arXiv subjects

Bernd Gärtner

Publications and source records attributed to Bernd Gärtner.

At least 19 recordsLinked to original sources

arXiv2025

Optimizing Symbol Visibility through Displacement

In information visualization, the position of symbols often encodes associated data values. When visualizing data elements with both a numerical and a categorical dimension, positioning in the categorical axis admits some flexibility. This flexibility can be exploited to reduce symbol overlap, and thereby increase legibility. In this paper, we initialize the algorithmic study of optimizing symbol legibility via a limited displacement of the symbols. Specifically, we consider closed unit square symbols that need to be placed at specified $y$-coordinates. We optimize the drawing order of the symbols as well as their $x$-displacement, constrained within a rectangular container, to maximize the minimum visible perimeter over all squares. If the container has width and height at most $2$, there is a point that stabs all squares. In this case, we prove that a staircase layout is arbitrarily close to optimality and can be computed in $O(n\log n)$ time. If the width is at most $2$, there is a vertical line that stabs all squares, and in this case, we design a 2-approximation algorithm (assuming fixed container height) that runs in $O(n\log n)$ time. As it turns out that a minimum visible perimeter of 2 is always achievable with a generic construction, we measure this approximation with respect to the visible perimeter exceeding 2. We show that, despite its simplicity, the algorithm gives asymptotically optimal results for certain instances.

cs.CG↗

arXiv2024

Computing Enclosing Depth

Enclosing depth is a recently introduced depth measure which gives a lower bound to many depth measures studied in the literature. So far, enclosing depth has only been studied from a combinatorial perspective. In this work, we give the first algorithms to compute the enclosing depth of a query point with respect to a data point set in any dimension. In the plane we are able to optimize the algorithm to get a runtime of O(n log n). In constant dimension, our algorithms still run in polynomial time.

cs.CG↗

arXiv2023

A Note on the Faces of the Dual Koch Arrangement

We analyze the faces of the dual Koch arrangement, which is the arrangement of $2^s + 1$ lines obtained by projective duality from the Koch chain $K_s$. In particular, we show that this line arrangement does not contain any $k$-gons for $k > 5$, and that the number of pentagons is $3 \cdot 2^{s-1} - 3$.

cs.CG↗

arXiv2022

A New Combinatorial Property of Geometric Unique Sink Orientations

A unique sink orientation (USO) is an orientation of the hypercube graph with the property that every face has a unique sink. A number of well-studied problems reduce in strongly polynomial time to finding the global sink of a USO; most notably, linear programming (LP) and the P-matrix linear complementarity problem (P-LCP). The former is not known to have a strongly polynomial-time algorithm, while the latter is not known to even have a polynomial-time algorithm, motivating the problem to find the global sink of a USO. Although, every known class of geometric USOs, arising from a concrete problem such as LP, is exponentially small, relative to the class of all USOs. Accordingly, geometric USOs exhibit additional properties that set them apart from general USOs, and it may be advantageous, if not necessary, to leverage these properties to find the global sink of a USO faster. Only a few such properties are known. In this paper, we establish a new combinatorial property of the USOs that arise from symmetric P-LCP, which includes the USOs that arise from linear and simple convex quadratic programming.

math.CO↗

arXiv2021

A Characterization of the Realizable Matoušek Unique Sink Orientations

The Matoušek LP-type problems were used by Matoušek to show that the Sharir-Welzl algorithm may require at least subexponential time. Later, Gärtner translated this result into the language of Unique Sink Orientations (USOs) and introduced the Matoušek USOs, the USOs equivalent to Matoušek's LP-type problems. He further showed that the Random Facet algorithm only requires quadratic time on the realizable subset of the Matoušek USOs, but without characterizing this subset. In this paper, we deliver this missing characterization and also provide concrete realizations for all realizable Matoušek USOs. Furthermore, we show that the realizable Matoušek USOs are exactly the orientations arising from simple extensions of cyclic-P-matroids.

