arXiv subjects

Ahmad Biniaz

Publications and source records attributed to Ahmad Biniaz.

At least 19 recordsLinked to original sources

arXiv2026

Closest Pair Queries in Vertical Slabs and Tight Bounds on the Number of Possible Answers

Let $S$ be a set of $n$ points in $\mathbb{R}^d$, where $d \geq 2$ is a constant, and let $H_1,H_2,\ldots,H_{m+1}$ be a sequence of vertical hyperplanes that are sorted by their first coordinates, such that exactly $n/m$ points of $S$ are between any two successive hyperplanes. Let $A(S,m)$ be the set of different closest pairs in the ${{m+1} \choose 2}$ vertical slabs that are bounded by $H_i$ and $H_j$, over all $1 \leq i < j \leq m+1$. We prove tight bounds for the largest possible size of $A(S,m)$, over all point sets of size $n$, and for all values of $1 \leq m \leq n$. As a result of these bounds, we obtain, for any constant $ε>0$, a data structure of size $O(n)$, such that for any vertical query slab $Q$, the closest pair in the set $Q \cap S$ can be reported in $O(n^{1/2+ε})$ time. Prior to this work, no linear space data structure with sublinear query time was known.

cs.CG↗

arXiv2025

Noncrossing Longest Paths and Cycles

Edge crossings in geometric graphs are sometimes undesirable as they could lead to unwanted situations such as collisions in motion planning and inconsistency in VLSI layout. Short geometric structures such as shortest perfect matchings, shortest spanning trees, shortest spanning paths, and shortest spanning cycles on a given point set are inherently noncrossing. However, the longest such structures need not be noncrossing. In fact, it is intuitive to expect many edge crossings in various geometric graphs that are longest. Recently, Álvarez-Rebollar, Cravioto-Lagos, Marín, Solé-Pi, and Urrutia (Graphs and Combinatorics, 2024) constructed a set of points for which the longest perfect matching is noncrossing. They raised several challenging questions in this direction. In particular, they asked whether the longest spanning path, on any finite set of points in the plane, must have a pair of crossing edges. They also conjectured that the longest spanning cycle must have a pair of crossing edges. In this paper, we give a negative answer to the question and also refute the conjecture. We present a framework for constructing arbitrarily large point sets for which the longest perfect matchings, the longest spanning paths, and the longest spanning cycles are noncrossing.

cs.CG↗

arXiv2025

Metric and Geometric Spanners that are Resilient to Degree-Bounded Edge Faults

Let $H$ be an edge-weighted graph, and let $G$ be a subgraph of $H$. We say that $G$ is an $f$-fault-tolerant $t$-spanner for $H$, if the following is true for any subset $F$ of at most $f$ edges of $G$: For any two vertices $p$ and $q$, the shortest-path distance between $p$ and $q$ in the graph $G \setminus F$ is at most $t$ times the shortest-path distance between $p$ and $q$ in the graph $H \setminus F$. Recently, Bodwin, Haeupler, and Parter generalized this notion to the case when $F$ can be any set of edges in $G$, as long as the maximum degree of $F$ is at most $f$. They gave constructions for general graphs $H$. We first consider the case when $H$ is a complete graph whose vertex set is an arbitrary metric space. We show that if this metric space contains a $t$-spanner with $m$ edges, then it also contains a graph $G$ with $O(fm)$ edges, that is resilient to edge faults of maximum degree $f$ and has stretch factor $O(ft)$. Next, we consider the case when $H$ is a complete graph whose vertex set is a metric space that admits a well-separated pair decomposition. We show that, if the metric space has such a decomposition of size $m$, then it contains a graph with at most $(2f+1)^2 m$ edges, that is resilient to edge faults of maximum degree $f$ and has stretch factor at most $1+\varepsilon$, for any given $\varepsilon > 0$. For example, if the vertex set is a set of $n$ points in $\mathbb{R}^d$ ($d$ being a constant) or a set of $n$ points in a metric space of bounded doubling dimension, then the spanner has $O(f^2 n)$ edges. Finally, for the case when $H$ is a complete graph on $n$ points in $\mathbb{R}^d$, we show how natural variants of the Yao- and $Θ$-graphs lead to graphs with $O(fn)$ edges, that are resilient to edge faults of maximum degree $f$ and have stretch factor at most $1+\varepsilon$, for any given $\varepsilon > 0$.

cs.CG↗

arXiv2025

An Improved Bound for Plane Covering Paths

A covering path for a finite set $P$ of points in the plane is a polygonal path such that every point of $P$ lies on a segment of the path. The vertices of the path need not be at points of $P$. A covering path is plane if its segments do not cross each other. Let $π(n)$ be the minimum number such that every set of $n$ points in the plane admits a plane covering path with at most $π(n)$ segments. We prove that $π(n)\le \lceil6n/7\rceil$. This improves the previous best-known upper bound of $\lceil 21n/22\rceil$, due to Biniaz (SoCG 2023). Our proof is constructive and yields a simple $O(n\log n)$-time algorithm for computing a plane covering path.

