arXiv subjects

Sándor Kisfaludi-Bak

Publications and source records attributed to Sándor Kisfaludi-Bak.

At least 37 records · Page 2Linked to original sources

arXiv2023

Separator Theorem and Algorithms for Planar Hyperbolic Graphs

The hyperbolicity of a graph, informally, measures how close a graph is (metrically) to a tree. Hence, it is intuitively similar to treewidth, but the measures are formally incomparable. Motivated by the broad study of algorithms and separators on planar graphs and their relation to treewidth, we initiate the study of planar graphs of bounded hyperbolicity. Our main technical contribution is a novel balanced separator theorem for planar $δ$-hyperbolic graphs that is substantially stronger than the classic planar separator theorem. For any fixed $δ\geq 0$, we can find balanced separator that induces either a single geodesic (shortest) path or a single geodesic cycle in the graph. An important advantage of our separator is that the union of our separator (vertex set $Z$) with any subset of the connected components of $G - Z$ induces again a planar $δ$-hyperbolic graph, which would not be guaranteed with an arbitrary separator. Our construction runs in near-linear time and guarantees that size of separator is $\mathrm{poly}(δ) \cdot \log n$. As an application of our separator theorem and its strong properties, we obtain two novel approximation schemes on planar $δ$-hyperbolic graphs. We prove that Maximum Independent Set and the Traveling Salesperson problem have a near-linear time FPTAS for any constant $δ$, running in $n\, \mathrm{polylog}(n) \cdot 2^{\mathcal{O}(δ^2)} \cdot \varepsilon^{-\mathcal{O}(δ)}$ time. We also show that our approximation scheme for Maximum Independent Set has essentially the best possible running time under the Exponential Time Hypothesis (ETH). This immediately follows from our third contribution: we prove that Maximum Independent Set has no $n^{o(δ)}$-time algorithm on planar $δ$-hyperbolic graphs, unless ETH fails.

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arXiv2023

A Quadtree, a Steiner Spanner, and Approximate Nearest Neighbours in Hyperbolic Space

We propose a data structure in $d$-dimensional hyperbolic space that can be considered a natural counterpart to quadtrees in Euclidean spaces. Based on this data structure we propose a so-called L-order for hyperbolic point sets, which is an extension of the Z-order defined in Euclidean spaces. Using these quadtrees and the L-order we build geometric spanners. Near-linear size $(1+ε)$-spanners do not exist in hyperbolic spaces, but we are able to create a Steiner spanner that achieves a spanning ratio of $1+ε$ with $\mathcal O_{d,ε}(n)$ edges, using a simple construction that can be maintained dynamically. As a corollary we also get a $(2+ε)$-spanner (in the classical sense) of the same size, where the spanning ratio $2+ε$ is almost optimal among spanners of subquadratic size. Finally, we show that our Steiner spanner directly provides a solution to the approximate nearest neighbour problem: given a point set $P$ in $d$-dimensional hyperbolic space we build the data structure in $\mathcal O_{d,ε}(n\log n)$ time, using $\mathcal O_{d,ε}(n)$ space. Then for any query point $q$ we can find a point $p\in P$ that is at most $1+ε$ times farther from $q$ than its nearest neighbour in $P$ in $\mathcal O_{d,ε}(\log n)$ time. Moreover, the data structure is dynamic and can handle point insertions and deletions with update time $\mathcal O_{d,ε}(\log n)$.

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arXiv2023

An ETH-Tight Exact Algorithm for Euclidean TSP

We study exact algorithms for Euclidean TSP in $\mathbb{R}^d$. In the early 1990s algorithms with $n^{O(\sqrt{n})}$ running time were presented for the planar case, and some years later an algorithm with $n^{O(n^{1-1/d})}$ running time was presented for any $d\geq 2$. Despite significant interest in subexponential exact algorithms over the past decade, there has been no progress on Euclidean TSP, except for a lower bound stating that the problem admits no $2^{O(n^{1-1/d-ε})}$ algorithm unless ETH fails. Up to constant factors in the exponent, we settle the complexity of Euclidean TSP by giving a $2^{O(n^{1-1/d})}$ algorithm and by showing that a $2^{o(n^{1-1/d})}$ algorithm does not exist unless ETH fails.

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arXiv2022

Computing Smallest Convex Intersecting Polygons

A polygon C is an intersecting polygon for a set O of objects in the plane if C intersects each object in O, where the polygon includes its interior. We study the problem of computing the minimum-perimeter intersecting polygon and the minimum-area convex intersecting polygon for a given set O of objects. We present an FPTAS for both problems for the case where O is a set of possibly intersecting convex polygons in the plane of total complexity n. Furthermore, we present an exact polynomial-time algorithm for the minimum-perimeter intersecting polygon for the case where O is a set of n possibly intersecting segments in the plane. So far, polynomial-time exact algorithms were only known for the minimum perimeter intersecting polygon of lines or of disjoint segments.

