arXiv subjects

Anil Maheshwari

Publications and source records attributed to Anil Maheshwari.

At least 37 records · Page 2Linked to original sources

arXiv2024

On 1-Planar Graphs with Bounded Cop-Number

Cops and Robbers is a type of pursuit-evasion game played on a graph where a set of cops try to capture a single robber. The cops first choose their initial vertex positions, and later the robber chooses a vertex. The cops and robbers make their moves in alternate turns: in the cops' turn, every cop can either choose to move to an adjacent vertex or stay on the same vertex, and likewise the robber in his turn. If the cops can capture the robber in a finite number of rounds, the cops win, otherwise the robber wins. The cop-number of a graph is the minimum number of cops required to catch a robber in the graph. It has long been known that graphs embedded on surfaces (such as planar graphs and toroidal graphs) have a small cop-number. Recently, Durocher et al. [Graph Drawing, 2023] investigated the problem of cop-number for the class of $1$-planar graphs, which are graphs that can be embedded in the plane such that each edge is crossed at most once. They showed that unlike planar graphs which require just three cops, 1-planar graphs have an unbounded cop-number. On the positive side, they showed that maximal 1-planar graphs require only three cops by crucially using the fact that the endpoints of every crossing in an embedded maximal 1-planar graph induce a $K_4$. In this paper, we show that the cop-number remains bounded even under the relaxed condition that the endpoints induce at least three edges. More precisely, let an $\times$-crossing of an embedded 1-planar graph be a crossing whose endpoints induce a matching; i.e., there is no edge connecting the endpoints apart from the crossing edges themselves. We show that any 1-planar graph that can be embedded without $\times$-crossings has cop-number at most 21. Moreover, any 1-planar graph that can be embedded with at most $γ$ $\times$-crossings has cop-number at most $γ+ 21$.

cs.DM↗

arXiv2024

Sparse graphs with local covering conditions on edges

In 1988, Erdős suggested the question of minimizing the number of edges in a connected $n$-vertex graph where every edge is contained in a triangle. Shortly after, Catlin, Grossman, Hobbs, and Lai resolved this in a stronger form. In this paper, we study a natural generalization of the question of Erdős in which we replace `triangle' with `clique of order $k$' for ${k\ge 3}$. We completely resolve this generalized question with the characterization of all extremal graphs. Motivated by applications in data science, we also study another generalization of the question of Erdős where every edge is required to be in at least $\ell$ triangles for $\ell\ge 2$ instead of only one triangle. We completely resolve this problem for $\ell = 2$.

math.CO↗

arXiv2024

Computing shortest paths amid non-overlapping weighted disks

In this article, we present an approximation algorithm for solving the Weighted Region Problem amidst a set of $ n $ non-overlapping weighted disks in the plane. For a given parameter $ \varepsilon \in (0,1]$, the length of the approximate path is at most $ (1 +\varepsilon) $ times larger than the length of the actual shortest path. The algorithm is based on the discretization of the space by placing points on the boundary of the disks. Using such a discretization we can use Dijkstra's algorithm for computing a shortest path in the geometric graph obtained in (pseudo-)polynomial time.

cs.CG↗

arXiv2024

Online Class Cover Problem

In this paper, we study the online class cover problem where a (finite or infinite) family $\cal F$ of geometric objects and a set ${\cal P}_r$ of red points in $\mathbb{R}^d$ are given a prior, and blue points from $\mathbb{R}^d$ arrives one after another. Upon the arrival of a blue point, the online algorithm must make an irreversible decision to cover it with objects from $\cal F$ that do not cover any points of ${\cal P}_r$. The objective of the problem is to place a minimum number of objects. When $\cal F$ consists of axis-parallel unit squares in $\mathbb{R}^2$, we prove that the competitive ratio of any deterministic online algorithm is $Ω(\log |{\cal P}_r|)$, and also propose an $O(\log |{\cal P}_r|)$-competitive deterministic algorithm for the problem.

