arXiv subjects

Adrian Dumitrescu

Publications and source records attributed to Adrian Dumitrescu.

At least 19 recordsLinked to original sources

arXiv2026

Doubling Sunlight for a Human Mars Base With Orbiting Solar Reflectors

The Sun's faintness at Mars' orbit makes producing energy, melting ice, and staying warm more difficult. Orbiting solar reflectors (OSRs) can augment sunlight at Mars, but the area of OSRs needed to double sunlight at a Mars base is not known. Here, we analyze Sun-synchronous Mars orbits to find the OSR area that doubles insolation to a Mars base. We show that the reflectors can deliver sunlight and maintain a stable orbit via attitude control and solar-sail propulsion, with no propellant. We also show that these Mars orbits can be reached via solar sailing from low Earth orbit, reducing delivery cost. Doubling sunlight is viable with an orbiting solar reflector areal density of $20 {\rm g/m^2}$, a 7-fold improvement over flight-proven solar sails. However, designing a spacecraft with low areal density, agile maneuverability (for accurate pointing), high tension in the sail membrane (to smooth wrinkles), and mass-manufacturability (to enable a constellation of reflectors) would be challenging. Still lower areal density of $<5 {\rm g/m^2}$ would be needed for more ambitious applications, such as sublimating the CO$_2$ ice at Mars' south pole to aid in terraforming.

astro-ph.EP↗

arXiv2026

Ordered Yao graphs: maximum degree, edge density, and clique numbers

For a positive integer $k$ and an ordered set of $n$ points in the plane, define its \textit{k-sector ordered Yao graphs} as follows. Divide the plane around each point into $k$ equal sectors and draw an edge from each point to its closest predecessor in each of the $k$ sectors. We analyze several natural parameters of these graphs. Our main results are as follows: I Let $d_k(n)$ be the maximum integer so that for every $n$-element point set in the plane, there exists an order such that the corresponding $k$-sector ordered Yao graph has maximum degree at least $d_k(n)$. We show that $d_k(n)=n-1$ if $k=4$ or $k \ge 6$, and provide some estimates for the remaining values of $k$. Namely, we show that $d_1(n) = Θ( \log {n} )$; $\frac{1}{2}(n-1) \le d_3(n) \le 5\left\lceil\frac{n}{6}\right\rceil-1$; $\frac{2}{3}(n-1) \le d_5(n) \le n-1$; II Let $e_k(n)$ be the minimum integer so that for every $n$-element point set in the plane, there exists an order such that the corresponding $k$-sector ordered Yao graph has at most $e_k(n)$ edges. Then $e_k(n)=\left\lceil\frac{k}{2}\right\rceil\cdot n-o(n)$. III Let $w_k$ be the minimum integer so that for every point set in the plane, there exists an order such that the corresponding $k$-sector ordered Yao graph has clique number at most $w_k$. Then $\lceil\frac{k}{2}\rceil \le w_k\le \lceil\frac{k}{2}\rceil+1$. All the orders mentioned above can be constructed effectively.

math.CO↗

arXiv2026

Rado's covering problem for cubes and balls: a semi-survey

What is the largest constant $c\in [0,1]$ with the property that every finite collection $\mathcal{C}$ of axis-parallel squares in the plane admits a disjoint sub-collection $\mathcal{S}$ occupying at least a fraction $c$ of the area covered by $\mathcal{C}$? This problem was first raised by T.~Radó in 1928, who was motivated by a classical covering lemma in real analysis due to Vitali. R.~Rado later generalized the problem from axis-parallel squares in the plane to homothetic copies of any given convex body $K$ in $\mathbb{R}^d$, where now we are looking for an optimal constant $F(K)$. Our utmost interest is for cubes and balls in the high-dimensional regime $d\rightarrow \infty$. The estimates that we currently have for cubes are much more precise than those for balls: namely if $Q^d$ is a $d$-dimensional cube, then \[ (e^{-1}+o(1))\frac{2^{-d}}{d \log{d}} \leq F(Q^d)\leq 2^{-d}, \] while denoting $B^d$ a $d$-dimensional Euclidean ball, then \[ (1+ε_d)3^{-d}\leq F(B^d)\leq 2.447^{-d}, \] where $ε_d>0$ vanishes exponentially fast as $d\rightarrow \infty$. The latter upper bound is obtained here by using the Kabatiansky--Levenshtein bound for the sphere packing problem.

