arXiv subjects

Joachim Spoerhase

Publications and source records attributed to Joachim Spoerhase.

32 records · Page 2Linked to original sources

arXiv2018

Approximating Node-Weighted k-MST on Planar Graphs

We study the problem of finding a minimum weight connected subgraph spanning at least $k$ vertices on planar, node-weighted graphs. We give a $(4+\eps)$-approximation algorithm for this problem. We achieve this by utilizing the recent LMP primal-dual $3$-approximation for the node-weighted prize-collecting Steiner tree problem by Byrka et al (SWAT'16) and adopting an approach by Chudak et al. (Math.\ Prog.\ '04) regarding Lagrangian relaxation for the edge-weighted variant. In particular, we improve the procedure of picking additional vertices (tree merging procedure) given by Sadeghian (2013) by taking a constant number of recursive steps and utilizing the limited guessing procedure of Arora and Karakostas (Math.\ Prog.\ '06). More generally, our approach readily gives a $(\nicefrac{4}{3}\cdot r+\eps)$-approximation on any graph class where the algorithm of Byrka et al.\ for the prize-collecting version gives an $r$-approximation. We argue that this can be interpreted as a generalization of an analogous result by Könemann et al. (Algorithmica~'11) for partial cover problems. Together with a lower bound construction by Mestre (STACS'08) for partial cover this implies that our bound is essentially best possible among algorithms that utilize an LMP algorithm for the Lagrangian relaxation as a black box. In addition to that, we argue by a more involved lower bound construction that even using the LMP algorithm by Byrka et al.\ in a \emph{non-black-box} fashion could not beat the factor $\nicefrac{4}{3}\cdot r$ when the tree merging step relies only on the solutions output by the LMP algorithm.

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arXiv2018

Constant-Factor Approximation for Ordered k-Median

We study the Ordered k-Median problem, in which the solution is evaluated by first sorting the client connection costs and then multiplying them with a predefined non-increasing weight vector (higher connection costs are taken with larger weights). Since the 1990s, this problem has been studied extensively in the discrete optimization and operations research communities and has emerged as a framework unifying many fundamental clustering and location problems such as k-Median and k-Center. This generality, however, renders the problem intriguing from the algorithmic perspective and obtaining non-trivial approximation algorithms was an open problem even for simple topologies such as trees. Recently, Aouad and Segev were able to obtain an O(log n) approximation algorithm for Ordered k-Median using a sophisticated local-search approach and the concept of surrogate models thereby extending the result by Tamir (2001) for the case of a rectangular weight vector, also known as k-Facility p-Centrum. In this paper, we provide an LP-rounding constant-factor approximation algorithm for the Ordered k-Median problem. We first provide a new analysis of the rounding process by Charikar and Li (2012) for k-Median, when applied to a fractional solution obtained from solving an LP relaxation over a non-metric, truncated cost vector, resulting in an elegant 15-approximation for the rectangular case. Then, we show that a simple weight bucketing can be applied to the general case resulting in O(log n) rectangles and hence in a constant-factor approximation in quasi-polynomial time. Finally, we show that also the clever distance bucketing by Aouad and Segev can be combined with the objective-oblivious version of our LP-rounding for the rectangular case, and that it results in a true, polynomial time, constant-factor approximation algorithm.

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arXiv2017

Bi-Factor Approximation Algorithms for Hard Capacitated $k$-Median Problems

The $k$-Facility Location problem is a generalization of the classical problems $k$-Median and Facility Location. The goal is to select a subset of at most $k$ facilities that minimizes the total cost of opened facilities and established connections between clients and opened facilities. We consider the hard-capacitated version of the problem, where a single facility may only serve a limited number of clients and creating multiple copies of a facility is not allowed. We construct approximation algorithms slightly violating the capacities based on rounding a fractional solution to the standard LP. It is well known that the standard LP (even in the case of uniform capacities and opening costs) has unbounded integrality gap if we only allow violating capacities by a factor smaller than $2$, or if we only allow violating the number of facilities by a factor smaller than $2$. In this paper, we present the first constant-factor approximation algorithms for the hard-capacitated variants of the problem. For uniform capacities, we obtain a $(2+\varepsilon)$-capacity violating algorithm with approximation ratio $O(1/\varepsilon^2)$; our result has not yet been improved. Then, for non-uniform capacities, we consider the case of $k$-Median, which is equivalent to $k$-Facility Location with uniform opening cost of the facilities. Here, we obtain a $(3+\varepsilon)$-capacity violating algorithm with approximation ratio $O(1/\varepsilon)$.

