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arXiv 2021年

作者： Bandyapadhyay, Sayan Fomin, Fedor V. Golovach, Petr A. Simonov, Kirill Department of Informatics University of Bergen Norway Algorithms and Complexity Group TU Wien Austria

We develop new algorithmic methods with provable guarantees for feature selection in regard to categorical data clustering. While feature selection is one of the most common approaches to reduce dimensionality in practice, most of the known feature selection methods are heuristics. We study the following mathematical model. We assume that there are some inadvertent (or undesirable) features of the input data that unnecessarily increase the cost of clustering. Consequently, we want to select a subset of the original features from the data such that there is a small-cost clustering on the selected features. More precisely, for given integers (the number of irrelevant features) and k (the number of clusters), budget B, and a set of n categorical data points (represented by m-dimensional vectors whose elements belong to a finite set of values Σ), we want to select m − relevant features such that the cost of any optimal k-clustering on these features does not exceed B. Here the cost of a cluster is the sum of Hamming distances (0-distances) between the selected features of the elements of the cluster and its center. The clustering cost is the total sum of the costs of the clusters. We use the framework of parameterized complexity to identify how the complexity of the problem depends on parameters k, B, and |Σ|. Our main result is an algorithm that solves the Feature Selection problem in time f(k, B, |Σ|) · mg(k,|Σ|) · n2 for some functions f and g. In other words, the problem is fixed-parameter tractable parameterized by B when |Σ| and k are constants. Our algorithm for Feature Selection is based on a solution to a more general problem, Constrained Clustering with Outliers. In this problem, we want to delete a certain number of outliers such that the remaining points could be clustered around centers satisfying specific constraints. One interesting fact about Constrained Clustering with Outliers is that besides Feature Selection, it encompasses many other fundamental pro

关键词： Clustering algorithms

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Bounding and Computing Obstacle Numbers of Graphs 30

Bounding and Computing Obstacle Numbers of Graphs

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30th Annual European Symposium on algorithms, ESA 2022

作者： Balko, Martin Chaplick, Steven Ganian, Robert Gupta, Siddharth Hoffmann, Michael Valtr, Pavel Wolff, Alexander Faculty of Mathematics and Physics Charles University Prague Czech Republic Maastricht University Netherlands Algorithms and Complexity Group TU Wien Austria Department of Computer Science University of Warwick Coventry United Kingdom Department of Computer Science ETH Zürich Switzerland Institut für Informatik Universität Würzburg Germany

ISBN: (纸本)9783959772471

An obstacle representation of a graph G consists of a set of pairwise disjoint simply-connected closed regions and a one-to-one mapping of the vertices of G to points such that two vertices are adjacent in G if and only if the line segment connecting the two corresponding points does not intersect any obstacle. The obstacle number of a graph is the smallest number of obstacles in an obstacle representation of the graph in the plane such that all obstacles are simple polygons. It is known that the obstacle number of each n-vertex graph is O(n log n) [Balko, Cibulka, and Valtr, 2018] and that there are n-vertex graphs whose obstacle number is Ω(n/(log log n)2) [Dujmovic and Morin, 2015]. We improve this lower bound to Ω(n/ log log n) for simple polygons and to Ω(n) for convex polygons. To obtain these stronger bounds, we improve known estimates on the number of n-vertex graphs with bounded obstacle number, solving a conjecture by Dujmovic and Morin. We also show that if the drawing of some n-vertex graph is given as part of the input, then for some drawings Ω(n2) obstacles are required to turn them into an obstacle representation of the graph. Our bounds are asymptotically tight in several instances. We complement these combinatorial bounds by two complexity results. First, we show that computing the obstacle number of a graph G is fixed-parameter tractable in the vertex cover number of G. Second, we show that, given a graph G and a simple polygon P, it is NP-hard to decide whether G admits an obstacle representation using P as the only obstacle. © 2022 Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. All rights reserved.