cs.CG↗

arXiv2024

Euclidean Maximum Matchings in the Plane---Local to Global

Let $M$ be a perfect matching on a set of points in the plane where every edge is a line segment between two points. We say that $M$ is globally maximum if it is a maximum-length matching on all points. We say that $M$ is $k$-local maximum if for any subset $M'=\{a_1b_1,\dots,a_kb_k\}$ of $k$ edges of $M$ it holds that $M'$ is a maximum-length matching on points $\{a_1,b_1,\dots,a_k,b_k\}$. We show that local maximum matchings are good approximations of global ones. Let $μ_k$ be the infimum ratio of the length of any $k$-local maximum matching to the length of any global maximum matching, over all finite point sets in the Euclidean plane. It is known that $μ_k\geqslant \frac{k-1}{k}$ for any $k\geqslant 2$. We show the following improved bounds for $k\in\{2,3\}$: $\sqrt{3/7}\leqslantμ_2< 0.93 $ and $\sqrt{3}/2\leqslantμ_3< 0.98$. We also show that every pairwise crossing matching is unique and it is globally maximum. Towards our proof of the lower bound for $μ_2$ we show the following result which is of independent interest: If we increase the radii of pairwise intersecting disks by factor $2/\sqrt{3}$, then the resulting disks have a common intersection.

cs.CG↗

arXiv2023

Distance-Preserving Graph Compression Techniques

We study the problem of distance-preserving graph compression for weighted paths and trees. The problem entails a weighted graph $G = (V, E)$ with non-negative weights, and a subset of edges $E^{\prime} \subset E$ which needs to be removed from G (with their endpoints merged as a supernode). The goal is to redistribute the weights of the deleted edges in a way that minimizes the error. The error is defined as the sum of the absolute differences of the shortest path lengths between different pairs of nodes before and after contracting $E^{\prime}$. Based on this error function, we propose optimal approaches for merging any subset of edges in a path and a single edge in a tree. Previous works on graph compression techniques aimed at preserving different graph properties (such as the chromatic number) or solely focused on identifying the optimal set of edges to contract. However, our focus in this paper is on achieving optimal edge contraction (when the contracted edges are provided as input) specifically for weighted trees and paths.

cs.DS↗

arXiv2021

Approximating bottleneck spanning trees on partitioned tuples of points

We present approximation algorithms for the following NP-hard optimization problems related to bottleneck spanning trees in metric spaces. 1. The disjoint bottleneck spanning tree problem: Given $n$ pairs of points in a metric space, find two disjoint trees each containing exactly one point from each pair and minimize the largest edge length (over all edges of both trees). It is known that approximating this problem by a factor better than 2 is NP-hard. We present a 4-approximation algorithm for this problem. This improves upon the previous best known approximation ratio of $9$. Our algorithm extends to a $(3k-2)$-approximation for a more general case where points are partitioned into $k$-tuples and we seek $k$ disjoint trees. 2. The generalized bottleneck spanning tree problem: Given $n$ points in some metric space that are partitioned into clusters of size at most 2, find a tree that contains exactly one point from each cluster and minimizes the largest edge length. We show that it is NP-hard to approximate this problem by a factor better than 2, and present a 3-approximation algorithm. 3. The partitioned bottleneck spanning tree problem: Given $kn$ points in some metric space, find $k$ trees each containing exactly $n$ points and minimize the largest edge length (over all edges of the $k$ trees). We show that it is NP-hard to approximate this problem by a factor better than 2 for any $k\ge 2$. We present an $α$-approximation algorithm for this problem where $α=2$ for $k=2,3$ and $α=3$ for $k\ge 4$. Towards obtaining these approximation ratios we present tight upper bounds on the edge lengths of $k$ equal-size disjoint trees that can be obtained from the nodes of a given tree. This result is of independent interest. Our hardness proofs imply that it is NP-hard to approximate the non-metric version of the above problems within any constant factor.