math.MG↗

arXiv2026

Finding Small Complete Subgraphs Efficiently

(I) We revisit the algorithmic problem of finding all triangles in a graph $G=(V,E)$ with $n$ vertices and $m$ edges. According to a result of Chiba and Nishizeki (1985), this task can be achieved by a combinatorial algorithm running in $O(m α) = O(m^{3/2})$ time, where $α= α(G)$ is the graph arboricity. We provide a new very simple combinatorial algorithm for finding all triangles in a graph and show that is amenable to the same running time analysis. We derive these worst-case bounds from first principles and with very simple proofs that do not rely on classic results due to Nash-Williams from the 1960s. Our experimental results show that our simple algorithm for triangle listing is substantially faster in practice than that of Chiba and Nishizeki on all examples of real-world graphs we tried. (II) We extend our arguments to the problem of finding all small complete subgraphs of a given fixed size. We show that the dependency on $m$ and $α$ in the running time $O(α^{\ell-2} \cdot m)$ of the algorithm of Chiba and Nishizeki for listing all copies of $K_\ell$, where $\ell \geq 3$, is asymptotically tight. (III) We give improved arboricity-sensitive running times for counting and/or detection of copies of $K_\ell$, for small $\ell \geq 4$. A key ingredient in our algorithms is, once again, the algorithm of Chiba and Nishizeki. Our new algorithms are faster than all previous algorithms in certain high-range arboricity intervals for every $\ell \geq 7$.

cs.DS↗

arXiv2026

Maximal Distortion of Geodesic Diameters in Polygonal Domains

For a polygon $P$ with holes in the plane, we denote by $\varrho(P)$ the ratio between the geodesic and the Euclidean diameters of $P$. It is shown that over all convex polygons with $h$~convex holes, the supremum of $\varrho(P)$ is between $Ω(h^{1/3})$ and $O(h^{1/2})$. The upper bound improves to $\varrho(P)\leq O(1+\min\{h^{3/4}Δ,h^{1/2}Δ^{1/2}\})$ if the Euclidean diameter of every hole is most $Δ$ times the Euclidean diameter of $P$; and to $O(1)$ if every hole is a \emph{fat} convex polygon. Furthermore, we show that the function $g(h)=\sup_P \varrho(P)$ over convex polygons with $h$ convex holes has the same growth rate as an analogous quantity over geometric triangulations with $h$ vertices when $h\rightarrow \infty$.

cs.CG↗

arXiv2026

On multiplicities of interpoint distances

Given a set $X\subseteq\mathbb{R}^2$ of $n$ points and a distance $d>0$, the multiplicity of $d$ is the number of times the distance $d$ appears between points in $X$. Let $a_1(X) \geq a_2(X) \geq \cdots \geq a_m(X)$ denote the multiplicities of the $m$ distances determined by $X$ and let $a(X)=\left(a_1(X),\dots,a_m(X)\right)$. In this paper, we study several questions from Erdős's time regarding distance multiplicities. Among other results, we show that: (1) If $X$ is convex or ``not too convex'', then there exists a distance other than the diameter that has multiplicity at most $n$. (2) There exists a set $X \subseteq \mathbb{R}^2$ of $n$ points, such that many distances occur with high multiplicity. In particular, at least $n^{Ω(1/\log\log{n})}$ distances have superlinear multiplicity in $n$. (3) For any (not necessarily fixed) integer $1\leq k\leq\log{n}$, there exists $X\subseteq\mathbb{R}^2$ of $n$ points, such that the difference between the $k^{\text{th}}$ and $(k+1)^{\text{th}}$ largest multiplicities is at least $Ω(\frac{n\log{n}}{k})$. Moreover, the distances in $X$ with the largest $k$ multiplicities can be prescribed. (4) For every $n\in\mathbb{N}$, there exists $X\subseteq\mathbb{R}^2$ of $n$ points, not all collinear or cocircular, such that $a(X)= (n-1,n-2,\ldots,1)$. There also exists $Y\subseteq\mathbb{R}^2$ of $n$ points with pairwise distinct distance multiplicities and $a(Y) \neq (n-1,n-2,\ldots,1)$.

math.CO↗

arXiv2026

Covering Complete Geometric Graphs by Monotone Paths

Given a set $A$ of $n$ points (vertices) in general position in the plane, the \emph{complete geometric graph} $K_n[A]$ consists of all $\binom{n}{2}$ segments (edges) between the elements of $A$. It is known that the edge set of every complete geometric graph on $n$ vertices can be partitioned into $O(n^{3/2})$ crossing-free paths (or matchings). We strengthen this result under various additional assumptions on the point set. In particular, we prove that for a set $A$ of $n$ \emph{randomly} selected points, uniformly distributed in $[0,1]^2$, with probability tending to $1$ as $n\rightarrow\infty$, the edge set of $K_n[A]$ can be covered by $O(n\log n)$ crossing-free paths and by $O(n\sqrt{\log n})$ crossing-free matchings. On the other hand, we construct $n$-element point sets such that covering the edge set of $K_n[A]$ requires a quadratic number of monotone paths.