math.CO↗

arXiv2021

The Crossing Tverberg Theorem

Tverberg's theorem is one of the cornerstones of discrete geometry. It states that, given a set $X$ of at least $(d+1)(r-1)+1$ points in $\mathbb R^d$, one can find a partition $X=X_1\cup \ldots \cup X_r$ of $X$, such that the convex hulls of the $X_i$, $i=1,\ldots,r$, all share a common point. In this paper, we prove a strengthening of this theorem that guarantees a partition which, in addition to the above, has the property that the boundaries of full-dimensional convex hulls have pairwise nonempty intersections. Possible generalizations and algorithmic aspects are also discussed. As a concrete application, we show that any $n$ points in the plane in general position span $\lfloor n/3\rfloor$ vertex-disjoint triangles that are pairwise crossing, meaning that their boundaries have pairwise nonempty intersections; this number is clearly best possible. A previous result of Rebollar et al.\ guarantees $\lfloor n/6\rfloor$ pairwise crossing triangles. Our result generalizes to a result about simplices in $\mathbb R^d,d\ge2$.

cs.CG↗

arXiv2021

A Subexponential Algorithm for ARRIVAL

The ARRIVAL problem is to decide the fate of a train moving along the edges of a directed graph, according to a simple (deterministic) pseudorandom walk. The problem is in $NP \cap coNP$ but not known to be in $P$. The currently best algorithms have runtime $2^{Θ(n)}$ where $n$ is the number of vertices. This is not much better than just performing the pseudorandom walk. We develop a subexponential algorithm with runtime $2^{O(\sqrt{n}\log n)}$. We also give a polynomial-time algorithm if the graph is almost acyclic. Both results are derived from a new general approach to solve ARRIVAL instances.

cs.DS↗

arXiv2019

Random Sampling with Removal

We study randomized algorithms for constrained optimization, in abstract frameworks that include, in strictly increasing generality: convex programming; LP-type problems; violator spaces; and a setting we introduce, consistent spaces. Such algorithms typically involve a step of finding the optimal solution for a random sample of the constraints. They exploit the condition that, in finite dimension $δ$, this sample optimum violates a provably small expected fraction of the non-sampled constraints, with the fraction decreasing in the sample size~$r$. We extend such algorithms by considering the technique of removal, where a fixed number $k$ of constraints are removed from the sample according to a fixed rule, with the goal of improving the solution quality. This may have the effect of increasing the number of violated non-sampled constraints. We study this increase, and bound it in a variety of general settings. This work is motivated by, and extends, results on removal as proposed for chance-constrained optimization. For many relevant values of $r$, $δ$, and $k$, we prove matching upper and lower bounds for the expected number of constraints violated by a random sample, after the removal of $k$ constraints. For a large range of values of $k$, the new upper bounds improve the previously best bounds for LP-type problems, which moreover had only been known in special cases, and not in the generality we consider. Moreover, we show that our results extend from finite to infinite spaces, for chance-constrained optimization.

cs.CG↗

arXiv2019

Phase Transition in Democratic Opinion Dynamics

Consider a community where initially, each individual is positive or negative regarding a reform proposal. In each round, individuals gather randomly in fixed rooms of different sizes, and all individuals in a room agree on the majority opinion in the room (with ties broken in favor of the negative opinion). The Galam model---introduced in statistical physics, specifically sociophysics---approximates this basic random process. We approach the model from a more mathematical perspective and study the threshold behavior and the consensus time of the model.

cs.DS↗

arXiv2018

ARRIVAL: Next Stop in CLS

We study the computational complexity of ARRIVAL, a zero-player game on $n$-vertex switch graphs introduced by Dohrau, Gärtner, Kohler, Matoušek, and Welzl. They showed that the problem of deciding termination of this game is contained in $\text{NP} \cap \text{coNP}$. Karthik C. S. recently introduced a search variant of ARRIVAL and showed that it is in the complexity class PLS. In this work, we significantly improve the known upper bounds for both the decision and the search variants of ARRIVAL. First, we resolve a question suggested by Dohrau et al. and show that the decision variant of ARRIVAL is in $\text{UP} \cap \text{coUP}$. Second, we prove that the search variant of ARRIVAL is contained in CLS. Third, we give a randomized $\mathcal{O}(1.4143^n)$-time algorithm to solve both variants. Our main technical contributions are (a) an efficiently verifiable characterization of the unique witness for termination of the ARRIVAL game, and (b) an efficient way of sampling from the state space of the game. We show that the problem of finding the unique witness is contained in CLS, whereas it was previously conjectured to be FPSPACE-complete. The efficient sampling procedure yields the first algorithm for the problem that has expected runtime $\mathcal{O}(c^n)$ with $c<2$.

