nearly flat, acutely triangulated convex cap C

has an edge-unfolding to a non-overlapping polygon in the plane.

A convex cap is the intersection of the surface of a convex polyhedron and a halfspace.

"Nearly flat" means that every face normal forms a

sufficiently small angle phi<Phi with the z-axis orthogonal to the halfspace bounding plane.

The size of Phi depends on the acuteness gap:

if every triangle angle is at most pi/2-alpha,

then Phi ~= 0.3 sqrt{alpha} suffices; e.g., for alpha=3 deg, Phi ~= 4 deg.

Even if C is closed to a polyhedron by adding

the convex polygonal base under C,

this polyhedron can be edge-unfolded without overlap.

The proof relies on some recently developed concepts,

angle-monotone and radially monotone curves.

The proof is constructive, leading to a polynomial-time algorithm

for finding the edge-cuts, at worst O(n^2); a version has been implemented.

In this paper we focus on MaxDBS for the class of unit disk graphs. We provide a polynomial-time constant-factor approximation algorithm for the problem. The approximation ratio of our algorithm does not depend on the diameter $d$. Even though the algorithm itself is simple, its analysis is rather involved. We combine tools from the theory of hypergraphs with bounded VC-dimension, $k$-quasi planar graphs, fractional Helly theorems and several geometric properties of unit disk graphs.

the $k$-fold cover of $X$ and $r$ consists of all points in $R^d$

that have $k$ or more points of $X$ within distance $r$.

We consider two filtrations ---

one in scale obtained by fixing $k$ and increasing $r$,

and the other in depth obtained by fixing $r$ and decreasing $k$

--- and we compute the persistence diagrams of both.

While standard methods suffice for the filtration in scale,

we need novel geometric and topological concepts for the filtration in depth.

In particular, we introduce a rhomboid tiling in $R^{d+1}$

whose horizontal integer slices are the order-$k$ Delaunay mosaics of $X$,

and construct a zigzag module from Delaunay mosaics that is isomorphic

to the persistence module of the multi-covers.

Here r = 0, 1, … is considered a fixed parameter.

The only case for which the bound was known previously was for r ≤ 3.

Our lower bound is also valid for the expected number of pivoting steps in the following applications:

(i) The Random-Edge algorithm on linear programs with n constraints in d = n−r variables;

(ii) the directed random walk on a grid polytope of corank r with n facets; and

(iii) the Random-Edge algorithm on an r-dimensional grid unique sink orientation of size n.

Terrain Guarding can be seen as a special case of the famous art gallery problem where one has to place at most $k$ guards on a terrain made of $n$ vertices in order to fully see it.

In 2010, King and Krohn showed that Terrain Guarding is NP-complete [SODA '10, SIAM J. Comput. '11] thereby solving a long-standing open question.

They observe that their proof does not settle the complexity of Orthogonal Terrain Guarding where the terrain only consists of horizontal or vertical segments; those terrains are called rectilinear or orthogonal.

Recently, Ashok et al. [SoCG'17] presented an FPT algorithm running in time $k^{O(k)}n^{O(1)}$ for \textsc{Dominating Set} in the visibility graphs of rectilinear terrains without 180-degree vertices.

They ask if Orthogonal Terrain Guarding is in P or NP-hard.

In the same paper, they give a subexponential-time algorithm running in $n^{O(\sqrt n)}$ (actually even $n^{O(\sqrt k)}$) for the general Terrain Guarding and notice that the hardness proof of King and Krohn only disproves a running time $2^{o(n^{1/4})}$ under the ETH.

Hence, there is a significant gap between their $2^{O(n^{1/2} \log n)}$-algorithm and the no $2^{o(n^{1/4})}$ ETH-hardness implied by King and Krohn's result.

In this paper, we answer those two remaining questions.

We adapt the gadgets of King and Krohn to rectilinear terrains in order to prove that even Orthogonal Terrain Guarding is NP-complete.

Then, we show how their reduction from Planar 3-SAT (as well as our adaptation for rectilinear terrains) can actually be made linear (instead of quadratic).

any ball is at most U . The objective function that we would like to minimize is the cardinality of B_0.

