robots to complete the search process while considering the geometric properties of the connected orthogonal search area.

We study some variants of the k-Watchman Routes problem, the cooperative version of the classic Watchman Routes problem in a simple polygon. The watchmen may be required to see the whole polygon, or some pre-determined quota of area within the polygon, and we want to minimize the maximum length traveled by any watchman. While the single watchman version of the problem has received much attention is rather well understood, it is not the case for multiple watchmen version. We provide the first tight approximability results for the anchored k-Watchman Routes problem in a simple polygon, assuming k is fixed, by a fully-polynomial time approximation scheme. The basis for the FPTAS is provided by an exact dynamic programming algorithm. If k is a variable, we give constant-factor approximations.

We study three fundamental geometric optimization problems-INDEPENDENT SET, PIERCING SET, and DOMINATING SET-on sets of axis-parallel segments in the plane. We consider special cases in which the segments are either unit length or are anchored on an inclined line (a line with slope-1). When the segments are anchored on both sides, we prove that all three problems are NP-complete2; NP-completeness was known for the corresponding problems with axis-parallel rectangles anchored on an inclined line (Correa et al., 2015 [1], Mudgal and Pandit, 2015 [2], Pandit, 2017, [3]). Further, we prove that the DOMINATING SET problem with unit segments in the plane is NP-complete. When the input segments are anchored on one side of the inclined line, known polynomial-time algorithms can be used to solve the INDEPENDENT SET and PIERCING SET problems. We also discuss approximation and the discrete variants of some of the problems.

Assume that a target is known to be present at an unknown point among a finite set of locations in the plane. We search for it using a mobile robot that has imperfect sensing capabilities. It takes time for the robot to move between locations and search a location; we have a total time budget within which to conduct the search. We study the problem of computing a search path/strategy for the robot that maximizes the probability of detection of the target. Considering non-uniform travel times between points (e.g., based on the distance between them) is crucial for search and rescue applications; such problems have been investigated to a limited extent due to their inherent complexity. In this paper, we describe fast algorithms with performance guarantees for this search problem and some variants, complement them with complexity results, and perform experiments to characterize their performance.

Motivated by the problem of estimating bottleneck capacities on the Internet, we formulate and study the problem of vantage point selection. We are given a graph G=(V, E) whose edges E have unknown capacity values that are to be discovered. Probes from a vantage point, i.e, a vertex v ∈ V, along shortest paths from v to all other vertices, reveal bottleneck edge capacities along each path. Our goal is to select k vantage points from V that reveal the maximum number of bottleneck edge capacities. We consider both a non-adaptive setting where all k vantage points are selected before any bottleneck capacity is revealed, and an adaptive setting where each vantage point selection instantly reveals bottleneck capacities along all shortest paths starting from that point. In the non-adaptive setting, by considering a relaxed model where edge capacities are drawn from a random permutation (which still leaves the problem of maximizing the expected number of revealed edges NP-hard), we are able to give a 1-1/e approximate algorithm. In the adaptive setting we work with the least permissive model where edge capacities are arbitrarily fixed but unknown. We compare with the best solution for the particular input instance (i.e. by enumerating all choices of k tuples), and provide both lower bounds on instance optimal approximation algorithms and upper bounds for trees and planar graphs.

We introduce the contiguous art gallery problem which is to guard the boundary of a simple polygon with a minimum number of guards such that each guard covers exactly one contiguous portion of the boundary. Art gallery problems are often NP-hard. In particular, it is NP-hard to minimize the number of guards to see the boundary of a simple polygon, without the contiguity constraint. This paper is a merge of three concurrent works [17, 53, 62] each showing that (surprisingly) the contiguous art gallery problem is solvable in polynomial time. The common idea of all three approaches is developing a greedy function that maps a point on the boundary to the furthest point on the boundary so that the contiguous interval along the boundary between them could be guarded by one guard. Repeatedly applying this function immediately leads to an OPT+1 approximation. By studying this greedy algorithm, we present three different approaches that achieve an optimal solution. The first and second approach apply this greedy algorithm from different points on the boundary that could be found in advance or on the fly while traversing along the boundary (respectively). The third approach represents this function as a piecewise linear rational function, which can be reduced to an abstract arc cover problem involving infinite families of arcs. We identify other problems that can be represented by similar functions, and solve them via the third approach. From the combinatorial point of view, we show that any n-vertex polygon can be guarded by at most left perpendicular n-2/2 right perpendicular. guards. This bound is tight because there are polygons that require this many guards.

