arXiv subjects

Lucas Meijer

Publications and source records attributed to Lucas Meijer.

8 recordsLinked to original sources

arXiv2026

The Nesting Bird Box Problem is ER-complete: Sharp Hardness Results for the Hidden Set Problem

In the (Nesting) Bird Box Problem we are given a polygonal domain P and a number k and we want to know if there is a set B of k points inside P such that no two points in B can see each other. The underlying idea is that each point represents a birdhouse and many birds only use a birdhouse if there is no other occupied birdhouse in its vicinity. We say two points a,b see each other if the open segment ab intersects neither the exterior of P nor any vertex of P. We show that the Nesting Bird Box problem is ER-complete. The complexity class ER can be defined by the set of problems that are polynomial time equivalent to finding a solution to the equation $p(x) = 0$, with $x\in R^n$ and $p\in $Z[X_1,...,X_n]$. The proof builds on the techniques developed in the original ER-completeness proof of the Art Gallery problem. However our proof is significantly shorter for two reasons. First, we can use recently developed tools that were not available at the time. Second, we consider polygonal domains with holes instead of simple polygons.

cs.CG↗

arXiv2025

First-Order Logic and Twin-Width for Some Geometric Graphs

For some geometric graph classes, tractability of testing first-order formulas is precisely characterised by the graph parameter twin-width. This was first proved for interval graphs among others in [BCKKLT, IPEC '22], where the equivalence is called delineation, and more generally holds for circle graphs, rooted directed path graphs, and $H$-graphs when $H$ is a forest. Delineation is based on the key idea that geometric graphs often admit natural vertex orderings, allowing to use the very rich theory of twin-width for ordered graphs. Answering two questions raised in their work, we prove that delineation holds for intersection graphs of non-degenerate axis-parallel unit segment graphs, but fails for visibility graphs of 1.5D terrains. We also prove delineation for intersection graphs of circular arcs.

cs.DM↗

arXiv2025

Devil's Games and $\text{Q}\mathbb{R}$: Continuous Games complete for the First-Order Theory of the Reals

We introduce the complexity class Quantified Reals ($\text{Q}\mathbb{R}$). Let FOTR be the set of true sentences in the first-order theory of the reals. A language $L$ is in $\text{Q}\mathbb{R}$, if there is a polynomial time reduction from $L$ to FOTR. This seems the first time this complexity class is studied. We show that $\text{Q}\mathbb{R}$ can also be defined using real Turing machines. It is known that deciding FOTR requires at least exponential time unconditionally [Berman, 1980]. We focus on devil's games with two defining properties: (1) Players (human and devil) alternate turns and (2) each turn has a continuum of options. First, we show that FOTRINV is $\text{Q}\mathbb{R}$-complete. FOTRINV has only inversion and addition constraints and all variables are in a compact interval. FOTRINV is a stepping stone for further reductions. Second, we show that the Packing Game is $\text{Q}\mathbb{R}$-complete. In the Packing Game we are given a container and two sets of pieces. One set of pieces for the human and one set for the devil. The human and the devil alternate by placing a piece into the container. Both rotations and translations are allowed. The first player that cannot place a piece loses. Third, we show that the Planar Extension Game is $\text{Q}\mathbb{R}$-complete. We are given a partially drawn plane graph and the human and the devil alternate by placing vertices and the corresponding edges in a straight-line manner. The vertices and edges to be placed are prescribed before hand. The first player that cannot place a vertex loses. Finally, we show that the Order Type Game is $\text{Q}\mathbb{R}$-complete. We are given an order-type together with a linear order. The human and the devil alternate in placing a point in the Euclidean plane following the linear order. The first player that cannot place a point correctly loses.

cs.CG↗

arXiv2025

On Equivalent Characterizations of the Polynomial Hierarchy in Abstract Models of Computation

We investigate machine models similar to Turing machines that are augmented with the operations of a first-order structure $\mathcal{R}$, and we show that under weak conditions on $\mathcal{R}$, the complexity class $\Sigma_k \mathcal{R}$ may be characterized in four equivalent ways: (1) by polynomial-time algorithms implemented on $\mathcal{R}$-machines together with witness strings, (2) by the $\Sigma_k\mathcal{R}$-complete problem $\Sigma_k\text{SAT}(\mathcal{R})$, (3) by the $k$th existential fragment of second-order metafinite logic over $\mathcal{R}$ via descriptive complexity, and (4) via oracles. By characterizing $\Sigma_k\mathcal{R}$ in these four ways, we extend previous work and embed it in one coherent framework. In addition, we derive similar results for $\exists_k \mathcal{R}$, the constant-free Boolean part of $\Sigma_k\mathcal{R}$, by showing that $\exists_k\mathcal{R}$ may be characterized in four analogous ways. Some conditions on $\mathcal{R}$ must be assumed in order to achieve the above quaternity because there are infinite-vocabulary structures for which $\text{NP}(\mathcal{R}) = \Sigma_1 \mathcal{R}$ does not have a complete problem. Surprisingly, even in these cases, we show that $\text{NP}(\mathcal{R})$ does have a characterization in terms of existential second-order metafinite logic, suggesting that descriptive complexity theory is well suited to working with infinite-vocabulary structures, such as real vector spaces.