We consider this problem in arbitrary metric spaces as well as Euclidean spaces of constant dimension. In the metric setting, even the uncapacitated version of the problem is hard to approximate to within a logarithmic factor. In the Euclidean setting, the best known approximation guarantee in dimensions 3 and higher is logarithmic. Thus we focus on obtaining “bi-criteria” approximations. In particular, we are allowed to expand the balls in our solution by some factor, but optimal solutions do not have that flexibility. Our main result is that allowing constant

factor expansion of the input balls suffices to obtain constant approximations for these problems. In fact, in the Euclidean setting, only (1 + ε) factor expansion is sufficient for any ε > 0, with the approximation factor being a polynomial in 1/ε. We obtain these results using a unified scheme for rounding the natural LP relaxation; this scheme may be useful for other capacitated covering problems. We also complement these bi-criteria approximations by obtaining hardness of approximation results that shed light on our understanding of these problems.

It was recently shown by [Sidiropoulos \& Sridhar, SoCG 2017] that several problems admit improved solutions when the input is a pointset in Euclidean space with fractal dimension smaller than the ambient dimension.

In this paper we prove nearly-matching lower bounds, thus establishing nearly-optimal bounds for various problems as a function of the fractal dimension.

More specifically, we show that for any set of $n$ points in $d$-dimensional Euclidean space, of fractal dimension $\delta\in (1,d)$, for any $\eps>0$ and $c\geq 1$, any $c$-spanner must have treewidth at least $\Omega \left( \frac{n^{1-1/(\delta - \epsilon)}}{c^{d-1}} \right)$, matching the previous upper bound.

The construction used to prove this lower bound on the treewidth of spanners, can also be used to derive lower bounds on the running time of algorithms for various problems, assuming the Exponential Time Hypothesis.

We provide two prototypical results of this type:

\begin{itemize}

\item For any $\delta \in (1,d)$ and any $\eps >0$, $d$-dimensional Euclidean TSP on $n$ points with fractal dimension at most $\delta$ cannot be solved in time $2^{O\left(n^{1-1/(\delta - \eps)} \right)}$.

The best-known upper bound is $2^{O(n^{1-1/\delta} \log n)}$.

\item For any $\delta \in (1,d)$ and any $\eps >0$, the problem of finding $k$-pairwise non-intersecting $d$-dimensional unit balls/axis parallel unit cubes with centers having fractal dimension at most $\delta$ cannot be solved in time $f(k)n^{O \left(k^{1-1/(\delta - \eps)}\right)}$ for any computable function $f$.

The best-known upper bound is $n^{O(k^{1-1/\delta} \log n)}$.

\end{itemize}

The above results nearly match previously known upper bounds from [Sidiropoulos \& Sridhar, SoCG 2017], and generalize analogous lower bounds for the case of ambient dimension due to [Marx \& Sidiropoulos, SoCG 2014].

on the orientable surface of genus $g$. An unpublished result by Robertson and Seymour implies that for every $t$, every graph of

sufficiently large genus contains as a minor a projective $t\times t$ grid or one of the following so-called $t$-Kuratowski

graphs: $K_{3,t}$, or $t$ copies of $K_5$ or $K_{3,3}$ sharing at most $2$ common vertices. We show that the $\mathbb{Z}_2$-genus

of graphs in these families is unbounded in $t$; in fact, equal to their genus. Together, this implies that the genus of a graph is

bounded from above by a function of its $\mathbb{Z}_2$-genus, solving a problem posed by Schaefer and \v{S}tefankovi\v{c}, and

giving an approximate version of the Hanani-Tutte theorem on surfaces.

In this paper we give approximation and fixed-parameter tractable (FPT) algorithms for minimum-distortion embeddings into the metric of a subdivision of some fixed graph $H$, or, equivalently, into any fixed 1-dimensional simplicial complex. More precisely, we study the following problem: For given graphs $G$, $H$ and integer $c$, is it possible to embed $G$ with distortion $c$ into a graph homeomorphic to $H$?

Then embedding into the line is the special case $H=K_2$, and embedding into the cycle is the case $H=K_3$, where $K_k$ denotes the complete graph on $k$ vertices.