math.CO↗

arXiv2025

A Couple of Simple Algorithms for $k$-Dispersion

Given a set $P$ of $n$ points in $\mathbf{R}^d$, and a positive integer $k \leq n$, the $k$-dispersion problem is that of selecting $k$ of the given points so that the minimum inter-point distance among them is maximized (under Euclidean distances). Among others, we show the following: (I) Given a set $P$ of $n$ points in the plane, and a positive integer $k \geq 2$, the $k$-dispersion problem can be solved by an algorithm running in $O\left(n^{k-1} \log{n}\right)$ time. This extends an earlier result for $k=3$, due to Horiyama, Nakano, Saitoh, Suetsugu, Suzuki, Uehara, Uno, and Wasa (2021) to arbitrary $k$. In particular, it improves on previous running times for small $k$. (II) Given a set $P$ of $n$ points in $\mathbf{R}^3$, and a positive integer $k \geq 2$, the $k$-dispersion problem can be solved by an algorithm running in $O\left(n^{k-1} \log{n}\right)$ time, if $k$ is even; and $O\left(n^{k-1} \log^2{n}\right)$ time, if $k$ is odd. For $k \geq 4$, no combinatorial algorithm running in $o(n^k)$ time was known for this problem. (III) Let $P$ be a set of $n$ random points uniformly distributed in $[0,1]^2$. Then under suitable conditions, a $0.99$-approximation for $k$-dispersion can be computed in $O(n)$ time with high probability.

cs.CG↗

arXiv2025

Lattice and Non-lattice Piercing of Axis-Parallel Rectangles

Given a family ${\mathcal F}$ of shapes in the plane, we study what is the lowest possible density of a point set $P$ that pierces (``intersects'', ``hits'') all translates of each shape in ${\mathcal F}$. For instance, if ${\mathcal F}$ consists of two axis-parallel rectangles the best known piercing set, i.e., one with the lowest density, is a lattice. Given a finite family ${\mathcal F}$ of axis-parallel rectangles, we present an algorithm for finding an optimal ${\mathcal F}$-piercing lattice. The algorithm runs in time polynomial in the number of rectangles and the maximum aspect ratio of the rectangles in the family. No prior algorithms for this problem were known. On the other hand, we show that for every $n \geq 3$, there exists a family of $n$ axis-parallel rectangles for which the best piercing density achieved by a lattice is separated by a positive (constant) gap from the optimal piercing density for the respective family. Finally, we show that the best lattice can be sometimes worse by $20\%$ than the optimal piercing set.

cs.CG↗

arXiv2025

Maximizing the Maximum Degree in Ordered Nearest Neighbor Graphs

For an ordered point set in a Euclidean space or, more generally, in an abstract metric space, the ordered Nearest Neighbor Graph is obtained by connecting each of the points to its closest predecessor by a directed edge. We show that for every set of $n$ points in $\mathbb{R}^d$, there exists an order such that the corresponding ordered Nearest Neighbor Graph has maximum degree at least $\log{n}/(4d)$. Apart from the $1/(4d)$ factor, this bound is the best possible. As for the abstract setting, we show that for every $n$-element metric space, there exists an order such that the corresponding ordered Nearest Neighbor Graph has maximum degree $Ω(\sqrt{\log{n}/\log\log{n}})$.

math.CO↗

arXiv2025

Note on the Number of Almost Ordinary Triangles

Let $X$ be a set of $n$ points in the plane, not all on a line. According to the Gallai-Sylvester theorem, $X$ always spans an \emph{ordinary line}, i.e., one that passes through precisely 2 elements of $X$. Given an integer $c\ge 2,$ a \emph{line} spanned by $X$ is called \emph{$c$-ordinary} if it passes through at most $c$ points of $X$. A \emph{triangle} spanned by 3 noncollinear points of $X$ is called \emph{$c$-ordinary} if all 3 lines determined by its sides are \emph{$c$-ordinary}. Motivated by a question of Erd\H os, Fulek \emph{et al.}~\cite{FMN+17} proved that there exists an absolute constant $c > 2$ such that if $X$ cannot be covered by 2 lines, then it determines at least one $c$-ordinary triangle. Moreover, the number of such triangles grows at least linearly in $n$. They raised the question whether the true growth rate of this function is superlinear. We prove that if $X$ cannot be covered by 2 lines, and no line passes through more than $n-t(n)$ points of $X$, for some function $t(n)\rightarrow\infty,$ then the number of $17$-ordinary triangles spanned by $X$ is at least constant times $n \cdot t(n)$, i.e., superlinear in $n$. We also show that the assumption $t(n)\rightarrow\infty$ is necessary. If we further assume that no line passes through more than $n/2-t(n)$ points of $X$, then the number of $17$-ordinary triangles grows superquadratically in $n$. This statement does not hold if $t(n)$ is bounded. We close this paper with some algorithmic results. In particular, we provide a $O(n^{2.372})$ time algorithm for counting all $c$-ordinary triangles in an $n$-element point set, for any $c<n$.