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arXiv2017

Approximation Schemes for Geometric Coverage Problems

In their seminal work, Mustafa and Ray (2009) showed that a wide class of geometric set cover (SC) problems admit a PTAS via local search -- this is one of the most general approaches known for such problems. Their result applies if a naturally defined "exchange graph" for two feasible solutions is planar and is based on subdividing this graph via a planar separator theorem due to Frederickson (1987). Obtaining similar results for the related maximum k-coverage problem (MC) seems non-trivial due to the hard cardinality constraint. In fact, while Badanidiyuru, Kleinberg, and Lee (2012) have shown (via a different analysis) that local search yields a PTAS for two-dimensional real halfspaces, they only conjectured that the same holds true for dimension three. Interestingly, at this point it was already known that local search provides a PTAS for the corresponding set cover case and this followed directly from the approach of Mustafa and Ray. In this work we provide a way to address the above-mentioned issue. First, we propose a color-balanced version of the planar separator theorem. The resulting subdivision approximates locally in each part the global distribution of the colors. Second, we show how this roughly balanced subdivision can be employed in a more careful analysis to strictly obey the hard cardinality constraint. More specifically, we obtain a PTAS for any "planarizable" instance of MC and thus essentially for all cases where the corresponding SC instance can be tackled via the approach of Mustafa and Ray. As a corollary, we confirm the conjecture of Badanidiyuru, Kleinberg, and Lee regarding real half spaces in dimension three. We feel that our ideas could also be helpful in other geometric settings involving a cardinality constraint.

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arXiv2016

Colored Non-Crossing Euclidean Steiner Forest

Given a set of $k$-colored points in the plane, we consider the problem of finding $k$ trees such that each tree connects all points of one color class, no two trees cross, and the total edge length of the trees is minimized. For $k=1$, this is the well-known Euclidean Steiner tree problem. For general $k$, a $kρ$-approximation algorithm is known, where $ρ\le 1.21$ is the Steiner ratio. We present a PTAS for $k=2$, a $(5/3+\varepsilon)$-approximation algorithm for $k=3$, and two approximation algorithms for general~$k$, with ratios $O(\sqrt n \log k)$ and $k+\varepsilon$.

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arXiv2016

New Algorithms for Maximum Disjoint Paths Based on Tree-Likeness

We study the classical NP-hard problems of finding maximum-size subsets from given sets of $k$ terminal pairs that can be routed via edge-disjoint paths (MaxEDP) or node-disjoint paths (MaxNDP) in a given graph. The approximability of MaxEDP/NDP is currently not well understood; the best known lower bound is $Ω(\log^{1/2-ε}{n})$, assuming NP$~\not\subseteq~$ZPTIME$(n^{\mathrm{poly}\log n})$. This constitutes a significant gap to the best known approximation upper bound of $O(\sqrt{n})$ due to Chekuri et al. (2006) and closing this gap is currently one of the big open problems in approximation algorithms. In their seminal paper, Raghavan and Thompson (Combinatorica, 1987) introduce the technique of randomized rounding for LPs; their technique gives an $O(1)$-approximation when edges (or nodes) may be used by $O(\frac{\log n}{\log\log n})$ paths. In this paper, we strengthen the above fundamental results. We provide new bounds formulated in terms of the feedback vertex set number $r$ of a graph, which measures its vertex deletion distance to a forest. In particular, we obtain the following. * For MaxEDP, we give an $O(\sqrt{r}\cdot \log^{1.5}{kr})$-approximation algorithm. As $r\leq n$, up to logarithmic factors, our result strengthens the best known ratio $O(\sqrt{n})$ due to Chekuri et al. * Further, we show how to route $Ω(\mathrm{OPT})$ pairs with congestion $O(\frac{\log{kr}}{\log\log{kr}})$, strengthening the bound obtained by the classic approach of Raghavan and Thompson. * For MaxNDP, we give an algorithm that gives the optimal answer in time $(k+r)^{O(r)}\cdot n$. If $r$ is at most triple-exponential in $k$, this improves the best known algorithm for MaxNDP with parameter $k$, by Kawarabayashi and Wollan (STOC 2010). We complement these positive results by proving that MaxEDP is NP-hard even for $r=1$, and MaxNDP is W$[1]$-hard for parameter $r$.