For this problem we give

- An approximation algorithm, which in time $f(H) \poly(n)$, for some function $f$, either correctly decides that there is no embedding of $G$ with distortion $c$ into any graph homeomorphic to $H$, or finds an embedding with distortion $\poly(c)$;

- An exact algorithm, which in time $f'(H, c) \poly(n)$, for some function $f'$, either correctly decides that there is no embedding of $G$ with distortion $c$ into any graph homeomorphic to $H$, or finds an embedding with distortion $c$.

Prior to our work, $\poly(\OPT)$-approximation or FPT algorithms were known only for embedding into paths and trees of bounded degrees.

We also obtain similar bounds for orthogonal line segment intersection reporting queries, vertical ray stabbing, and vertical stabbing-max, improving previous bounds, respectively, of Blelloch [SODA 2008] and Mortensen [SODA 2003], of Tao (2014), and of Agarwal, Arge, and Yi [SODA 2005] and Nekrich [ISAAC 2011].

\begin{enumerate}

\item every tree of size $n$ (with arbitrarily large degree) has a straight-line drawing with area $n2^{O(\sqrt{\log\log n\log\log\log n})}$, improving the longstanding $O(n\log n)$ bound;

\item every tree of size $n$ (with arbitrarily large degree) has a straight-line upward drawing with area $n\sqrt{\log n}(\log\log n)^{O(1)}$, improving the longstanding $O(n\log n)$ bound;

\item every binary tree of size $n$ has a straight-line orthogonal drawing with area $n2^{O(\log^*n)}$, improving the previous $O(n\log\log n)$ bound by Shin, Kim, and Chwa (1996) and Chan, Goodrich, Kosaraju, and Tamassia (1996);

\item every binary tree of size $n$ has a straight-line order-preserving drawing with area $n2^{O(\log^*n)}$, improving the previous $O(n\log\log n)$ bound by Garg and Rusu (2003);

\item every binary tree of size $n$ has a straight-line orthogonal order-preserving drawing with area $n2^{O(\sqrt{\log n})}$, improving the $O(n^{3/2})$ previous bound by Frati (2007).

\end{enumerate}

plane. The intersection graph of any family of plane convex sets is a string graph,

but not all string graphs can be obtained in this way. We prove the following

structure theorem conjectured by Jansson and Uzzell: The vertex set of almost

all string graphs on n vertices can be partitioned into five cliques such

that some pair of them is not connected by any edge. We also

show that every graph with the above property is an intersection graph of plane

convex sets. As a corollary, we obtain that almost all string graphs on n

vertices are intersection graphs of plane convex sets.

neighbor queries for a set $S$ of point sites in a static simple polygon

$P$. Our data structure allows us to insert a new site in $S$, delete a site

from $S$, and ask for the site in $S$ closest to an arbitrary query point

$q \in P$. All distances are measured using the geodesic distance, that is,

the length of the shortest path that is completely contained in $P$. Our

data structure achieves polylogarithmic update and query times, and uses

$O(n\log^3n\log m + m)$ space, where $n$ is the number of sites in $S$ and

$m$ is the number of vertices in $P$. The crucial ingredient for our data

structure is an implicit representation of a vertical shallow cutting of the

geodesic distance functions. We show that such an implicit representation

exists, and that we can compute it efficiently.

These are the most popular and practical approaches to comparing such curves.

We offer the first data structures and query algorithms for ANN with arbitrarily good approximation factor, at the expense of increasing space usage and preprocessing time over existing methods. Query time complexity is comparable or significantly improved by our algorithms; our algorithm is especially efficient when the length of the curves is bounded.

More precisely, an instance of this problem consists of (i) a graph $G$ whose vertices are partitioned into clusters and whose inter-cluster edges are partitioned into bundles, and (ii) a region $R$ of on a 2-dimensional compact surface $M$ without boundary given as the union of a set of pairwise disjoint discs corresponding to the clusters and a set of pairwise non-intersecting ``pipes'' corresponding to the bundles, connecting certain pairs of these discs.