关键词： Graph theory

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Segment Intersection Representations, Level Planarity and Constrained Ordering Problems

arXiv

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arXiv 2025年

作者： Fink, Simon D. Pfretzschner, Matthias Stumpf, Peter Algorithms and Complexity Group Technische Universität Wien Austria Faculty of Computer Science and Mathematics University of Passau Germany Department of Theoretical Computer Science Faculty of Information Technology Czech Technical University Prague Czech Republic Department of Applied Mathematics Charles University Prague Czech Republic

In the Segment Intersection Graph Representation Problem, we want to represent the vertices of a graph as straight line segments in the plane such that two segments cross if and only if there is an edge between the corresponding vertices. This problem is NP-hard (even ∃R-complete [19]) in the general case [13] and remains so if we restrict the segments to be axis-aligned, i.e., horizontal and vertical [12]. A long standing open question for the latter variant is its complexity when the order of segments along one axis (say the vertical order of horizontal segments) is already given [12,14]. We resolve this question by giving efficient solutions using two very different approaches that are interesting on their own. First, using a graph-drawing perspective, we relate the problem to a variant of the well-known Level Planarity problem, where vertices have to lie on preassigned horizontal levels. In our case, each level also carries consecutivity constraints on its vertices;this Level Planarity variant is known to have a quadratic solution. Second, we use an entirely combinatorial approach and show that both problems can equivalently be formulated as a linear ordering problem subject to certain consecutivity constraints. While the complexity of such problems varies greatly, we show that in this case the constraints are well-structured in a way that allows a direct quadratic solution. Thus, we obtain three different-but-equivalent perspectives on this problem: the initial geometric one, one from planar graph drawing and a purely combinatorial one. © 2025, CC BY.

关键词： Graph theory

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Space-Efficient Parameterized algorithms on Graphs of Low Shrubdepth

arXiv

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arXiv 2023年

作者： Bergougnoux, Benjamin Chekan, Vera Ganian, Robert Kanté, Mamadou Moustapha Mnich, Matthias Oum, Sang-Il Pilipczuk, Michal van Leeuwen, Erik Jan Institute of Informatics University of Warsaw Poland Humboldt-Universität zu Berlin Germany Algorithms and Complexity Group TU Wien Vienna Austria Université Clermont Auvergne Clermont Auvergne INP LIMOS CNRS Clermont-Ferrand France Hamburg University of Technology Institute for Algorithms and Complexity Hamburg Germany Daejeon Korea Republic of Department of Mathematical Sciences KAIST Daejeon Korea Republic of Dept. Information and Computing Sciences Utrecht University Netherlands

Dynamic programming on various graph decompositions is one of the most fundamental techniques used in parameterized complexity. Unfortunately, even if we consider concepts as simple as path or tree decompositions, such dynamic programming uses space that is exponential in the decomposition’s width, and there are good reasons to believe that this is necessary. However, it has been shown that in graphs of low treedepth it is possible to design algorithms which achieve polynomial space complexity without requiring worse time complexity than their counterparts working on tree decompositions of bounded width. Here, treedepth is a graph parameter that, intuitively speaking, takes into account both the depth and the width of a tree decomposition of the graph, rather than the width alone. Motivated by the above, we consider graphs that admit clique expressions with bounded depth and label count, or equivalently, graphs of low shrubdepth. Here, shrubdepth is a bounded-depth analogue of cliquewidth, in the same way as treedepth is a bounded-depth analogue of treewidth. We show that also in this setting, bounding the depth of the decomposition is a deciding factor for improving the space complexity. More precisely, we prove that on n-vertex graphs equipped with a tree-model (a decomposition notion underlying shrubdepth) of depth d and using k labels, Independent Set can be solved in time 2O(dk) · nO(1) using O(dk2 log n) space;Max Cut can be solved in time nO(dk) using O(dk log n) space;and Dominating Set can be solved in time 2O(dk) · nO(1) using nO(1) space via a randomized algorithm. We also establish a lower bound, conditional on a certain assumption about the complexity of Longest Common Subsequence, which shows that at least in the case of Independent Set the exponent of the parametric factor in the time complexity has to grow with d if one wishes to keep the space complexity polynomial. Copyright © 2023, The Authors. All rights reserved.