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arXiv2022

Dynamic Time Warping Under Translation: Approximation Guided by Space-Filling Curves

The Dynamic Time Warping (DTW) distance is a popular measure of similarity for a variety of sequence data. For comparing polygonal curves $π, σ$ in $\mathbb{R}^d$, it provides a robust, outlier-insensitive alternative to the Fréchet distance. However, like the Fréchet distance, the DTW distance is not invariant under translations. Can we efficiently optimize the DTW distance of $π$ and $σ$ under arbitrary translations, to compare the curves' shape irrespective of their absolute location? There are surprisingly few works in this direction, which may be due to its computational intricacy: For the Euclidean norm, this problem contains as a special case the geometric median problem, which provably admits no exact algebraic algorithm (that is, no algorithm using only addition, multiplication, and $k$-th roots). We thus investigate exact algorithms for non-Euclidean norms as well as approximation algorithms for the Euclidean norm: - For the $L_1$ norm in $\mathbb{R}^d$, we provide an $\mathcal{O}(n^{2(d+1)})$-time algorithm, i.e., an exact polynomial-time algorithm for constant $d$. Here and below, $n$ bounds the curves' complexities. - For the Euclidean norm in $\mathbb{R}^2$, we show that a simple problem-specific insight leads to a $(1+\varepsilon)$-approximation in time $\mathcal{O}(n^3/\varepsilon^2)$. We then show how to obtain a subcubic $\widetilde{\mathcal{O}}(n^{2.5}/\varepsilon^2)$ time algorithm with significant new ideas; this time comes close to the well-known quadratic time barrier for computing DTW for fixed translations. Technically, the algorithm is obtained by speeding up repeated DTW distance estimations using a dynamic data structure for maintaining shortest paths in weighted planar digraphs. Crucially, we show how to traverse a candidate set of translations using space-filling curves in a way that incurs only few updates to the data structure.

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arXiv2022

Towards Sub-Quadratic Diameter Computation in Geometric Intersection Graphs

We initiate the study of diameter computation in geometric intersection graphs from the fine-grained complexity perspective. A geometric intersection graph is a graph whose vertices correspond to some shapes in $d$-dimensional Euclidean space, such as balls, segments, or hypercubes, and whose edges correspond to pairs of intersecting shapes. The diameter of a graph is the largest distance realized by a pair of vertices in the graph. Computing the diameter in near-quadratic time is possible in several classes of intersection graphs [Chan and Skrepetos 2019], but it is not at all clear if these algorithms are optimal, especially since in the related class of planar graphs the diameter can be computed in $\widetilde{\mathcal{O}}(n^{5/3})$ time [Cabello 2019, Gawrychowski et al. 2021]. In this work we (conditionally) rule out sub-quadratic algorithms in several classes of intersection graphs, i.e., algorithms of running time $\mathcal{O}(n^{2-δ})$ for some $δ>0$. In particular, there are no sub-quadratic algorithms already for fat objects in small dimensions: unit balls in $\mathbb{R}^3$ or congruent equilateral triangles in $\mathbb{R}^2$. For unit segments and congruent equilateral triangles, we can even rule out strong sub-quadratic approximations already in $\mathbb{R}^2$. It seems that the hardness of approximation may also depend on dimensionality: for axis-parallel unit hypercubes in~$\mathbb{R}^{12}$, distinguishing between diameter 2 and 3 needs quadratic time (ruling out $(3/2-\varepsilon)$- approximations), whereas for axis-parallel unit squares, we give an algorithm that distinguishes between diameter $2$ and $3$ in near-linear time. Note that many of our lower bounds match the best known algorithms up to sub-polynomial factors.