cs.LO↗

arXiv2025

Oracle Separations for RPH

While theoretical computer science primarily works with discrete models of computation, like the Turing machine and the wordRAM, there are many scenarios in which introducing real computation models is more adequate. We want to compare real models of computation with discrete models of computation. We do this by means of oracle separation results. We define the notion of a real Turing machine as an extension of the (binary) Turing machine by adding a real tape. Using those machines, we define and study the real polynomial hierarchy RPH. We are interested in RPH as the first level of the hierarchy corresponds to the well-known complexity class ER. It is known that $NP \subseteq ER \subseteq PSPACE$ and furthermore $PH \subseteq RPH \subseteq PSPACE$. We are interested to know if any of those inclusions are tight. In the absence of unconditional separations of complexity classes, we turn to oracle separation. We develop a technique that allows us to transform oracle separation results from the binary world to the real world. As applications, we show there are oracles such that: - $RPH^O$ proper subset of $PSPACE^O$, - $\Sigma_{k+1}^O$ not contained in $\Sigma_kR^O$, for all $k\geq 0$, - $\Sigma_kR^O$ proper subset of $\Sigma_{k+1}R^O$, for all $k\geq 0$, - $BQP^O$ not contained in $RPH^O$. Our results hint that ER is strictly contained in PSPACE and that there is a separation between the different levels of the real polynomial hierarchy. We also bound the power of real computations by showing that NP-hard problems are unlikely to be solvable using polynomial time on a realRAM. Furthermore, our oracle separations hint that polynomial-time quantum computing cannot be simulated on an efficient real Turing machine.

cs.CC↗

arXiv2024

Clustering with Few Disks to Minimize the Sum of Radii

Given a set of $n$ points in the Euclidean plane, the $k$-MinSumRadius problem asks to cover this point set using $k$ disks with the objective of minimizing the sum of the radii of the disks. After a long line of research on related problems, it was finally discovered that this problem admits a polynomial time algorithm [GKKPV~'12]; however, the running time of this algorithm is $O(n^{881})$, and its relevance is thereby mostly of theoretical nature. A practically and structurally interesting special case of the $k$-MinSumRadius problem is that of small $k$. For the $2$-MinSumRadius problem, a near-quadratic time algorithm with expected running time $O(n^2 \log^2 n \log^2 \log n)$ was given over 30 years ago [Eppstein~'92]. We present the first improvement of this result, namely, a near-linear time algorithm to compute the $2$-MinSumRadius that runs in expected $O(n \log^2 n \log^2 \log n)$ time. We generalize this result to any constant dimension $d$, for which we give an $O(n^{2-1/(\lceil d/2\rceil + 1) + \varepsilon})$ time algorithm. Additionally, we give a near-quadratic time algorithm for $3$-MinSumRadius in the plane that runs in expected $O(n^2 \log^2 n \log^2 \log n)$ time. All of these algorithms rely on insights that uncover a surprisingly simple structure of optimal solutions: we can specify a linear number of lines out of which one separates one of the clusters from the remaining clusters in an optimal solution.

cs.CG↗

arXiv2023

3-Coloring in Time O(1.3217^n)

We propose a new algorithm for 3-coloring that runs in time O(1.3217^n). For this algorithm, we make use of the time O(1.3289^n) algorithm for 3-coloring by Beigel and Eppstein. They described a structure in all graphs, whose vertices could be colored relatively easily. In this paper, we improve upon this structure and present new ways to determine how the involved vertices reduce the runtime of the algorithm.

cs.DS↗

arXiv2022

Sometimes Two Irrational Guards are Needed

In the art gallery problem, we are given a closed polygon $P$, with rational coordinates and an integer $k$. We are asked whether it is possible to find a set (of guards) $G$ of size $k$ such that any point $p\in P$ is seen by a point in $G$. We say two points $p$, $q$ see each other if the line segment $pq$ is contained inside $P$. It was shown by Abrahamsen, Adamaszek, and Miltzow that there is a polygon that can be guarded with three guards, but requires four guards if the guards are required to have rational coordinates. In other words, an optimal solution of size three might need to be irrational. We show that an optimal solution of size two might need to be irrational. Note that it is well-known that any polygon that can be guarded with one guard has an optimal guard placement with rational coordinates. Hence, our work closes the gap on when irrational guards are possible to occur.

cs.CG↗