arXiv subjects

Anastasios Sidiropoulos

Publications and source records attributed to Anastasios Sidiropoulos.

At least 37 records · Page 2Linked to original sources

arXiv2017

Routing Symmetric Demands in Directed Minor-Free Graphs with Constant Congestion

The problem of routing in graphs using node-disjoint paths has received a lot of attention and a polylogarithmic approximation algorithm with constant congestion is known for undirected graphs [Chuzhoy and Li 2016] and [Chekuri and Ene 2013]. However, the problem is hard to approximate within polynomial factors on directed graphs, for any constant congestion [Chuzhoy, Kim and Li 2016]. Recently, [Chekuri, Ene and Pilipczuk 2016] have obtained a polylogarithmic approximation with constant congestion on directed planar graphs, for the special case of symmetric demands. We extend their result by obtaining a polylogarithmic approximation with constant congestion on arbitrary directed minor-free graphs, for the case of symmetric demands.

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arXiv2017

On constant multi-commodity flow-cut gaps for directed minor-free graphs

The multi-commodity flow-cut gap is a fundamental parameter that affects the performance of several divide \& conquer algorithms, and has been extensively studied for various classes of undirected graphs. It has been shown by Linial, London and Rabinovich \cite{linial1994geometry} and by Aumann and Rabani \cite{aumann1998log} that for general $n$-vertex graphs it is bounded by $O(\log n)$ and the Gupta-Newman-Rabinovich-Sinclair conjecture \cite{gupta2004cuts} asserts that it is $O(1)$ for any family of graphs that excludes some fixed minor. The flow-cut gap is poorly understood for the case of directed graphs. We show that for uniform demands it is $O(1)$ on directed series-parallel graphs, and on directed graphs of bounded pathwidth. These are the first constant upper bounds of this type for some non-trivial family of directed graphs. We also obtain $O(1)$ upper bounds for the general multi-commodity flow-cut gap on directed trees and cycles. These bounds are obtained via new embeddings and Lipschitz quasipartitions for quasimetric spaces, which generalize analogous results form the metric case, and could be of independent interest. Finally, we discuss limitations of methods that were developed for undirected graphs, such as random partitions, and random embeddings.

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arXiv2017

Temporal Hierarchical Clustering

We study hierarchical clusterings of metric spaces that change over time. This is a natural geometric primitive for the analysis of dynamic data sets. Specifically, we introduce and study the problem of finding a temporally coherent sequence of hierarchical clusterings from a sequence of unlabeled point sets. We encode the clustering objective by embedding each point set into an ultrametric space, which naturally induces a hierarchical clustering of the set of points. We enforce temporal coherence among the embeddings by finding correspondences between successive pairs of ultrametric spaces which exhibit small distortion in the Gromov-Hausdorff sense. We present both upper and lower bounds on the approximability of the resulting optimization problems.

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arXiv2017

Polylogarithmic approximation for minimum planarization (almost)

In the minimum planarization problem, given some $n$-vertex graph, the goal is to find a set of vertices of minimum cardinality whose removal leaves a planar graph. This is a fundamental problem in topological graph theory. We present a $\log^{O(1)} n$-approximation algorithm for this problem on general graphs with running time $n^{O(\log n/\log\log n)}$. We also obtain a $O(n^\varepsilon)$-approximation with running time $n^{O(1/\varepsilon)}$ for any arbitrarily small constant $\varepsilon > 0$. Prior to our work, no non-trivial algorithm was known for this problem on general graphs, and the best known result even on graphs of bounded degree was a $n^{Ω(1)}$-approximation [Chekuri and Sidiropoulos 2013]. As an immediate corollary, we also obtain improved approximation algorithms for the crossing number problem on graphs of bounded degree. Specifically, we obtain $O(n^{1/2+\varepsilon})$-approximation and $n^{1/2} \log^{O(1)} n$-approximation algorithms in time $n^{O(1/\varepsilon)}$ and $n^{O(\log n/\log\log n)}$ respectively. The previously best-known result was a polynomial-time $n^{9/10}\log^{O(1)} n$-approximation algorithm [Chuzhoy 2011]. Our algorithm introduces several new tools including an efficient grid-minor construction for apex graphs, and a new method for computing irrelevant vertices. Analogues of these tools were previously available only for exact algorithms. Our work gives efficient implementations of these ideas in the setting of approximation algorithms, which could be of independent interest.