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.

We consider two very basic problems – one in unsupervised and one in supervised learning. In the former, we are given a set of points and have to label half of the points red and half the points blue so as to maximize the red–blue separation, i.e., the length of a shortest bichromatic edge. In the latter, the data (points in the plane) are already labeled red and blue, and we seek a linear classifier (a separator of the two given point sets) that can be described using the smallest integers. We give algorithms for both problems. Our solutions are simple; the main contribution of the paper is highlighting the problems and their algorithmic solutions, which, to our knowledge, have not been presented previously, despite the problems being fundamental to the field. We also consider related problems.

We investigate the DISPERSIVE ART GALLERY PROBLEM with vertex guards and rectangular visibility (r-visibility) for a class of orthogonal polygons that reflect the properties of real-world floor plans: these office-like polygons consist of rectangular rooms and corridors. In the dispersive variant of the Art Gallery Problem, the objective is not to minimize the number of guards but to maximize the minimum geodesic L-1-distance between any two guards, called the dispersion distance. Our main contributions are as follows. We prove that determining whether a vertex guard set can achieve a dispersion distance of 4 in office-like polygons is NP-complete, where vertices of the polygon are restricted to integer coordinates. Additionally, we present a simple worst-case optimal algorithm that guarantees a dispersion distance of 3 in polynomial time. Our complexity result extends to polyominoes, resolving an open question posed by Rieck and Scheffer [27]. When vertex coordinates are allowed to be rational, we establish analogous results, proving that achieving a dispersion distance of 2 + epsilon is NP-hard for any epsilon > 0, while the classic Art Gallery Problem remains solvable in polynomial time for this class of polygons. Furthermore, we give a straightforward polynomial-time algorithm that computes worst-case optimal solutions with a dispersion distance 2. On the other hand, for the more restricted class of hole-free independent office-like polygons, we propose a dynamic programming approach that computes optimal solutions. Moreover, we demonstrate that the problem is practically tractable for arbitrary orthogonal polygons. To this end, we compare solvers based on SAT, CP, and MIP formulations. Notably, SAT solvers efficiently compute optimal solutions for randomly generated instances with up to 1600 vertices in under 15 s.

The well-known Watchman Route problem seeks a shortest route in a polygonal domain from which every point of the domain can be seen. In this paper, we study the cooperative variant of the problem, namely the k-Watchmen Routes problem, in a simple polygon P. We look at both the version in which the k watchmen must collectively see all of P, and the quota version in which they must see a predetermined fraction of P's area. We give an exact pseudopolynomial time algorithm for the k-Watchmen Routes problem in a simple orthogonal polygon P with the constraint that watchmen must move on axis-parallel segments, and there is a given common starting point on the boundary. Further, we give a fully polynomial-time approximation scheme and a constant-factor approximation for unconstrained movement. For the quota version, we give a constant-factor approximation in a simple polygon, utilizing the solution to the (single) Quota Watchman Route problem.

Given a geometric domain $P$, visibility-based search problems seek routes for one or more mobile agents (``watchmen'') to move within $P$ in order to be able to see a portion (or all) of $P$, while optimizing objectives, such as the length(s) of the route(s), the size (e.g., area or volume) of the portion seen, the probability of detecting a target distributed within $P$ according to a prior distribution, etc. The classic watchman route problem seeks a shortest route for an observer, with omnidirectional vision, to see all of $P$. In this paper we study bicriteria optimization problems for a single mobile agent within a polygonal domain $P$ in the plane, with the criteria of route length and area seen. Specifically, we address the problem of computing a minimum length route that sees at least a specified area of $P$ (minimum length, for a given area quota). We also study the problem of computing a length-constrained route that sees as much area as possible. We provide hardness results and approximation algorithms. In particular, for a simple polygon $P$ we provide the first fully polynomial-time approximation scheme for the problem of computing a shortest route seeing an area quota, as well as a (slightly more efficient) polynomial dual approximation. We also consider polygonal domains $P$ (with holes) and the special case of a planar domain consisting of a union of lines. Our results yield the first approximation algorithms for computing a time-optimal search route in $P$ to guarantee some specified probability of detection of a static target within $P$, randomly distributed in $P$ according to a given prior distribution.