cs.CG↗

arXiv2025

On Separating Path and Tree Systems in Graphs

We explore the concept of separating systems of vertex sets of graphs. A separating system of a set $X$ is a collection of subsets of $X$ such that for any pair of distinct elements in $X$, there exists a set in the separating system that contains exactly one of the two elements. A separating system of the vertex set of a graph $G$ is called a vertex-separating path (tree) system of $G$ if the elements of the separating system are paths (trees) in the graph $G$. In this paper, we focus on the size of the smallest vertex-separating path (tree) system for different types of graphs, including trees, grids, and maximal outerplanar graphs.

cs.DM↗

arXiv2025

Contiguous Boundary Guarding

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 $\lfloor \frac{n-2}{2}\rfloor$ guards. This bound is tight because there are polygons that require this many guards.

cs.CG↗

arXiv2025

Polychromatic Coloring of Tuples in Hypergraphs

A hypergraph $H$ consists of a set $V$ of vertices and a set $E$ of hyperedges that are subsets of $V$. A $t$-tuple of $H$ is a subset of $t$ vertices of $V$. A $t$-tuple $k$-coloring of $H$ is a mapping of its $t$-tuples into $k$ colors. A coloring is called $(t,k,f)$-polychromatic if each hyperedge of $E$ that has at least $f$ vertices contains tuples of all the $k$ colors. Let $f_H(t,k)$ be the minimum $f$ such that $H$ has a $(t,k,f)$-polychromatic coloring. For a family of hypergraphs $\cal{H}$ let $f_{\cal{H}}(t,k)$ be the maximum $f_H(t,k)$ over all hypergraphs $H$ in $\cal{H}$. We present several bounds on $f_{\cal{H}}(t,k)$ for $t\ge 2$. - Let $\cal{H}$ be the family of hypergraphs $H$ that is obtained by taking any set $P$ of points in $\Re^2$, setting $V:=P$ and $E:=\{d\cap P\colon d\text{ is a disk in }\Re^2\}$. We prove that $f_\cal{H}(2,k)\le 3.7^k$, that is, the pairs of points (2-tuples) can be $k$-colored such that any disk containing at least $3.7^k$ points has pairs of all colors. - For the family $\mathcal{H}$ of shrinkable hypergraphs of VC-dimension at most $d$ we prove that $ f_\cal{H}(d{+}1,k) \leq c^k$ for some constant $c=c(d)$. We also prove that every hypergraph with $n$ vertices and with VC-dimension at most $d$ has a $(d{+}1)$-tuple $T$ of depth at least $\frac{n}{c}$, i.e., any hyperedge that contains $T$ also contains $\frac{n}{c}$ other vertices. - For the relationship between $t$-tuple coloring and vertex coloring in any hypergraph $H$ we establish the inequality $\frac{1}{e}\cdot tk^{\frac{1}{t}}\le f_H(t,k)\le f_H(1,tk^{\frac{1}{t}})$. For the special case of $k=2$, we prove that $t+1\le f_H(t,2)\le\max\{f_H(1,2), t+1\}$; this improves upon the previous best known upper bound. - We generalize some of our results to higher dimensions, other shapes, pseudo-disks, and also study the relationship between tuple coloring and epsilon nets.

cs.CG↗

arXiv2024

Minimum Plane Bichromatic Spanning Trees

For a set of red and blue points in the plane, a minimum bichromatic spanning tree (MinBST) is a shortest spanning tree of the points such that every edge has a red and a blue endpoint. A MinBST can be computed in $O(n\log n)$ time where $n$ is the number of points. In contrast to the standard Euclidean MST, which is always plane (noncrossing), a MinBST may have edges that cross each other. However, we prove that a MinBST is quasi-plane, that is, it does not contain three pairwise crossing edges, and we determine the maximum number of crossings. Moreover, we study the problem of finding a minimum plane bichromatic spanning tree (MinPBST) which is a shortest bichromatic spanning tree with pairwise noncrossing edges. This problem is known to be NP-hard. The previous best approximation algorithm, due to Borgelt et al. (2009), has a ratio of $O(\sqrt{n})$. It is also known that the optimum solution can be computed in polynomial time in some special cases, for instance, when the points are in convex position, collinear, semi-collinear, or when one color class has constant size. We present an $O(\log n)$-factor approximation algorithm for the general case.