cs.CG↗

arXiv2021

Exact and Approximation Algorithms for Many-To-Many Point Matching in the Plane

Given two sets $S$ and $T$ of points in the plane, of total size $n$, a {many-to-many} matching between $S$ and $T$ is a set of pairs $(p,q)$ such that $p\in S$, $q\in T$ and for each $r\in S\cup T$, $r$ appears in at least one such pair. The {cost of a pair} $(p,q)$ is the (Euclidean) distance between $p$ and $q$. In the {minimum-cost many-to-many matching} problem, the goal is to compute a many-to-many matching such that the sum of the costs of the pairs is minimized. This problem is a restricted version of minimum-weight edge cover in a bipartite graph, and hence can be solved in $O(n^3)$ time. In a more restricted setting where all the points are on a line, the problem can be solved in $O(n\log n)$ time [Colannino, Damian, Hurtado, Langerman, Meijer, Ramaswami, Souvaine, Toussaint; Graphs Comb., 2007]. However, no progress has been made in the general planar case in improving the cubic time bound. In this paper, we obtain an $O(n^2\cdot poly(\log n))$ time exact algorithm and an $O( n^{3/2}\cdot poly(\log n))$ time $(1+ε)$-approximation in the planar case. Our results affirmatively address an open problem posed in [Colannino et al., Graphs Comb., 2007].

cs.CG↗

arXiv2021

A linear-time algorithm for semitotal domination in strongly chordal graphs

In a graph $G=(V,E)$ with no isolated vertex, a dominating set $D \subseteq V$, is called a semitotal dominating set if for every vertex $u \in D$ there is another vertex $v \in D$, such that distance between $u$ and $v$ is at most two in $G$. Given a graph $G=(V,E)$ without isolated vertices, the Minimum Semitotal Domination problem is to find a minimum cardinality semitotal dominating set of $G$. The semitotal domination number, denoted by $γ_{t2}(G)$, is the minimum cardinality of a semitotal dominating set of $G$. The decision version of the problem remains NP-complete even when restricted to chordal graphs, chordal bipartite graphs, and planar graphs. Galby et al. in [6] proved that the problem can be solved in polynomial time for bounded MIM-width graphs which includes many well known graph classes, but left the complexity of the problem in strongly chordal graphs unresolved. Henning and Pandey in [20] also asked to resolve the complexity status of the problem in strongly chordal graphs. In this paper, we resolve the complexity of the problem in strongly chordal graphs by designing a linear-time algorithm for the problem.

math.CO↗

arXiv2021

Computing Maximum Independent Set on Outerstring Graphs and Their Relatives

A graph $G$ with $n$ vertices is called an outerstring graph if it has an intersection representation of a set of $n$ curves inside a disk such that one endpoint of every curve is attached to the boundary of the disk. Given an outerstring graph representation, the Maximum Independent Set (MIS) problem of the underlying graph can be computed in $O(s^3)$ time, where $s$ is the number of segments in the representation (Keil et al., Comput. Geom., 60:19--25, 2017). If the strings are of constant size (e.g., line segments, L-shapes, etc.), then the algorithm takes $O(n^3)$ time. In this paper, we examine the fine-grained complexity of the MIS problem on some well-known outerstring representations. We show that solving the MIS problem on grounded segment and grounded square-L representations is at least as hard as solving MIS on circle graph representations. Note that no $O(n^{2-δ})$-time algorithm, $δ>0$, is known for the MIS problem on circle graphs. For the grounded string representations where the strings are $y$-monotone simple polygonal paths of constant length with segments at integral coordinates, we solve MIS in $O(n^2)$ time and show this to be the best possible under the strong exponential time hypothesis (SETH). For the intersection graph of $n$ L-shapes in the plane, we give a $(4\cdot \log OPT)$-approximation algorithm for MIS (where $OPT$ denotes the size of an optimal solution), improving the previously best-known $(4\cdot \log n)$-approximation algorithm of Biedl and Derka (WADS 2017).