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arXiv2017

Computing the Gromov-Hausdorff Distance for Metric Trees

The Gromov-Hausdorff (GH) distance is a natural way to measure distance between two metric spaces. We prove that it is $\mathrm{NP}$-hard to approximate the Gromov-Hausdorff distance better than a factor of $3$ for geodesic metrics on a pair of trees. We complement this result by providing a polynomial time $O(\min\{n, \sqrt{rn}\})$-approximation algorithm for computing the GH distance between a pair of metric trees, where $r$ is the ratio of the longest edge length in both trees to the shortest edge length. For metric trees with unit length edges, this yields an $O(\sqrt{n})$-approximation algorithm.

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arXiv2017

Temporal Clustering

We study the problem of clustering sequences of unlabeled point sets taken from a common metric space. Such scenarios arise naturally in applications where a system or process is observed in distinct time intervals, such as biological surveys and contagious disease surveillance. In this more general setting existing algorithms for classical (i.e.~static) clustering problems are not applicable anymore. We propose a set of optimization problems which we collectively refer to as 'temporal clustering'. The quality of a solution to a temporal clustering instance can be quantified using three parameters: the number of clusters $k$, the spatial clustering cost $r$, and the maximum cluster displacement $δ$ between consecutive time steps. We consider spatial clustering costs which generalize the well-studied $k$-center, discrete $k$-median, and discrete $k$-means objectives of classical clustering problems. We develop new algorithms that achieve trade-offs between the three objectives $k$, $r$, and $δ$. Our upper bounds are complemented by inapproximability results.

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arXiv2017

Algorithmic interpretations of fractal dimension

We study algorithmic problems on subsets of Euclidean space of low fractal dimension. These spaces are the subject of intensive study in various branches of mathematics, including geometry, topology, and measure theory. There are several well-studied notions of fractal dimension for sets and measures in Euclidean space. We consider a definition of fractal dimension for finite metric spaces which agrees with standard notions used to empirically estimate the fractal dimension of various sets. We define the fractal dimension of some metric space to be the infimum $δ>0$, such that for any $ε> 0$, for any ball $B$ of radius $r\geq 2ε$, and for any $ε$-net $N$ (that is, for any maximal $ε$-packing), we have $|B\cap N|=O((r/ε)^δ)$. Using this definition we obtain faster algorithms for a plethora of classical problems on sets of low fractal dimension in Euclidean space. Our results apply to exact and fixed-parameter algorithms, approximation schemes, and spanner constructions. Interestingly, the dependence of the performance of these algorithms on the fractal dimension nearly matches the currently best-known dependence on the standard Euclidean dimension. Thus, when the fractal dimension is strictly smaller than the ambient dimension, our results yield improved solutions in all of these settings.

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arXiv2016

The limited blessing of low dimensionality: when $1-1/d$ is the best possible exponent for $d$-dimensional geometric problems

We are studying $d$-dimensional geometric problems that have algorithms with $1-1/d$ appearing in the exponent of the running time, for example, in the form of $2^{n^{1-1/d}}$ or $n^{k^{1-1/d}}$. This means that these algorithms perform somewhat better in low dimensions, but the running time is almost the same r all large values $d$ of the dimension. Our main result is showing that for some of these problems the dependence on $1-1/d$ is best possible under a standard complexity assumption. We show that, assuming the Exponential Time Hypothesis, --- $d$-dimensional Euclidean TSP on $n$ points cannot be solved in time $2^{O(n^{1-1/d-ε})}$ for any $ε>0$, and --- the problem of finding a set of $k$ pairwise nonintersecting $d$-dimensional unit balls/axis parallel unit cubes cannot be solved in time $f(k)n^{o(k^{1-1/d})}$ for any computable function $f$. These lower bounds essentially match the known algorithms for these problems. To obtain these results, we first prove lower bounds on the complexity of Constraint Satisfaction Problems (CSPs) whose constraint graphs are $d$-dimensional grids. We state the complexity results on CSPs in a way to make them convenient starting points for problem-specific reductions to particular $d$-dimensional geometric problems and to be reusable in the future for further results of similar flavor.