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arXiv2016

On Monotone Drawings of Trees

A crossing-free straight-line drawing of a graph is monotone if there is a monotone path between any pair of vertices with respect to some direction. We show how to construct a monotone drawing of a tree with $n$ vertices on an $O(n^{1.5}) \times O(n^{1.5})$ grid whose angles are close to the best possible angular resolution. Our drawings are convex, that is, if every edge to a leaf is substituted by a ray, the (unbounded) faces form convex regions. It is known that convex drawings are monotone and, in the case of trees, also crossing-free. A monotone drawing is strongly monotone if, for every pair of vertices, the direction that witnesses the monotonicity comes from the vector that connects the two vertices. We show that every tree admits a strongly monotone drawing. For biconnected outerplanar graphs, this is easy to see. On the other hand, we present a simply-connected graph that does not have a strongly monotone drawing in any embedding.

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arXiv2014

Improved Approximation Algorithms for Box Contact Representations

We study the following geometric representation problem: Given a graph whose vertices correspond to axis-aligned rectangles with fixed dimensions, arrange the rectangles without overlaps in the plane such that two rectangles touch if the graph contains an edge between them. This problem is called \textsc{Contact Representation of Word Networks} (\textsc{Crown}) since it formalizes the geometric problem behind drawing word clouds in which semantically related words are close to each other. \textsc{Crown} is known to be NP-hard, and there are approximation algorithms for certain graph classes for the optimization version, \textsc{Max-Crown}, in which realizing each desired adjacency yields a certain profit. We present the first $O(1)$-approximation algorithm for the general case, when the input is a complete weighted graph, and for the bipartite case. Since the subgraph of realized adjacencies is necessarily planar, we also consider several planar graph classes (namely stars, trees, outerplanar, and planar graphs), improving upon the known results. For some graph classes, we also describe improvements in the unweighted case, where each adjacency yields the same profit. Finally, we show that the problem is APX-hard on bipartite graphs of bounded maximum degree.

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arXiv2014

Network Design Problems with Bounded Distances via Shallow-Light Steiner Trees

In a directed graph $G$ with non-correlated edge lengths and costs, the \emph{network design problem with bounded distances} asks for a cost-minimal spanning subgraph subject to a length bound for all node pairs. We give a bi-criteria $(2+\varepsilon,O(n^{0.5+\varepsilon}))$-approximation for this problem. This improves on the currently best known linear approximation bound, at the cost of violating the distance bound by a factor of at most~$2+\varepsilon$. In the course of proving this result, the related problem of \emph{directed shallow-light Steiner trees} arises as a subproblem. In the context of directed graphs, approximations to this problem have been elusive. We present the first non-trivial result by proposing a $(1+\varepsilon,O(|R|^{\varepsilon}))$-ap\-proxi\-ma\-tion, where $R$ are the terminals. Finally, we show how to apply our results to obtain an $(α+\varepsilon,O(n^{0.5+\varepsilon}))$-approximation for \emph{light-weight directed $α$-spanners}. For this, no non-trivial approximation algorithm has been known before. All running times depends on $n$ and $\varepsilon$ and are polynomial in $n$ for any fixed $\varepsilon>0$.

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arXiv2012

Approximating Minimum Manhattan Networks in Higher Dimensions

We study the minimum Manhattan network problem, which is defined as follows. Given a set of points called \emph{terminals} in $\R^d$, find a minimum-length network such that each pair of terminals is connected by a set of axis-parallel line segments whose total length is equal to the pair's Manhattan (that is, $L_1$-) distance. The problem is NP-hard in 2D and there is no PTAS for 3D (unless ${\cal P}={\cal NP}$). Approximation algorithms are known for 2D, but not for 3D. We present, for any fixed dimension $d$ and any $\eps>0$, an $O(n^\eps)$-approximation algorithm. For 3D, we also give a $4(k-1)$-approximation algorithm for the case that the terminals are contained in the union of $k \ge 2$ parallel planes.