cs.CG↗

arXiv2021

Constant Delay Lattice Train Schedules

The following geometric vehicle scheduling problem has been considered: given continuous curves $f_1, \ldots, f_n : \mathbb{R} \rightarrow \mathbb{R}^2$, find non-negative delays $t_1, \ldots, t_n$ minimizing $\max \{ t_1, \ldots, t_n \}$ such that, for every distinct $i$ {and $j$} and every time $t$, $| f_j (t - t_j) - f_i (t - t_i) | > \ell$, where~$\ell$ is a given safety distance. We study a variant of this problem where we consider trains (rods) of fixed length $\ell$ that move at constant speed and sets of train lines (tracks), each of which consisting of an axis-parallel line-segment with endpoints in the integer lattice $\mathbb{Z}^d$ and of a direction of movement (towards $\infty$ {or $- \infty$}). We are interested in upper bounds on the maximum delay we need to introduce on any line to avoid collisions, but more specifically on universal upper bounds that apply no matter the set of train lines. We show small universal constant upper bounds for $d = 2$ and any given $\ell$ and also for $d = 3$ and $\ell = 1$. Through clique searching, we are also able to show that several of these upper bounds are tight.

cs.CG↗

arXiv2020

A Simple Randomized $O(n \log n)$--Time Closest-Pair Algorithm in Doubling Metrics

Consider a metric space $(P,dist)$ with $N$ points whose doubling dimension is a constant. We present a simple, randomized, and recursive algorithm that computes, in $O(N \log N)$ expected time, the closest-pair distance in $P$. To generate recursive calls, we use previous results of Har-Peled and Mendel, and Abam and Har-Peled for computing a sparse annulus that separates the points in a balanced way.

cs.CG↗

arXiv2020

Maximum Bipartite Subgraph of Geometric Intersection Graphs

We study the Maximum Bipartite Subgraph (MBS) problem, which is defined as follows. Given a set $S$ of $n$ geometric objects in the plane, we want to compute a maximum-size subset $S'\subseteq S$ such that the intersection graph of the objects in $S'$ is bipartite. We first give a simple $O(n)$-time algorithm that solves the MBS problem on a set of $n$ intervals. We also give an $O(n^2)$-time algorithm that computes a near-optimal solution for the problem on circular-arc graphs. We show that the MBS problem is NP-hard on geometric graphs for which the maximum independent set is NP-hard (hence, it is NP-hard even on unit squares and unit disks). On the other hand, we give a PTAS for the problem on unit squares and unit disks. Moreover, we show fast approximation algorithms with small-constant factors for the problem on unit squares, unit disks and unit-height rectangles. Finally, we study a closely related geometric problem, called Maximum Triangle-free Subgraph (TFS), where the objective is the same as that of MBS except the intersection graph induced by the set $S'$ needs to be triangle-free only (instead of being bipartite).

cs.DM↗

arXiv2019

Linear Size Planar Manhattan Network for Convex Point Sets

Let $G = (V, E)$ be an edge-weighted geometric graph such that every edge is horizontal or vertical. The weight of an edge $uv \in E$ is its length. Let $ W_G (u,v)$ denote the length of a shortest path between a pair of vertices $u$ and $v$ in $G$. The graph $G$ is said to be a Manhattan network for a given point set $ P $ in the plane if $P \subseteq V$ and $\forall p,q \in P$, $ W_G (p,q)=|pq|_1$. In addition to $ P$, graph $G$ may also include a set $T$ of Steiner points in its vertex set $V$. In the Manhattan network problem, the objective is to construct a Manhattan network of small size for a set of $ n $ points. This problem was first considered by Gudmundsson et al.\cite{gudmundsson2007small}. They give a construction of a Manhattan network of size $Θ(n \log n)$ for general point set in the plane. We say a Manhattan network is planar if it can be embedded in the plane without any edge crossings. In this paper, we construct a linear size planar Manhattan network for convex point set in linear time using $\mathcal{ O}(n)$ Steiner points. We also show that, even for convex point set, the construction in Gudmundsson et al. \cite{gudmundsson2007small} needs $Ω(n \log n)$ Steiner points and the network may not be planar.