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arXiv2022

Computing list homomorphisms in geometric intersection graphs

A homomorphism from a graph $G$ to a graph $H$ is an edge-preserving mapping from $V(G)$ to $V(H)$. Let $H$ be a fixed graph with possible loops. In the list homomorphism problem, denoted by \textsc{LHom}($H$), the instance is a graph $G$, whose every vertex is equipped with a subset of $V(H)$, called list. We ask whether there exists a homomorphism from $G$ to $H$, such that every vertex from $G$ is mapped to a vertex from its list. We study the complexity of the \textsc{LHom}($H$) problem in intersection graphs of various geometric objects. In particular, we are interested in answering the question for what graphs $H$ and for what types of geometric objects, the \textsc{LHom}($H$) problem can be solved in time subexponential in the number of vertices of the instance. We fully resolve this question for string graphs, i.e., intersection graphs of continuous curves in the plane. Quite surprisingly, it turns out that the dichotomy exactly coincides with the analogous dichotomy for graphs excluding a fixed path as an induced subgraph [Okrasa, Rzążewski, STACS 2021]. Then we turn our attention to subclasses of string graphs, defined as intersections of fat objects. We observe that the (non)existence of subexponential-time algorithms in such classes is closely related to the size $\mathrm{mrc}(H)$ of a maximum reflexive clique in $H$, i.e., maximum number of pairwise adjacent vertices, each of which has a loop. We study the maximum value of $\mathrm{mrc}(H)$ that guarantees the existence of a subexponential-time algorithm for \textsc{LHom}($H$) in intersection graphs of (i) convex fat objects, (ii) fat similarly-sized objects, and (iii) disks. In the first two cases we obtain optimal results, by giving matching algorithms and lower bounds. Finally, we discuss possible extensions of our results to weighted generalizations of \textsc{LHom}($H$).

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arXiv2021

Clique-Based Separators for Geometric Intersection Graphs

Let $F$ be a set of $n$ objects in the plane and let $G(F)$ be its intersection graph. A balanced clique-based separator of $G(F)$ is a set $S$ consisting of cliques whose removal partitions $G(F)$ into components of size at most $δn$, for some fixed constant $δ<1$. The weight of a clique-based separator is defined as $\sum_{C\in S}\log (|C|+1)$. Recently De Berg et al. (SICOMP 2020) proved that if $S$ consists of convex fat objects, then $G(F)$ admits a balanced clique-based separator of weight $O(\sqrt{n})$. We extend this result in several directions, obtaining the following results. Map graphs admit a balanced clique-based separator of weight $O(\sqrt{n})$, which is tight in the worst case. Intersection graphs of pseudo-disks admit a balanced clique-based separator of weight $O(n^{2/3}\log n)$. If the pseudo-disks are polygonal and of total complexity $O(n)$ then the weight of the separator improves to $O(\sqrt{n}\log n)$. Intersection graphs of geodesic disks inside a simple polygon admit a balanced clique-based separator of weight $O(n^{2/3}\log n)$. Visibility-restricted unit-disk graphs in a polygonal domain with $r$ reflex vertices admit a balanced clique-based separator of weight $O(\sqrt{n}+r\log(n/r))$, which is tight in the worst case. These results immediately imply sub-exponential algorithms for MAXIMUM INDEPENDENT SET (and, hence, VERTEX COVER), for FEEDBACK VERTEX SET, and for $q$-COLORING for constant $q$ in these graph classes.

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arXiv2021

Online Search for a Hyperplane in High-Dimensional Euclidean Space

We consider the online search problem in which a server starting at the origin of a $d$-dimensional Euclidean space has to find an arbitrary hyperplane. The best-possible competitive ratio and the length of the shortest curve from which each point on the $d$-dimensional unit sphere can be seen are within a constant factor of each other. We show that this length is in $Ω(d)\cap O(d^{3/2})$.

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arXiv2020

A Framework for ETH-Tight Algorithms and Lower Bounds in Geometric Intersection Graphs

We give an algorithmic and lower-bound framework that facilitates the construction of subexponential algorithms and matching conditional complexity bounds. It can be applied to intersection graphs of similarly-sized fat objects, yielding algorithms with running time $2^{O(n^{1-1/d})}$ for any fixed dimension $d \geq 2$ for many well known graph problems, including Independent Set, $r$-Dominating Set for constant $r$, and Steiner Tree. For most problems, we get improved running times compared to prior work; in some cases, we give the first known subexponential algorithm in geometric intersection graphs. Additionally, most of the obtained algorithms are representation-agnostic, i.e., they work on the graph itself and do not require the geometric representation. Our algorithmic framework is based on a weighted separator theorem and various treewidth techniques. The lower bound framework is based on a constructive embedding of graphs into d-dimensional grids, and it allows us to derive matching $2^{Ω(n^{1-1/d})}$ lower bounds under the Exponential Time Hypothesis even in the much more restricted class of $d$-dimensional induced grid graphs.