cs.CC↗

arXiv2017

(Biased) Majority Rule Cellular Automata

Consider a graph $G=(V,E)$ and a random initial vertex-coloring, where each vertex is blue independently with probability $p_{b}$, and red with probability $p_r=1-p_b$. In each step, all vertices change their current color synchronously to the most frequent color in their neighborhood and in case of a tie, a vertex conserves its current color; this model is called majority model. If in case of a tie a vertex always chooses blue color, it is called biased majority model. We are interested in the behavior of these deterministic processes, especially in a two-dimensional torus (i.e., cellular automaton with (biased) majority rule). In the present paper, as a main result we prove both majority and biased majority cellular automata exhibit a threshold behavior with two phase transitions. More precisely, it is shown that for a two-dimensional torus $T_{n,n}$, there are two thresholds $0\leq p_1, p_2\leq 1$ such that $p_b \ll p_1$, $p_1 \ll p_b \ll p_2$, and $p_2 \ll p_b$ result in monochromatic configuration by red, stable coexistence of both colors, and monochromatic configuration by blue, respectively in $\mathcal{O}(n^2)$ number of steps

cs.FL↗

arXiv2017

Majority Model on Random Regular Graphs

Consider a graph $G=(V,E)$ and an initial random coloring where each vertex $v \in V$ is blue with probability $P_b$ and red otherwise, independently from all other vertices. In each round, all vertices simultaneously switch their color to the most frequent color in their neighborhood and in case of a tie, a vertex keeps its current color. The main goal of the present paper is to analyze the behavior of this basic and natural process on the random $d$-regular graph $\mathbb{G}_{n,d}$. It is shown that for all $ε>0$, $P_b \le 1/2-ε$ results in final complete occupancy by red in $\mathcal{O}(\log_d\log n)$ rounds with high probability, provided that $d\geq c/ε^2$ for a suitable constant $c$. Furthermore, we show that with high probability, $\mathbb{G}_{n,d}$ is immune; i.e., the smallest dynamic monopoly is of linear size. A dynamic monopoly is a subset of vertices that can take over in the sense that a commonly chosen initial color eventually spreads throughout the whole graph, irrespective of the colors of other vertices. This answers an open question of Peleg.

cs.DS↗

arXiv2017

ARRIVAL: A zero-player graph game in NP $\cap$ coNP

Suppose that a train is running along a railway network, starting from a designated origin, with the goal of reaching a designated destination. The network, however, is of a special nature: every time the train traverses a switch, the switch will change its position immediately afterwards. Hence, the next time the train traverses the same switch, the other direction will be taken, so that directions alternate with each traversal of the switch. Given a network with origin and destination, what is the complexity of deciding whether the train, starting at the origin, will eventually reach the destination? It is easy to see that this problem can be solved in exponential time, but we are not aware of any polynomial-time method. In this short paper, we prove that the problem is in NP $\cap$ coNP. This raises the question whether we have just failed to find a (simple) polynomial-time solution, or whether the complexity status is more subtle, as for some other well-known (two-player) graph games.

cs.CC↗

arXiv2017

Stochastic continuum armed bandit problem of few linear parameters in high dimensions

We consider a stochastic continuum armed bandit problem where the arms are indexed by the $\ell_2$ ball $B_{d}(1+ν)$ of radius $1+ν$ in $\mathbb{R}^d$. The reward functions $r :B_{d}(1+ν) \rightarrow \mathbb{R}$ are considered to intrinsically depend on $k \ll d$ unknown linear parameters so that $r(\mathbf{x}) = g(\mathbf{A} \mathbf{x})$ where $\mathbf{A}$ is a full rank $k \times d$ matrix. Assuming the mean reward function to be smooth we make use of results from low-rank matrix recovery literature and derive an efficient randomized algorithm which achieves a regret bound of $O(C(k,d) n^{\frac{1+k}{2+k}} (\log n)^{\frac{1}{2+k}})$ with high probability. Here $C(k,d)$ is at most polynomial in $d$ and $k$ and $n$ is the number of rounds or the sampling budget which is assumed to be known beforehand.

stat.ML↗

arXiv2017

Algorithms for Learning Sparse Additive Models with Interactions in High Dimensions