关键词： Dynamic programming

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On Families of Planar DAGs with Constant Stack Number

arXiv

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arXiv 2021年

作者： Nöllenburg, Martin Pupyrev, Sergey Algorithms and Complexity Group TU Wien Austria Meta Menlo ParkCA United States

A k-stack layout (or k-page book embedding) of a graph consists of a total order of the vertices, and a partition of the edges into k sets of non-crossing edges with respect to the vertex order. The stack number of a graph is the minimum k such that it admits a k-stack layout. In this paper we study a long-standing problem regarding the stack number of planar directed acyclic graphs (DAGs), for which the vertex order has to respect the orientation of the edges. We investigate upper and lower bounds on the stack number of several families of planar graphs: We improve the constant upper bounds on the stack number of single-source and monotone outerplanar DAGs and of outerpath DAGs, and improve the constant upper bound for upward planar 3-trees. Further, we provide computer-aided lower bounds for upward (outer-) planar DAGs. Copyright © 2021, The Authors. All rights reserved.

关键词： Directed graphs

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How to find a good explanation for clustering?

arXiv

引用

arXiv 2021年

作者： Bandyapadhyay, Sayan Fomin, Fedor V. Golovach, Petr A. Lochet, William Purohit, Nidhi Simonov, Kirill Department of Informatics University of Bergen Norway Algorithms and Complexity Group TU Wien Vienna Austria

k-means and k-median clustering are powerful unsupervised machine learning techniques. However, due to complicated dependences on all the features, it is challenging to interpret the resulting cluster assignments. Moshkovitz, Dasgupta, Rashtchian, and Frost [ICML 2020] proposed an elegant model of explainable k-means and k-median clustering. In this model, a decision tree with k leaves provides a straightforward characterization of the data set into clusters. We study two natural algorithmic questions about explainable clustering. (1) For a given clustering, how to find the "best explanation" by using a decision tree with k leaves? (2) For a given set of points, how to find a decision tree with k leaves minimizing the k-means/median objective of the resulting explainable clustering? To address the first question, we introduce a new model of explainable clustering. Our model, inspired by the notion of outliers in robust statistics, is the following. We are seeking a small number of points (outliers) whose removal makes the existing clustering well-explainable. For addressing the second question, we initiate the study of the model of Moshkovitz et al. from the perspective of multivariate complexity. Our rigorous algorithmic analysis sheds some light on the influence of parameters like the input size, dimension of the data, the number of outliers, the number of clusters, and the approximation ratio, on the computational complexity of explainable clustering. © 2021, CC BY.

关键词： Decision trees

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Selecting Matchings via Multiwinner Voting: How Structure Defeats a Large Candidate Space

arXiv

引用

arXiv 2021年

作者： Boehmer, Niclas Brill, Markus Schmidt-Kraepelin, Ulrike Research Group Algorithmics and Computational Complexity TU Berlin Germany Research Group Efficient Algorithms TU Berlin Germany

Given a set of agents with approval preferences over each other, we study the task of finding k matchings fairly representing everyone’s preferences. We model the problem as an approval-based multiwinner election where the set of candidates consists of matchings of the agents, and agents’ preferences over each other are lifted to preferences over matchings. Due to the exponential number of candidates in such elections, standard algorithms for classical sequential voting rules (such as those proposed by Thiele and Phragmén) are rendered inefficient. We show that the computational tractability of these rules can be regained by exploiting the structure of the approval preferences. Moreover, we establish algorithmic results and axiomatic guarantees that go beyond those obtainable in the general multiwinner setting. Assuming that approvals are symmetric, we show that Proportional Approval Voting (PAV), a well-established but computationally intractable voting rule, becomes polynomial-time computable, and its sequential variant (seq-PAV), which does not provide any proportionality guarantees in general, fulfills a rather strong guarantee known as extended justified representation. Some of our algorithmic results extend to other types of compactly representable elections with an exponential candidate space. Copyright © 2021, The Authors. All rights reserved.