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arXiv2016

Quasimetric embeddings and their applications

We study generalizations of classical metric embedding results to the case of quasimetric spaces; that is, spaces that do not necessarily satisfy symmetry. Quasimetric spaces arise naturally from the shortest-path distances on directed graphs. Perhaps surprisingly, very little is known about low-distortion embeddings for quasimetric spaces. Random embeddings into ultrametric spaces are arguably one of the most successful geometric tools in the context of algorithm design. We extend this to the quasimetric case as follows. We show that any $n$-point quasimetric space supported on a graph of treewidth $t$ admits a random embedding into quasiultrametric spaces with distortion $O(t \log^2 n)$, where quasiultrametrics are a natural generalization of ultrametrics. This result allows us to obtain $t\log^{O(1)} n$-approximation algorithms for the Directed Non-Bipartite Sparsest-Cut and the Directed Multicut problems on $n$-vertex graphs of treewidth $t$, with running time polynomial in both $n$ and $t$. The above results are obtained by considering a generalization of random partitions to the quasimetric case, which we refer to as random quasipartitions. Using this definition and a construction of [Chuzhoy and Khanna 2009] we derive a polynomial lower bound on the distortion of random embeddings of general quasimetric spaces into quasiultrametric spaces. Finally, we establish a lower bound for embedding the shortest-path quasimetric of a graph $G$ into graphs that exclude $G$ as a minor. This lower bound is used to show that several embedding results from the metric case do not have natural analogues in the quasimetric setting.

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arXiv2016

Layouts of Expander Graphs

Bourgain and Yehudayoff recently constructed $O(1)$-monotone bipartite expanders. By combining this result with a generalisation of the unraveling method of Kannan, we construct 3-monotone bipartite expanders, which is best possible. We then show that the same graphs admit 3-page book embeddings, 2-queue layouts, 4-track layouts, and have simple thickness 2. All these results are best possible.

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arXiv2016

Constant-factor approximations for asymmetric TSP on nearly-embeddable graphs

In the Asymmetric Traveling Salesperson Problem (ATSP) the goal is to find a closed walk of minimum cost in a directed graph visiting every vertex. We consider the approximability of ATSP on topologically restricted graphs. It has been shown by [Oveis Gharan and Saberi 2011] that there exists polynomial-time constant-factor approximations on planar graphs and more generally graphs of constant orientable genus. This result was extended to non-orientable genus by [Erickson and Sidiropoulos 2014]. We show that for any class of \emph{nearly-embeddable} graphs, ATSP admits a polynomial-time constant-factor approximation. More precisely, we show that for any fixed $k\geq 0$, there exist $α, β>0$, such that ATSP on $n$-vertex $k$-nearly-embeddable graphs admits a $α$-approximation in time $O(n^β)$. The class of $k$-nearly-embeddable graphs contains graphs with at most $k$ apices, $k$ vortices of width at most $k$, and an underlying surface of either orientable or non-orientable genus at most $k$. Prior to our work, even the case of graphs with a single apex was open. Our algorithm combines tools from rounding the Held-Karp LP via thin trees with dynamic programming. We complement our upper bounds by showing that solving ATSP exactly on graphs of pathwidth $k$ (and hence on $k$-nearly embeddable graphs) requires time $n^{Ω(k)}$, assuming the Exponential-Time Hypothesis (ETH). This is surprising in light of the fact that both TSP on undirected graphs and Minimum Cost Hamiltonian Cycle on directed graphs are FPT parameterized by treewidth.

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arXiv2015

How to Walk Your Dog in the Mountains with No Magic Leash

We describe a $O(\log n )$-approximation algorithm for computing the homotopic \Frechet distance between two polygonal curves that lie on the boundary of a triangulated topological disk. Prior to this work, algorithms were known only for curves on the Euclidean plane with polygonal obstacles. A key technical ingredient in our analysis is a $O(\log n)$-approximation algorithm for computing the minimum height of a homotopy between two curves. No algorithms were previously known for approximating this parameter. Surprisingly, it is not even known if computing either the homotopic \Frechet distance, or the minimum height of a homotopy, is in NP.