We are to decide whether $G$ can be embedded inside $M$ so that the vertices in every cluster are drawn in the corresponding disc, the edges in every bundle pass only through its corresponding pipe, and every edge crosses the boundary of each disc at most once.

shape~$P$ can be placed in any orientation inside a convex shape~$Q$

in the plane, then $P$ can also be turned continuously

through~$360^{\circ}$ inside~$Q$. We also prove a lower bound of

$\Omega(m n^{2})$ on the number of combinatorially distinct maximal

placements of a convex $m$-gon~$P$ in a convex $n$-gon~$Q$. This

matches the upper bound proven by Agarwal, Amenta, and Sharir in 1998.

Almost three decades ago, an elegant polynomial-time algorithm was found for Maximum Clique on unit disk graphs [Discrete Mathematics '90].

Since then, it has been an intriguing open question whether or not this tractability can be extended to the more general disk graphs.

We show the rather surprising result that a disjoint union of cycles is the complement of a disk graph if and only if at most one of those cycles is of odd length.

From that, we derive the first QPTAS and subexponential algorithm for Maximum Clique on disk graphs.

In stark contrast, Maximum Clique in the intersection graph of ellipses or triangles is very unlikely to have such algorithms.

$$|A/A+A| \gg |A|^{3/2 + 1/26}.$$

This improves a result of Shkredov.

We show that these techniques can be refined and extended to instances of much bigger size and different type, based on an array of modifications and parallelizations in combination with more efficient geometric encodings and

data structures. As a result, we are able to solve MWT instances with up to 30,000,000 uniformly distributed points in less than 4 minutes to provable optimality. Moreover, we can compute optimal solutions for a vast array of other

benchmark instances that are {\em not} uniformly distributed, including normally distributed instances (up to 30,000,00 points), all point sets in the TSPLIB (up to 85,900 points), and VLSI instances with up to 744,710 points. This demonstrates that from a practical point of view, MWT instances can be handled quite well, despite their theoretical difficulty.

Interestingly, no previous authors studied line simplification under these measures in its pure form, namely: for a given $\eps>0$, choose a minimum size subsequence of the vertices of the input such that the Hausdorff or Fr\'echet distance between the input and output polylines is at most $\eps$.

We analyze how the well-known Douglas-Peucker and Imai-Iri simplification

algorithms perform compared to the optimum possible, also in the situation where

the algorithms are given considerably larger values than $\eps$.

Furthermore, we show that computing an optimal simplification using the undirected Hausdorff distance is NP-hard.

The same holds when using the directed Hausdorff distance from the input to the output polyline, whereas the reverse can be computed in polynomial time.

Finally, we give a polynomial time algorithm to compute the optimal simplification under the Fr\'echet distance.

In this paper, we consider this problem in the adversarial model where $S$ and $R$ are points on a line. We improve the analysis of the deterministic Robust Matching Algorithm (Nayyar and Raghvendra FOCS'17) from $O(\log^2 n)$ to an optimal $\Theta(\log n)$. Previously, only a randomized algorithm (under the weaker oblivious adversary model) achieves a $O(\log n)$ (Gupta and Lewi, ICALP'12) competitive ratio. The well-known Work Function Algorithm (WFA) has a competitive ratio of $O(n)$ and $\Omega(\log n)$ for the line metric. Therefore, WFA cannot achieve an asymptotically better competitive ratio than our algorithm.

However, to date, there is very limited theoretical understanding of this framework in terms of graph reconstruction. This paper makes a first step towards closing this gap. Specifically, first, leveraging existing theoretical understanding of persistence-guided discrete Morse cancellation, we provide a simplified version of the existing discrete Morse-based graph reconstruction algorithm. We then introduce a simple and natural noise model and show that the aforementioned framework can correctly reconstruct a graph under this noise model, in the sense that it has the same topology as the hidden ground-truth graph, and is also geometrically close. We also provide some experimental results for our simplified graph-reconstruction algorithm.

as input an $n$-vertex directed graph $G=(V,E)$,

and two distinguished vertices $s$ and $t$.

The problem is to determine whether there exists a path

from $s$ to $t$ in $G$.

This is a canonical complete problem for class NL.

Asano et al. proposed

an $\widetilde{O}(\sqrt{n})$ space and polynomial time algorithm

for the directed grid and planar graph reachability problem.

The main result of this paper is to show that

the directed graph reachability problem restricted to grid graphs can be solved in polynomial time using only $\widetilde{O}(n^{1/3})$ space.