math.CO↗

arXiv2025

Closed curve covering and multiagent TSP ratios

How efficiently can a closed curve of unit length in $\mathbb{R}^d$ be covered by $k$ closed curves so as to minimize the maximum length of the $k$ curves? We show that the maximum length is at most $2k^{-1} - \frac{1}{4} k^{-4}$ for all $k\geq 2$ and $d \geq 2$. As a first byproduct, we show that $k$ agents can traverse a Euclidean TSP instance significantly faster than a single agent. We thereby sharpen recent planar results by Berendsohn, Kim, and Kozma (2025) and extend these improvements to all dimensions. As a second byproduct, we obtain a linear time approximation algorithm with ratio $2 - \frac{1}{4} k^{-3}$ for covering any closed polygonal curve in $\mathbb{R}^d$ by $k$ closed curves so that the maximum length of an individual curve is minimized.

math.MG↗

arXiv2025

Arcs with increasing chords in $\mathbf{R}^d$

A curve $γ$ that connects $s$ and $t$ has the increasing chord property if $|bc| \leq |ad|$ whenever $a,b,c,d$ lie in that order on $γ$. For planar curves, the length of such a curve is known to be at most $2π/3 \cdot |st|$. Here we examine the question in higher dimensions and from the algorithmic standpoint and show the following: (I) The length of any $s-t$ curve with increasing chords in $\mathbf{R}^d$ is at most $2 \cdot \left( e/2 \cdot (d+4) \right)^{d-1} \cdot |st|$ for every $d \geq 3$. This is the first bound in higher dimensions. (II) Given a polygonal chain $P=(p_1, p_2, \dots, p_n)$ in $\mathbf{R}^d$, where $d \geq 4$, $k =\lfloor d/2 \rfloor$, it can be tested whether it satisfies the increasing chord property in $O\left(n^{2-1/(k+1)} {\rm polylog} (n) \right)$ expected time. This is the first subquadratic algorithm in higher dimensions.

cs.CG↗

arXiv2025

A Purely Geometric Variant of the Gale-Berlekamp Switching Game

We introduce the following variant of the Gale-Berlekamp switching game. Let $P$ be a set of n noncollinear points in the plane, each of them having weight $+1$ or $-1$. At each step, we pick a line $\ell$ passing through at least two points of $P$, and switch the sign of every point $p \in P\cap\ell$. The objective is to maximize the total weight of the elements of $P$. We show that one can always achieve that this quantity is at least $n - o(n)$, as $n\rightarrow\infty$, and at least $n/3$, for every $n$. Moreover, these can be attained by a polynomial time algorithm.

cs.CG↗

arXiv2025

The number of regular simplices in higher dimensions

We study the extremal function $S^k_d(n)$, defined as the maximum number of regular $(k-1)$-simplices spanned by $n$ points in $\mathbb{R}^d$. For any fixed $d\geq2k\geq6$, we determine the asymptotic behavior of $S^k_d(n)$ up to a lower-order term. In particular, when $k=3$, we determine the exact value of $S^3_d(n)$, for all even dimensions $d\geq6$ and sufficiently large $n$. This resolves a conjecture of Erd\H{o}s in a stronger form. The proof leverages techniques from hypergraph Tur\'an theory and linear algebra.