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arXiv2015

Metric embedding with outliers

We initiate the study of metric embeddings with \emph{outliers}. Given some metric space $(X,ρ)$ we wish to find a small set of outlier points $K \subset X$ and either an isometric or a low-distortion embedding of $(X\setminus K,ρ)$ into some target metric space. This is a natural problem that captures scenarios where a small fraction of points in the input corresponds to noise. For the case of isometric embeddings we derive polynomial-time approximation algorithms for minimizing the number of outliers when the target space is an ultrametric, a tree metric, or constant-dimensional Euclidean space. The approximation factors are 3, 4 and 2, respectively. For the case of embedding into an ultrametric or tree metric, we further improve the running time to $O(n^2)$ for an $n$-point input metric space, which is optimal. We complement these upper bounds by showing that outlier embedding into ultrametrics, trees, and $d$-dimensional Euclidean space for any $d\geq 2$ are all NP-hard, as well as NP-hard to approximate within a factor better than 2 assuming the Unique Game Conjecture. For the case of non-isometries we consider embeddings with small $\ell_{\infty}$ distortion. We present polynomial-time \emph{bi-criteria} approximation algorithms. Specifically, given some $ε> 0$, let $k_ε$ denote the minimum number of outliers required to obtain an embedding with distortion $ε$. For the case of embedding into ultrametrics we obtain a polynomial-time algorithm which computes a set of at most $3k_ε$ outliers and an embedding of the remaining points into an ultrametric with distortion $O(ε\log n)$. For embedding a metric of unit diameter into constant-dimensional Euclidean space we present a polynomial-time algorithm which computes a set of at most $2k_ε$ outliers and an embedding of the remaining points with distortion $O(\sqrtε)$.

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arXiv2015

Approximate Greedy Clustering and Distance Selection for Graph Metrics

$\newcommand{\eps}{\varepsilon}$ In this paper, we consider two important problems defined on finite metric spaces, and provide efficient new algorithms and approximation schemes for these problems on inputs given as graph shortest path metrics or high-dimensional Euclidean metrics. The first of these problems is the greedy permutation (or farthest-first traversal) of a finite metric space: a permutation of the points of the space in which each point is as far as possible from all previous points. We describe randomized algorithms to find $(1+\eps)$-approximate greedy permutations of any graph with $n$ vertices and $m$ edges in expected time $O(\eps^{-1}(m+n)\log n\log(n/\eps))$, and to find $(1+\eps)$-approximate greedy permutations of points in high-dimensional Euclidean spaces in expected time $O(\eps^{-2} n^{1+1/(1+\eps)^2 + o(1)})$. Additionally we describe a deterministic algorithm to find exact greedy permutations of any graph with $n$ vertices and treewidth $O(1)$ in worst-case time $O(n^{3/2}\log^{O(1)} n)$. The second of the two problems we consider is distance selection: given $k \in [ \binom{n}{2} ]$, we are interested in computing the $k$th smallest distance in the given metric space. We show that for planar graph metrics one can approximate this distance, up to a constant factor, in near linear time.

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arXiv2014

Beyond the Euler characteristic: Approximating the genus of general graphs

Computing the Euler genus of a graph is a fundamental problem in graph theory and topology. It has been shown to be NP-hard by [Thomassen '89] and a linear-time fixed-parameter algorithm has been obtained by [Mohar '99]. Despite extensive study, the approximability of the Euler genus remains wide open. While the existence of an $O(1)$-approximation is not ruled out, the currently best-known upper bound is a trivial $O(n/g)$-approximation that follows from bounds on the Euler characteristic. In this paper, we give the first non-trivial approximation algorithm for this problem. Specifically, we present a polynomial-time algorithm which given a graph $G$ of Euler genus $g$ outputs an embedding of $G$ into a surface of Euler genus $g^{O(1)}$. Combined with the above $O(n/g)$-approximation, our result also implies a $O(n^{1-α})$-approximation, for some universal constant $α>0$. Our approximation algorithm also has implications for the design of algorithms on graphs of small genus. Several of these algorithms require that an embedding of the graph into a surface of small genus is given as part of the input. Our result implies that many of these algorithms can be implemented even when the embedding of the input graph is unknown.