We study the problem of guarding the boundary of a simple polygon with a minimum number of guards such that each guard covers a contiguous portion of the boundary. First, we present a simple greedy algorithm for this problem that returns a guard set of size at most OPT + 1, where OPT is the number of guards in an optimal solution. Then, we present a polynomial-time exact algorithm. While the algorithm is not complicated, its correctness proof is rather involved. This result is interesting in the sense that guarding problems are typically NP-hard and, in particular, it is NP-hard to minimize the number of guards to see the boundary of a simple polygon, without the contiguous boundary guarding constraint. From the combinatorial point of view, we show that any n-vertex polygon can be guarded by at most ⌊n-2/2⌋ guards. This bound is tight because there are polygons that require this many guards.

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. 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.

We study the Dispersive Art Gallery Problem with vertex guards: Given a polygon 𝒫, with pairwise geodesic Euclidean vertex distance of at least 1, and a rational number ℓ; decide whether there is a set of vertex guards such that 𝒫 is guarded, and the minimum geodesic Euclidean distance between any two guards (the so-called dispersion distance) is at least ℓ. We show that it is NP-complete to decide whether a polygon with holes has a set of vertex guards with dispersion distance 2. On the other hand, we provide an algorithm that places vertex guards in simple polygons at dispersion distance at least 2. This result is tight, as there are simple polygons in which any vertex guard set has a dispersion distance of at most 2.

We present a new class of problems where the goal is to select a “fair” subgraph H of a given graph G = (V,E), such that H decomposes into many small components. A subgraph H c G is (P, d) fair if every vertex v ϵ P has the same degree d in H, where P c V and d > 0 are input parameters. These problems arise when the goal is to allow individuals to equally participate in activities in such a way that the connected components within an interaction graph, which models potential interactions among people, are of the smallest possible size, so that the spread of the contagion, and the difficulty of contact tracing in case of infection, is minimized. Within a preference graph that models the set of preferred choices for each individual when selecting among available options of where to conduct any particular type of activity (e.g., which gym to attend), we seek to compute the fair subgraph of assignments of individuals to these options, so that the number of people in each connected component (“interaction community”) of the resulting subgraph is minimized, and everyone is given the same number of options for every activity. We show that the fair subgraph selection problem is NP-hard, even for very restricted versions. We then formulate the problem as an integer program, and give a polynomial time computable lower bound on the optimal solution.

Let P be a polygon and C a set of shortcuts, where each shortcut is a directed straight-line segment connecting two vertices of P. A shortcut hull of P is another polygon that encloses P and whose oriented boundary is composed of elements from C. We require P and the output shortcut hull to be weakly simple polygons, which we define as a generalization of simple polygons. Shortcut hulls find their application in cartography, where a common task is to compute simplified representations of area features. We aim at a shortcut hull that has a small area and a small perimeter. Our optimization objective is to minimize a convex combination of these two criteria. If no holes in the shortcut hull are allowed, the problem admits a straight-forward solution via computation of shortest paths. For the more challenging case in which the shortcut hull may contain holes, we present a polynomial-time algorithm that is based on computing a constrained, weighted triangulation of the input polygon's exterior. We use this problem as a starting point for investigating further variants, e.g., restricting the number of edges or bends. We demonstrate that shortcut hulls can be used for the schematization of polygons.

Let $E=\{e_1,\ldots,e_n\}$ be a set of $C$-oriented disjoint segments in the plane, where $C$ is a given finite set of orientations that spans the plane, and let $s$ and $t$ be two points. %(We also require that for each orientation in $C$, its opposite orientation is also in $C$.) We seek a minimum-link $C$-oriented tour of $E$, that is, a polygonal path $\pi$ from $s$ to $t$ that visits the segments of $E$ in order, such that, the orientations of its edges are in $C$ and their number is minimum. We present an algorithm for computing such a tour in $O(|C|^2 \cdot n^2)$ time. This problem already captures most of the difficulties occurring in the study of the more general problem, in which $E$ is a set of not-necessarily-disjoint $C$-oriented polygons.