A function $f: \mathbb{R}^d \rightarrow \mathbb{R}$ is a Sparse Additive Model (SPAM), if it is of the form $f(\mathbf{x}) = \sum_{l \in \mathcal{S}}ϕ_{l}(x_l)$ where $\mathcal{S} \subset [d]$, $|\mathcal{S}| \ll d$. Assuming $ϕ$'s, $\mathcal{S}$ to be unknown, there exists extensive work for estimating $f$ from its samples. In this work, we consider a generalized version of SPAMs, that also allows for the presence of a sparse number of second order interaction terms. For some $\mathcal{S}_1 \subset [d], \mathcal{S}_2 \subset {[d] \choose 2}$, with $|\mathcal{S}_1| \ll d, |\mathcal{S}_2| \ll d^2$, the function $f$ is now assumed to be of the form: $\sum_{p \in \mathcal{S}_1}ϕ_{p} (x_p) + \sum_{(l,l^{\prime}) \in \mathcal{S}_2}ϕ_{(l,l^{\prime})} (x_l,x_{l^{\prime}})$. Assuming we have the freedom to query $f$ anywhere in its domain, we derive efficient algorithms that provably recover $\mathcal{S}_1,\mathcal{S}_2$ with finite sample bounds. Our analysis covers the noiseless setting where exact samples of $f$ are obtained, and also extends to the noisy setting where the queries are corrupted with noise. For the noisy setting in particular, we consider two noise models namely: i.i.d Gaussian noise and arbitrary but bounded noise. Our main methods for identification of $\mathcal{S}_2$ essentially rely on estimation of sparse Hessian matrices, for which we provide two novel compressed sensing based schemes. Once $\mathcal{S}_1, \mathcal{S}_2$ are known, we show how the individual components $ϕ_p$, $ϕ_{(l,l^{\prime})}$ can be estimated via additional queries of $f$, with uniform error bounds. Lastly, we provide simulation results on synthetic data that validate our theoretical findings.

cs.LG↗

arXiv2017

Pseudo Unique Sink Orientations

A unique sink orientation (USO) is an orientation of the $n$-dimensional cube graph ($n$-cube) such that every face (subcube) has a unique sink. The number of unique sink orientations is $n^{Θ(2^n)}$. If a cube orientation is not a USO, it contains a pseudo unique sink orientation (PUSO): an orientation of some subcube such that every proper face of it has a unique sink, but the subcube itself hasn't. In this paper, we characterize and count PUSOs of the $n$-cube. We show that PUSOs have a much more rigid structure than USOs and that their number is between $2^{Ω(2^{n-\log n})}$ and $2^{O(2^n)}$ which is negligible compared to the number of USOs. As tools, we introduce and characterize two new classes of USOs: border USOs (USOs that appear as facets of PUSOs), and odd USOs which are dual to border USOs but easier to understand.

math.CO↗

arXiv2016

Screening Rules for Convex Problems

We propose a new framework for deriving screening rules for convex optimization problems. Our approach covers a large class of constrained and penalized optimization formulations, and works in two steps. First, given any approximate point, the structure of the objective function and the duality gap is used to gather information on the optimal solution. In the second step, this information is used to produce screening rules, i.e. safely identifying unimportant weight variables of the optimal solution. Our general framework leads to a large variety of useful existing as well as new screening rules for many applications. For example, we provide new screening rules for general simplex and $L_1$-constrained problems, Elastic Net, squared-loss Support Vector Machines, minimum enclosing ball, as well as structured norm regularized problems, such as group lasso.

math.OC↗

arXiv2016

The Niceness of Unique Sink Orientations

Random Edge is the most natural randomized pivot rule for the simplex algorithm. Considerable progress has been made recently towards fully understanding its behavior. Back in 2001, Welzl introduced the concepts of \emph{reachmaps} and \emph{niceness} of Unique Sink Orientations (USO), in an effort to better understand the behavior of Random Edge. In this paper, we initiate the systematic study of these concepts. We settle the questions that were asked by Welzl about the niceness of (acyclic) USO. Niceness implies natural upper bounds for Random Edge and we provide evidence that these are tight or almost tight in many interesting cases. Moreover, we show that Random Edge is polynomial on at least $n^{Ω(2^n)}$ many (possibly cyclic) USO. As a bonus, we describe a derandomization of Random Edge which achieves the same asymptotic upper bounds with respect to niceness and discuss some algorithmic properties of the reachmap.

cs.DM↗