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arXiv2013

Fat Polygonal Partitions with Applications to Visualization and Embeddings

Let $\mathcal{T}$ be a rooted and weighted tree, where the weight of any node is equal to the sum of the weights of its children. The popular Treemap algorithm visualizes such a tree as a hierarchical partition of a square into rectangles, where the area of the rectangle corresponding to any node in $\mathcal{T}$ is equal to the weight of that node. The aspect ratio of the rectangles in such a rectangular partition necessarily depends on the weights and can become arbitrarily high. We introduce a new hierarchical partition scheme, called a polygonal partition, which uses convex polygons rather than just rectangles. We present two methods for constructing polygonal partitions, both having guarantees on the worst-case aspect ratio of the constructed polygons; in particular, both methods guarantee a bound on the aspect ratio that is independent of the weights of the nodes. We also consider rectangular partitions with slack, where the areas of the rectangles may differ slightly from the weights of the corresponding nodes. We show that this makes it possible to obtain partitions with constant aspect ratio. This result generalizes to hyper-rectangular partitions in $\mathbb{R}^d$. We use these partitions with slack for embedding ultrametrics into $d$-dimensional Euclidean space: we give a $\mathop{\rm polylog}(Δ)$-approximation algorithm for embedding $n$-point ultrametrics into $\mathbb{R}^d$ with minimum distortion, where $Δ$ denotes the spread of the metric, i.e., the ratio between the largest and the smallest distance between two points. The previously best-known approximation ratio for this problem was polynomial in $n$. This is the first algorithm for embedding a non-trivial family of weighted-graph metrics into a space of constant dimension that achieves polylogarithmic approximation ratio.

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arXiv2013

Approximation algorithms for Euler genus and related problems

The Euler genus of a graph is a fundamental and well-studied parameter in graph theory and topology. Computing it has been shown to be NP-hard by [Thomassen '89 & '93], and it is known to be fixed-parameter tractable. However, the approximability of the Euler genus is wide open. While the existence of an O(1)-approximation is not ruled out, only an O(sqrt(n))-approximation [Chen, Kanchi, Kanevsky '97] is known even in bounded degree graphs. In this paper we give a polynomial-time algorithm which on input a bounded-degree graph of Euler genus g, computes a drawing into a surface of Euler genus poly(g, log(n)). Combined with the upper bound from [Chen, Kanchi, Kanevsky '97], our result also implies a O(n^(1/2 - alpha))-approximation, for some constant alpha>0. Using our algorithm for approximating the Euler genus as a subroutine, we obtain, in a unified fashion, algorithms with approximation ratios of the form poly(OPT, log(n)) for several related problems on bounded degree graphs. These include the problems of orientable genus, crossing number, and planar edge and vertex deletion problems. Our algorithm and proof of correctness for the crossing number problem is simpler compared to the long and difficult proof in the recent breakthrough by [Chuzhoy 2011], while essentially obtaining a qualitatively similar result. For planar edge and vertex deletion problems our results are the first to obtain a bound of form poly(OPT, log(n)). We also highlight some further applications of our results in the design of algorithms for graphs with small genus. Many such algorithms require that a drawing of the graph is given as part of the input. Our results imply that in several interesting cases, we can implement such algorithms even when the drawing is unknown.

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arXiv2013

Minimum d-dimensional arrangement with fixed points

In the Minimum $d$-Dimensional Arrangement Problem (d-dimAP) we are given a graph with edge weights, and the goal is to find a 1-1 map of the vertices into $\mathbb{Z}^d$ (for some fixed dimension $d\geq 1$) minimizing the total weighted stretch of the edges. This problem arises in VLSI placement and chip design. Motivated by these applications, we consider a generalization of d-dimAP, where the positions of some of the vertices (pins) is fixed and specified as part of the input. We are asked to extend this partial map to a map of all the vertices, again minimizing the weighted stretch of edges. This generalization, which we refer to as d-dimAP+, arises naturally in these application domains (since it can capture blocked-off parts of the board, or the requirement of power-carrying pins to be in certain locations, etc.). Perhaps surprisingly, very little is known about this problem from an approximation viewpoint. For dimension $d=2$, we obtain an $O(k^{1/2} \cdot \log n)$-approximation algorithm, based on a strengthening of the spreading-metric LP for 2-dimAP. The integrality gap for this LP is shown to be $Ω(k^{1/4})$. We also show that it is NP-hard to approximate 2-dimAP+ within a factor better than $Ω(k^{1/4-\eps})$. We also consider a (conceptually harder, but practically even more interesting) variant of 2-dimAP+, where the target space is the grid $\mathbb{Z}_{\sqrt{n}} \times \mathbb{Z}_{\sqrt{n}}$, instead of the entire integer lattice $\mathbb{Z}^2$. For this problem, we obtain a $O(k \cdot \log^2{n})$-approximation using the same LP relaxation. We complement this upper bound by showing an integrality gap of $Ω(k^{1/2})$, and an $Ω(k^{1/2-\eps})$-inapproximability result. Our results naturally extend to the case of arbitrary fixed target dimension $d\geq 1$.

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