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arXiv2012

Polylogarithmic Approximation for Generalized Minimum Manhattan Networks

Given a set of $n$ terminals, which are points in $d$-dimensional Euclidean space, the minimum Manhattan network problem (MMN) asks for a minimum-length rectilinear network that connects each pair of terminals by a Manhattan path, that is, a path consisting of axis-parallel segments whose total length equals the pair's Manhattan distance. Even for $d=2$, the problem is NP-hard, but constant-factor approximations are known. For $d \ge 3$, the problem is APX-hard; it is known to admit, for any $\eps > 0$, an $O(n^\eps)$-approximation. In the generalized minimum Manhattan network problem (GMMN), we are given a set $R$ of $n$ terminal pairs, and the goal is to find a minimum-length rectilinear network such that each pair in $R$ is connected by a Manhattan path. GMMN is a generalization of both MMN and the well-known rectilinear Steiner arborescence problem (RSA). So far, only special cases of GMMN have been considered. We present an $O(\log^{d+1} n)$-approximation algorithm for GMMN (and, hence, MMN) in $d \ge 2$ dimensions and an $O(\log n)$-approximation algorithm for 2D. We show that an existing $O(\log n)$-approximation algorithm for RSA in 2D generalizes easily to $d>2$ dimensions.

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arXiv2011

Drawing Graphs with Vertices at Specified Positions and Crossings at Large Angles

Point-set embeddings and large-angle crossings are two areas of graph drawing that independently have received a lot of attention in the past few years. In this paper, we consider problems in the intersection of these two areas. Given the point-set-embedding scenario, we are interested in how much we gain in terms of computational complexity, curve complexity, and generality if we allow large-angle crossings as compared to the planar case. We investigate two drawing styles where only bends or both bends and edges must be drawn on an underlying grid. We present various results for drawings with one, two, and three bends per edge.

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arXiv2010

Maximum Betweenness Centrality: Approximability and Tractable Cases

The Maximum Betweenness Centrality problem (MBC) can be defined as follows. Given a graph find a $k$-element node set $C$ that maximizes the probability of detecting communication between a pair of nodes $s$ and $t$ chosen uniformly at random. It is assumed that the communication between $s$ and $t$ is realized along a shortest $s$--$t$ path which is, again, selected uniformly at random. The communication is detected if the communication path contains a node of $C$. Recently, Dolev et al. (2009) showed that MBC is NP-hard and gave a $(1-1/e)$-approximation using a greedy approach. We provide a reduction of MBC to Maximum Coverage that simplifies the analysis of the algorithm of Dolev et al. considerably. Our reduction allows us to obtain a new algorithm with the same approximation ratio for a (generalized) budgeted version of MBC. We provide tight examples showing that the analyses of both algorithms are best possible. Moreover, we prove that MBC is APX-complete and provide an exact polynomial-time algorithm for MBC on tree graphs.

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arXiv2010

An Optimal Algorithm for the Indirect Covering Subtree Problem

We consider the indirect covering subtree problem (Kim et al., 1996). The input is an edge weighted tree graph along with customers located at the nodes. Each customer is associated with a radius and a penalty. The goal is to locate a tree-shaped facility such that the sum of setup and penalty cost is minimized. The setup cost equals the sum of edge lengths taken by the facility and the penalty cost is the sum of penalties of all customers whose distance to the facility exceeds their radius. The indirect covering subtree problem generalizes the single maximum coverage location problem on trees where the facility is a node rather than a subtree. Indirect covering subtree can be solved in $O(n\log^2 n)$ time (Kim et al., 1996). A slightly faster algorithm for single maximum coverage location with a running time of $O(n\log^2n/\log\log n)$ has been provided (Spoerhase and Wirth, 2009). We achieve time $O(n\log n)$ for indirect covering subtree thereby providing the fastest known algorithm for both problems. Our result implies also faster algorithms for competitive location problems such as $(1,X)$-medianoid and $(1,p)$-centroid on trees. We complement our result by a lower bound of $Ω(n\log n)$ for single maximum coverage location and $(1,X)$-medianoid on a real-number RAM model showing that our algorithm is optimal in running time.

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