math.CO↗

arXiv2025

General Position Subset Selection in Line Arrangements

Given a set of points in the plane, the \textsc{General Position Subset Selection} problem is that of finding a maximum-size subset of points in general position, i.e., with no three points collinear. The problem is known to be ${\rm NP}$-complete and ${\rm APX}$-hard, and the best approximation ratio known is $Ω\left({\rm OPT}^{-1/2}\right) =Ω(n^{-1/2})$. Here we obtain better approximations in three specials cases: (I) A constant factor approximation for the case where the input set consists of lattice points and is \emph{dense}, which means that the ratio between the maximum and the minimum distance in $P$ is of the order of $Θ(\sqrt{n})$. (II) An $Ω\left((\log{n})^{-1/2}\right)$-approximation for the case where the input set is the set of vertices of a \emph{generic} $n$-line arrangement, i.e., one with $Ω(n^2)$ vertices. The scenario in (I) is a special case of that in (II). (III) An $Ω\left((\log{n})^{-1/2}\right)$-approximation for the case where the input set has at most $O(\sqrt{n})$ points collinear and can be covered by $O(\sqrt{n})$ lines. Our approximations rely on probabilistic methods and results from incidence geometry.

cs.CG↗

arXiv2024

Geometric Variants of the Gale--Berlekamp Switching Game

The Gale-Berlekamp switching game is played on the following device: $G_n=\{1,2,\ldots,n\} \times \{1,2,\ldots,n\}$ is an $n \times n$ array of lights is controlled by $2n$ switches, one for each row or column. Given an (arbitrary) initial configuration of the board, the objective is to have as many lights on as possible. Denoting the maximum difference (discrepancy) between the number of lights that are on minus the number of lights that are off by $F(n)$, it is known (Brown and Spencer, 1971) that $F(n)= Θ(n^{3/2})$, and more precisely, that $F(n) \geq \left( 1+ o(1) \right) \sqrt{\frac{2}π} n^{3/2} \approx 0.797 \ldots n^{3/2}$. Here we extend the game to other playing boards. For example: (i)~For any constant $c>1$, if $c n$ switches are conveniently chosen, then the maximum discrepancy for the square board is $Ω(n^{3/2})$. From the other direction, suppose we fix any set of $a$ column switches, $b$ row switches, where $a \geq b$ and $a+b=n$. Then the maximum discrepancy is at most $-b (n-b)$. (ii) A board $H \subset \{1,\ldots,n\}^2$, with area $A=|H|$, is \emph{dense} if $A \geq c (u+v)^2$, for some constant $c>0$, where $u= |\{x \colon (x,y) \in H\}|$ and $v=|\{y \colon (x,y) \in H\}|$. For a dense board of area $A$, we show that the maximum discrepancy is $Θ(A^{3/4})$. This result is a generalization of the Brown and Spencer result for the original game. (iii) If $H$ consists of the elements of $G_n$ below the hyperbola $xy=n$, then its maximum discrepancy is $Ω(n)$ and $O(n (\log n)^{1/2})$.

math.CO↗

arXiv2024

Subset selection problems in planar point sets

Given a finite point set satisfying condition $\mathcal{A}$, the subset selection problem asks, how large of a subset satisfying condition $\mathcal{B}$ can be extracted? In this paper, we make progress on three instances of subset selection problems in planar point sets. Let $n,s\in\mathbb{N}$ with $n\geq s$, and let $P\subseteq\mathbb{R}^2$ be a set of $n$ points, where at most $s$ points lie on the same line. Firstly, we select a general position subset of $P$. This problem was proposed by Erd\H{o}s under the regime when $s$ is a constant. For $s$ being non-constant, we give new lower and upper bounds on the maximum size of such a subset. In particular, we show that in the worst case such a set can have size at most $O(n^{5/6+o(1)}/\sqrt{s})$ when $3\leq s\leq n^{1/3}$ and $O(n/s)$ when $n^{1/3}\leq s\leq n$. Secondly, we select a monotone general position subset of $P$, that is, a subset in general position where the points are ordered from left to right and their $y$-coordinates are either non-decreasing or non-increasing. We present bounds on the maximum size of such a subset. In particular, when $s=\Omega(\sqrt{n})$, our upper and lower bounds differ at most by a logarithmic factor. Lastly, we select a subset of $P$ with pairwise distinct slopes. This problem was initially studied by Erd\H{o}s, Graham, Ruzsa, and Taylor on the grid. We show that for $s=O(\sqrt{n})$ such a subset of size $\Omega((n/\log{s})^{1/3})$ can always be found in $P$. When $s=\Theta(\sqrt{n})$, this matches a lower bound given by Zhang on the grid. As for the upper bound, we show that in the worst case such a subset has size at most $O(\sqrt{n})$ for $2\leq s\leq n^{3/8}$ and $O((n/s)^{4/5})$ for $n^{3/8}\leq s=O(\sqrt{n})$. The proofs use a wide range of tools such as incidence geometry, probabilistic methods, the hypergraph container method, and additive combinatorics.

math.CO↗