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

Improved Bounds for Covering Paths and Trees in the Plane

A covering path for a planar point set is a path drawn in the plane with straight-line edges such that every point lies at a vertex or on an edge of the path. A covering tree is defined analogously. Let $π(n)$ be the minimum number such that every set of $n$ points in the plane can be covered by a noncrossing path with at most $π(n)$ edges. Let $τ(n)$ be the analogous number for noncrossing covering trees. Dumitrescu, Gerbner, Keszegh, and Tóth (Discrete & Computational Geometry, 2014) established the following inequalities: \[\frac{5n}{9} - O(1) < π(n) < \left(1-\frac{1}{601080391}\right)n, \quad\text{and} \quad\frac{9n}{17} - O(1) < τ(n)\leqslant \left\lfloor\frac{5n}{6}\right\rfloor.\] We report the following improved upper bounds: \[π(n)\leqslant \left(1-\frac{1}{22}\right)n, \quad\text{and}\quad τ(n)\leqslant \left\lceil\frac{4n}{5}\right\rceil.\] In the same context we study rainbow polygons. For a set of colored points in the plane, a perfect rainbow polygon is a simple polygon that contains exactly one point of each color in its interior or on its boundary. Let $ρ(k)$ be the minimum number such that every $k$-colored point set in the plane admits a perfect rainbow polygon of size $ρ(k)$. Flores-Peñaloza, Kano, Martínez-Sandoval, Orden, Tejel, Tóth, Urrutia, and Vogtenhuber (Discrete Mathematics, 2021) proved that $20k/19 - O(1) <ρ(k) < 10k/7 + O(1).$ We report the improved upper bound $ρ(k)< 7k/5 + O(1)$. To obtain the improved bounds we present simple $O(n\log n)$-time algorithms that achieve paths, trees, and polygons with our desired number of edges.

cs.CG↗

arXiv2023

Token Swapping on Trees

The input to the token swapping problem is a graph with vertices $v_1, v_2, \ldots, v_n$, and $n$ tokens with labels $1, 2, \ldots, n$, one on each vertex. The goal is to get token $i$ to vertex $v_i$ for all $i= 1, \ldots, n$ using a minimum number of swaps, where a swap exchanges the tokens on the endpoints of an edge. We present some results about token swapping on a tree, also known as "sorting with a transposition tree": 1. An optimum swap sequence may need to perform a swap on a leaf vertex that has the correct token (a "happy leaf"), disproving a conjecture of Vaughan. 2. Any algorithm that fixes happy leaves -- as all known approximation algorithms for the problem do -- has approximation factor at least $4/3$. Furthermore, the two best-known 2-approximation algorithms have approximation factor exactly 2. 3. A generalized problem -- weighted coloured token swapping -- is NP-complete on trees, even when they are restricted to be subdivided stars, but solvable in polynomial time on paths and stars. In this version, tokens and vertices have colours, and colours have weights. The goal is to get every token to a vertex of the same colour, and the cost of a swap is the sum of the weights of the two tokens involved.

cs.CC↗

arXiv2022

Acute Tours in the Plane

We confirm the following conjecture of Fekete and Woeginger from 1997: for any sufficiently large even number $n$, every set of $n$ points in the plane can be connected by a spanning tour (Hamiltonian cycle) consisting of straight-line edges such that the angle between any two consecutive edges is at most $π/2$. Our proof is constructive and suggests a simple $O(n\log n)$-time algorithm for finding such a tour. The previous best-known upper bound on the angle is $2π/3$, and it is due to Dumitrescu, Pach and Tóth (2009).

cs.CG↗

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

A short proof of the non-biplanarity of $K_9$

Battle, Harary, and Kodama (1962) and independently Tutte (1963) proved that the complete graph with nine vertices is not biplanar. Aiming towards simplicity and brevity, in this note we provide a short proof of this claim.

math.CO↗

arXiv2020

Improved approximation ratios for two Euclidean maximum spanning tree problems

We study the following two maximization problems related to spanning trees in the Euclidean plane. It is not known whether or not these problems are NP-hard. We present approximation algorithms with better approximation ratios for both problems. The improved ratios are obtained mainly by employing the Steiner ratio, which has not been used in this context earlier. (i) Longest noncrossing spanning tree: Given a set of points in the plane, the goal is to find a maximum-length noncrossing spanning tree. Alon, Rajagopalan, and Suri (SoCG 1993) studied this problem for the first time and gave a $0.5$-approximation algorithm. Over the years, the approximation ratio has been successively improved to $0.502$, $0.503$, and to $0.512$ which is the current best ratio, due to Cabello et al.. We revisit this problem and improve the ratio further to $0.519$. The improvement is achieved by a collection of ideas, some from previous works and some new ideas (including the use of the Steiner ratio), along with a more refined analysis. (ii) Longest spanning tree with neighborhoods: Given a collection of regions (neighborhoods) in the plane, the goal is to select a point in each neighborhood so that the longest spanning tree on selected points has maximum length. We present an algorithm with approximation ratio $0.524$ for this problem. The previous best ratio, due to Chen and Dumitrescu, is $0.511$ which is in turn the first improvement beyond the trivial ratio $0.5$. Our algorithm is fairly simple, its analysis is relatively short, and it takes linear time after computing a diametral pair of points. The simplicity comes from the fact that our solution belongs to a set containing three stars and one double-star. The shortness and the improvement come from the use of the Steiner ratio.