关键词： Polynomial approximation

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Minimum Link Fencing

arXiv

引用

arXiv 2022年

作者： Bhore, Sujoy Klute, Fabian Löffler, Maarten Nöllenburg, Martin Terziadis, Soeren Villedieu, Anaïs Department of Computer Science & Engineering Indian Institute of Technology Bombay India Department of Information and Computing Sciences Utrecht University Netherlands Algorithms and Complexity Group TU Wien Austria

We study a variant of the geometric multicut problem, where we are given a set P of colored and pairwise interior-disjoint polygons in the plane. The objective is to compute a set of simple closed polygon boundaries (fences) that separate the polygons in such a way that any two polygons that are enclosed by the same fence have the same color, and the total number of links of all fences is minimized. We call this the minimum link fencing (MLF) problem and consider the natural case of bounded minimum link fencing (BMLF), where P contains a polygon Q that is unbounded in all directions and can be seen as an outer polygon. We show that BMLF is NP-hard in general and that it is XP-time solvable when each fence contains at most two polygons and the number of segments per fence is the parameter. Finally, we present an O(n log n)-time algorithm for the case that the convex hull of P \ {Q} does not intersect Q. © 2022, CC BY.

关键词： Computational geometry

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On Finding Hamiltonian Cycles in Barnette Graphs

arXiv

引用

arXiv 2022年

作者： Gh, Behrooz Bagheri Feder, Tomas Fleischner, Herbert Subi, Carlos Algorithms and Complexity Group Vienna University of Technology Favoritenstrasse 9-11 Vienna1040 Austria Computer Science Department Stanford University StanfordCA94305 United States

In this paper we deal with hamiltonicity in planar cubic graphs G having a facial 2−factor Q via (quasi) spanning trees of faces in G/Q and study the algorithmic complexity of finding such (quasi) spanning trees of faces. Moreover, we show that if Barnette’s Conjecture is false, then hamiltonicity in 3−connected planar cubic bipartite graphs is an NP-complete problem. © 2022, CC0.

关键词： Computational complexity

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Extending partial 1-planar drawings 47

Extending partial 1-planar drawings

引用

47th International Colloquium on Automata, Languages, and Programming, ICALP 2020

作者： Eiben, Eduard Ganian, Robert Hamm, Thekla Klute, Fabian Nöllenburg, Martin Department of Computer Science Royal Holloway University of London Egham United Kingdom Algorithms and Complexity Group TU Wien Austria

ISBN: (纸本)9783959771382

Algorithmic extension problems of partial graph representations such as planar graph drawings or geometric intersection representations are of growing interest in topological graph theory and graph drawing. In such an extension problem, we are given a tuple (G, H, H) consisting of a graph G, a connected subgraph H of G and a drawing H of H, and the task is to extend H into a drawing of G while maintaining some desired property of the drawing, such as planarity. In this paper we study the problem of extending partial 1-planar drawings, which are drawings in the plane that allow each edge to have at most one crossing. In addition we consider the subclass of IC-planar drawings, which are 1-planar drawings with independent crossings. Recognizing 1-planar graphs as well as IC-planar graphs is NP-complete and the NP-completeness easily carries over to the extension problem. Therefore, our focus lies on establishing the tractability of such extension problems in a weaker sense than polynomial-time tractability. Here, we show that both problems are fixed-parameter tractable when parameterized by the number of edges missing from H, i.e., the edge deletion distance between H and G. The second part of the paper then turns to a more powerful parameterization which is based on measuring the vertex+edge deletion distance between the partial and complete drawing, i.e., the minimum number of vertices and edges that need to be deleted to obtain H from G. © Eduard Eiben, Robert Ganian, Thekla Hamm, Fabian Klute, and Martin Nöllenburg;licensed under Creative Commons License CC-BY 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020).

关键词： Parameterization

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