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arXiv2020

On the Approximability of the Traveling Salesman Problem with Line Neighborhoods

We study the variant of the Euclidean Traveling Salesman problem where instead of a set of points, we are given a set of lines as input, and the goal is to find the shortest tour that visits each line. The best known upper and lower bounds for the problem in $\mathbb{R}^d$, with $d\ge 3$, are $\mathrm{NP}$-hardness and an $O(\log^3 n)$-approximation algorithm which is based on a reduction to the group Steiner tree problem. We show that TSP with lines in $\mathbb{R}^d$ is APX-hard for any $d\ge 3$. More generally, this implies that TSP with $k$-dimensional flats does not admit a PTAS for any $1\le k \leq d-2$ unless $\mathrm{P}=\mathrm{NP}$, which gives a complete classification of the approximability of these problems, as there are known PTASes for $k=0$ (i.e., points) and $k=d-1$ (hyperplanes). We are able to give a stronger inapproximability factor for $d=O(\log n)$ by showing that TSP with lines does not admit a $(2-ε)$-approximation in $d$ dimensions under the unique games conjecture. On the positive side, we leverage recent results on restricted variants of the group Steiner tree problem in order to give an $O(\log^2 n)$-approximation algorithm for the problem, albeit with a running time of $n^{O(\log\log n)}$.

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arXiv2020

A quasi-polynomial algorithm for well-spaced hyperbolic TSP

We study the traveling salesman problem in the hyperbolic plane of Gaussian curvature $-1$. Let $α$ denote the minimum distance between any two input points. Using a new separator theorem and a new rerouting argument, we give an $n^{O(\log^2 n)\max(1,1/α)}$ algorithm for Hyperbolic TSP. This is quasi-polynomial time if $α$ is at least some absolute constant, and it grows to $n^{O(\sqrt{n})}$ as $α$ decreases to $\log^2 n/\sqrt{n}$. (For even smaller values of $α$, we can use a planarity-based algorithm of Hwang et al. (1993), which gives a running time of $n^{O(\sqrt{n})}$.)

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arXiv2019

Hyperbolic intersection graphs and (quasi)-polynomial time

We study unit ball graphs (and, more generally, so-called noisy uniform ball graphs) in $d$-dimensional hyperbolic space, which we denote by $\mathbb{H}^d$. Using a new separator theorem, we show that unit ball graphs in $\mathbb{H}^d$ enjoy similar properties as their Euclidean counterparts, but in one dimension lower: many standard graph problems, such as Independent Set, Dominating Set, Steiner Tree, and Hamiltonian Cycle can be solved in $2^{O(n^{1-1/(d-1)})}$ time for any fixed $d\geq 3$, while the same problems need $2^{O(n^{1-1/d})}$ time in $\mathbb{R}^d$. We also show that these algorithms in $\mathbb{H}^d$ are optimal up to constant factors in the exponent under ETH. This drop in dimension has the largest impact in $\mathbb{H}^2$, where we introduce a new technique to bound the treewidth of noisy uniform disk graphs. The bounds yield quasi-polynomial ($n^{O(\log n)}$) algorithms for all of the studied problems, while in the case of Hamiltonian Cycle and $3$-Coloring we even get polynomial time algorithms. Furthermore, if the underlying noisy disks in $\mathbb{H}^2$ have constant maximum degree, then all studied problems can be solved in polynomial time. This contrasts with the fact that these problems require $2^{Ω(\sqrt{n})}$ time under ETH in constant maximum degree Euclidean unit disk graphs. Finally, we complement our quasi-polynomial algorithm for Independent Set in noisy uniform disk graphs with a matching $n^{Ω(\log n)}$ lower bound under ETH. This shows that the hyperbolic plane is a potential source of NP-intermediate problems.

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arXiv2019

How does object fatness impact the complexity of packing in d dimensions?

Packing is a classical problem where one is given a set of subsets of Euclidean space called objects, and the goal is to find a maximum size subset of objects that are pairwise non-intersecting. The problem is also known as the Independent Set problem on the intersection graph defined by the objects. Although the problem is NP-complete, there are several subexponential algorithms in the literature. One of the key assumptions of such algorithms has been that the objects are fat, with a few exceptions in two dimensions; for example, the packing problem of a set of polygons in the plane surprisingly admits a subexponential algorithm. In this paper we give tight running time bounds for packing similarly-sized non-fat objects in higher dimensions. We propose an alternative and very weak measure of fatness called the stabbing number, and show that the packing problem in Euclidean space of constant dimension $d \geq 3$ for a family of similarly sized objects with stabbing number $α$ can be solved in $2^{O(n^{1-1/d}α)}$ time. We prove that even in the case of axis-parallel boxes of fixed shape, there is no $2^{o(n^{1-1/d}α)}$ algorithm under ETH. This result smoothly bridges the whole range of having constant-fat objects on one extreme ($α=1$) and a subexponential algorithm of the usual running time, and having very "skinny" objects on the other extreme ($α=n^{1/d}$), where we cannot hope to improve upon the brute force running time of $2^{O(n)}$, and thereby characterizes the impact of fatness on the complexity of packing in case of similarly sized objects. We also study the same problem when parameterized by the solution size $k$, and give a $n^{O(k^{1-1/d}α)}$ algorithm, with an almost matching lower bound.