cs.CG↗

arXiv2020

Better approximation algorithms for maximum weight internal spanning trees in cubic graphs and claw-free graphs

Given a connected vertex-weighted graph $G$, the maximum weight internal spanning tree (MaxwIST) problem asks for a spanning tree of $G$ that maximizes the total weight of internal nodes. This problem is NP-hard and APX-hard, with the currently best known approximation factor $1/2$ (Chen et al., Algorithmica 2019). For the case of claw-free graphs, Chen et al. present an involved approximation algorithm with approximation factor $7/12$. They asked whether it is possible to improve these ratios, in particular for claw-free graphs and cubic graphs. We improve the approximation factors for the MaxwIST problem in cubic graphs and claw-free graphs. For cubic graphs we present an algorithm that computes a spanning tree whose total weight of internal vertices is at least $\frac{3}{4}-\frac{3}{n}$ times the total weight of all vertices, where $n$ is the number of vertices of $G$. This ratio is almost tight for large values of $n$. For claw-free graphs of degree at least three, we present an algorithm that computes a spanning tree whose total internal weight is at least $\frac{3}{5}-\frac{1}{n}$ times the total vertex weight. The degree constraint is necessary as this ratio may not be achievable if we allow vertices of degree less than three. With the above ratios, we immediately obtain better approximation algorithms with factors $\frac{3}{4}-ε$ and $\frac{3}{5}-ε$ for the MaxwIST problem in cubic graphs and claw-free graphs of degree at least three, for any $ε>0$. In addition to improving the approximation factors, the new algorithms are relatively short compared to that of Chen et al.. The new algorithms are fairly simple, and employ a variant of the depth-first search algorithm that selects a relatively-large-weight vertex in every branching step. Moreover, the new algorithms take linear time while previous algorithms for similar problem instances are super-linear.

cs.DS↗

arXiv2020

Compatible Paths on Labelled Point Sets

Let $P$ and $Q$ be finite point sets of the same cardinality in $\mathbb{R}^2$, each labelled from $1$ to $n$. Two noncrossing geometric graphs $G_P$ and $G_Q$ spanning $P$ and $Q$, respectively, are called compatible if for every face $f$ in $G_P$, there exists a corresponding face in $G_Q$ with the same clockwise ordering of the vertices on its boundary as in $f$. In particular, $G_P$ and $G_Q$ must be straight-line embeddings of the same connected $n$-vertex graph. Deciding whether two labelled point sets admit compatible geometric paths is known to be NP-complete. We give polynomial-time algorithms to find compatible paths or report that none exist in three scenarios: $O(n)$ time for points in convex position; $O(n^2)$ time for two simple polygons, where the paths are restricted to remain inside the closed polygons; and $O(n^2 \log n)$ time for points in general position if the paths are restricted to be monotone.

cs.CG↗

arXiv2019

Euclidean Bottleneck Bounded-Degree Spanning Tree Ratios

Inspired by the seminal works of Khuller et al. (STOC 1994) and Chan (SoCG 2003) we study the bottleneck version of the Euclidean bounded-degree spanning tree problem. A bottleneck spanning tree is a spanning tree whose largest edge-length is minimum, and a bottleneck degree-$K$ spanning tree is a degree-$K$ spanning tree whose largest edge-length is minimum. Let $β_K$ be the supremum ratio of the largest edge-length of the bottleneck degree-$K$ spanning tree to the largest edge-length of the bottleneck spanning tree, over all finite point sets in the Euclidean plane. It is known that $β_5=1$, and it is easy to verify that $β_2\ge 2$, $β_3\ge \sqrt{2}$, and $β_4>1.175$. It is implied by the Hamiltonicity of the cube of the bottleneck spanning tree that $β_2\le 3$. The degree-3 spanning tree algorithm of Ravi et al. (STOC 1993) implies that $β_3\le 2$. Andersen and Ras (Networks, 68(4):302-314, 2016) showed that $β_4\le \sqrt{3}$. We present the following improved bounds: $β_2\ge\sqrt{7}$, $β_3\le \sqrt{3}$, and $β_4\le \sqrt{2}$. As a result, we obtain better approximation algorithms for Euclidean bottleneck degree-3 and degree-4 spanning trees. As parts of our proofs of these bounds we present some structural properties of the Euclidean minimum spanning tree which are of independent interest.

cs.CG↗