cs.CG↗

arXiv2019

Packing Boundary-Anchored Rectangles and Squares

Consider a set $P$ of $n$ points on the boundary of an axis-aligned square $Q$. We study the boundary-anchored packing problem on $P$ in which the goal is to find a set of interior-disjoint axis-aligned rectangles in $Q$ such that each rectangle is anchored (has a corner at some point in $P$), each point in $P$ is used to anchor at most one rectangle, and the total area of the rectangles is maximized. Here, a rectangle is anchored at a point $p$ in $P$ if one of its corners coincides with $p$. In this paper, we show how to solve this problem in time linear in $n$, provided that the points of $P$ are given in sorted order along the boundary of $Q$. We also consider the problem for anchoring squares and give an $O(n^4)$-time algorithm when the points in $P$ lie on two opposite sides of $Q$.

cs.CG↗

arXiv2019

Color spanning Localized query

Let P be a set of n points and each of the points is colored with one of the k possible colors. We present efficient algorithms to pre-process P such that for a given query point q, we can quickly identify the smallest color spanning object of the desired type containing q. In this paper, we focus on (i) intervals, (ii) axis-parallel square, (iii) axis-parallel rectangle, (iv) equilateral triangle of fixed orientation and (v) circle, as our desired type of objects.

cs.CG↗

arXiv2019

Flip Distance to some Plane Configurations

We study an old geometric optimization problem in the plane. Given a perfect matching $M$ on a set of $n$ points in the plane, we can transform it to a non-crossing perfect matching by a finite sequence of flip operations. The flip operation removes two crossing edges from $M$ and adds two non-crossing edges. Let $f(M)$ and $F(M)$ denote the minimum and maximum lengths of a flip sequence on $M$, respectively. It has been proved by Bonnet and Miltzow (2016) that $f(M)=O(n^2)$ and by van Leeuwen and Schoone (1980) that $F(M)=O(n^3)$. We prove that $f(M)=O(nΔ)$ where $Δ$ is the spread of the point set, which is defined as the ratio between the longest and the shortest pairwise distances. This improves the previous bound if the point set has sublinear spread. For a matching $M$ on $n$ points in convex position we prove that $f(M)=n/2-1$ and $F(M)={{n/2} \choose 2}$; these bounds are tight. Any bound on $F(\cdot)$ carries over to the bichromatic setting, while this is not necessarily true for $f(\cdot)$. Let $M'$ be a bichromatic matching. The best known upper bound for $f(M')$ is the same as for $F(M')$, which is essentially $O(n^3)$. We prove that $f(M')\le n-2$ for points in convex position, and $f(M')= O(n^2)$ for semi-collinear points. The flip operation can also be defined on spanning trees. For a spanning tree $T$ on a convex point set we show that $f(T)=O(n\log n)$.

cs.CG↗

arXiv2018

On the Minimum Consistent Subset Problem

Let $P$ be a set of $n$ colored points in the plane. Introduced by Hart (1968), a consistent subset of $P$, is a set $S\subseteq P$ such that for every point $p$ in $P\setminus S$, the closest point of $p$ in $S$ has the same color as $p$. The consistent subset problem is to find a consistent subset of $P$ with minimum cardinality. This problem is known to be NP-complete even for two-colored point sets. Since the initial presentation of this problem, aside from the hardness results, there has not been a significant progress from the algorithmic point of view. In this paper we present the following algorithmic results: 1. The first subexponential-time algorithm for the consistent subset problem. 2. An $O(n\log n)$-time algorithm that finds a consistent subset of size two in two-colored point sets (if such a subset exists). Towards our proof of this running time we present a deterministic $O(n \log n)$-time algorithm for computing a variant of the compact Voronoi diagram; this improves the previously claimed expected running time. 3. An $O(n\log^2 n)$-time algorithm that finds a minimum consistent subset in two-colored point sets where one color class contains exactly one point; this improves the previous best known $O(n^2)$ running time which is due to Wilfong (SoCG 1991). 4. An $O(n)$-time algorithm for the consistent subset problem on collinear points; this improves the previous best known $O(n^2)$ running time. 5. A non-trivial $O(n^6)$-time dynamic programming algorithm for the consistent subset problem on points arranged on two parallel lines. To obtain these results, we combine tools from planar separators, additively-weighted Voronoi diagrams with respect to convex distance functions, point location in farthest-point Voronoi diagrams, range trees, paraboloid lifting, minimum covering of a circle with arcs, and several geometric transformations.

cs.CG↗