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arXiv2019

On One-Round Discrete Voronoi Games

Let $V$ be a multiset of $n$ points in $\mathbb{R}^d$, which we call voters, and let $k\geq 1$ and $\ell\geq 1$ be two given constants. We consider the following game, where two players $\mathcal{P}$ and $\mathcal{Q}$ compete over the voters in $V$: First, player $\mathcal{P}$ selects $k$ points in $\mathbb{R}^d$, and then player $\mathcal{Q}$ selects $\ell$ points in $\mathbb{R}^d$. Player $\mathcal{P}$ wins a voter $v\in V$ iff $\mathrm{dist}(v,P) \leq \mathrm{dist}(v,Q)$, where $\mathrm{dist}(v,P) := \min_{p\in P} \mathrm{dist}(v,p)$ and $\mathrm{dist}(v,Q)$ is defined similarly. Player $\mathcal{P}$ wins the game if he wins at least half the voters. The algorithmic problem we study is the following: given $V$, $k$, and $\ell$, how efficiently can we decide if player $\mathcal{P}$ has a winning strategy, that is, if $\mathcal{P}$ can select his $k$ points such that he wins the game no matter where $\mathcal{Q}$ places her points. Banik et al. devised a singly-exponential algorithm for the game in $\mathbb{R}^1$, for the case $k=\ell$. We improve their result by presenting the first polynomial-time algorithm for the game in $\mathbb{R}^1$. Our algorithm can handle arbitrary values of $k$ and $\ell$. We also show that if $d\geq 2$, deciding if player $\mathcal{P}$ has a winning strategy is $Σ_2^P$-hard when $k$ and $\ell$ are part of the input. Finally, we prove that for any dimension $d$, the problem is contained in the complexity class $\exists\forall \mathbb{R}$, and we give an algorithm that works in polynomial time for fixed $k$ and $\ell$.

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arXiv2018

Nearly ETH-Tight Algorithms for Planar Steiner Tree with Terminals on Few Faces

The Planar Steiner Tree problem is one of the most fundamental NP-complete problems as it models many network design problems. Recall that an instance of this problem consists of a graph with edge weights, and a subset of vertices (often called terminals); the goal is to find a subtree of the graph of minimum total weight that connects all terminals. A seminal paper by Erickson et al. [Math. Oper. Res., 1987] considers instances where the underlying graph is planar and all terminals can be covered by the boundary of $k$ faces. Erickson et al. show that the problem can be solved by an algorithm using $n^{O(k)}$ time and $n^{O(k)}$ space, where $n$ denotes the number of vertices of the input graph. In the past 30 years there has been no significant improvement of this algorithm, despite several efforts. In this work, we give an algorithm for Planar Steiner Tree with running time $2^{O(k)} n^{O(\sqrt{k})}$ using only polynomial space. Furthermore, we show that the running time of our algorithm is almost tight: we prove that there is no $f(k)n^{o(\sqrt{k})}$ algorithm for Planar Steiner Tree for any computable function $f$, unless the Exponential Time Hypothesis fails.

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arXiv2017

The Dominating Set Problem in Geometric Intersection Graphs

We study the parameterized complexity of dominating sets in geometric intersection graphs. In one dimension, we investigate intersection graphs induced by translates of a fixed pattern Q that consists of a finite number of intervals and a finite number of isolated points. We prove that Dominating Set on such intersection graphs is polynomially solvable whenever Q contains at least one interval, and whenever Q contains no intervals and for any two point pairs in Q the distance ratio is rational. The remaining case where Q contains no intervals but does contain an irrational distance ratio is shown to be NP-complete and contained in FPT (when parameterized by the solution size). In two and higher dimensions, we prove that Dominating Set is contained in W[1] for intersection graphs of semi-algebraic sets with constant description complexity. This generalizes known results from the literature. Finally, we establish W[1]-hardness for a large class of intersection graphs.

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arXiv2017

On the number of touching pairs in a set of planar curves

Given a set of planar curves (Jordan arcs), each pair of which meets -- either crosses or touches -- exactly once, we establish an upper bound on the number of touchings. We show that such a curve family has $O(t^2n)$ touchings, where $t$ is the number of faces in the curve arrangement that contains at least one endpoint of one of the curves. Our method relies on finding special subsets of curves called quasi-grids in curve families; this gives some structural insight into curve families with a high number of touchings.

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