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Weighted Model Counting with Twin-Width 25

Weighted Model Counting with Twin-Width

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25th International Conference on Theory and Applications of Satisfiability Testing, SAT 2022

作者： Ganian, Robert Pokrývka, Filip Schidler, André Simonov, Kirill Szeider, Stefan Algorithms and Complexity Group TU Wien Austria Masaryk University Brno Czech Republic

ISBN: (纸本)9783959772426

Bonnet et al. (FOCS 2020) introduced the graph invariant twin-width and showed that many NP-hard problems are tractable for graphs of bounded twin-width, generalizing similar results for other width measures, including treewidth and clique-width. In this paper, we investigate the use of twin-width for solving the propositional satisfiability problem (SAT) and propositional model counting. We particularly focus on Bounded-ones Weighted Model Counting (BWMC), which takes as input a CNF formula F along with a bound k and asks for the weighted sum of all models with at most k positive literals. BWMC generalizes not only SAT but also (weighted) model counting. We develop the notion of "signed" twin-width of CNF formulas and establish that BWMC is fixed-parameter tractable when parameterized by the certified signed twin-width of F plus k. We show that this result is tight: it is neither possible to drop the bound k nor use the vanilla twin-width instead if one wishes to retain fixed-parameter tractability, even for the easier problem SAT. Our theoretical results are complemented with an empirical evaluation and comparison of signed twin-width on various classes of CNF formulas. © Robert Ganian, Filip Pokrývka, André Schidler, Kirill Simonov, and Stefan Szeider.

关键词： Computational complexity

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GdMetriX - A NetworkX Extension For Graph Drawing Metrics 32

GdMetriX - A NetworkX Extension For Graph Drawing Metrics

引用

32nd International Symposium on Graph Drawing and Network Visualization, GD 2024

作者： Nöllenburg, Martin Röder, Sebastian Wallinger, Markus Algorithms and Complexity Group TU Wien Austria TU Wien Austria Chair for Efficient Algorithms Technical University of Munich Germany

ISBN: (纸本)9783959773430

networkX is a well-established Python library for network analysis. With gdMetriX, we aim to extend the functionality of networkX and provide common quality metrics used in the field of graph drawing, such as the number of crossings or the angular resolution. In addition, the package provides easy-to-use access to the graph datasets provided by the 'Graph Layout Benchmark Datasets' project from the Northeastern University Visualization Lab. © Martin Nöllenburg, Sebastian Röder, and Markus Wallinger.

关键词： Data visualization

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Parameterized complexity of Feature Selection for Categorical Data Clustering 46

Parameterized Complexity of Feature Selection for Categorica...

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46th International Symposium on Mathematical Foundations of Computer Science, MFCS 2021

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

ISBN: (纸本)9783959772013

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 problems r

关键词： Budget control

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Fixed-parameter evolutionary algorithms and the vertex cover problem

Fixed-parameter evolutionary algorithms and the vertex cover...

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11th Annual Genetic and Evolutionary Computation Conference, GECCO-2009

作者： Kratsch, Stefan Neumann, Frank Algorithms and Complexity Group Max-Planck-Institut für Informatik Saarbrücken Germany

ISBN: (纸本)9781605583259

In this paper, we consider multi-objective evolutionary algorithms for the Vertex Cover problem in the context of parameterized complexity. We relate the runtime of our algorithms to the input size and the cost of a minimum solution and point out that the search process of evolutionary algorithms creates partial solutions that are similar to the effect of a kernelization (i.e. a special type of preprocessing from parameterized complexity). Based on this, we show that evolutionary algorithms solve the vertex cover problem efficiently if the size of a minimum vertex cover is not too large, i.e. the expected runtime is bounded by O(f(OPT)nc), where c is a constant and f a function that only depends on OPT. This shows that evolutionary algorithms are randomized fixed-parameter tractable algorithms for the vertex cover problem. Copyright 2009 ACM.

关键词： Multiobjective optimization

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A Tight Subexponential-Time Algorithm for Two-Page Book Embedding 51

A Tight Subexponential-Time Algorithm for Two-Page Book Embe...

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51st International Colloquium on Automata, Languages, and Programming, ICALP 2024

作者： Ganian, Robert Müller, Haiko Ordyniak, Sebastian Paesani, Giacomo Rychlicki, Mateusz Algorithms and Complexity Group TU Wien Austria School of Computing University of Leeds United Kingdom

ISBN: (纸本)9783959773225

A book embedding of a graph is a drawing that maps vertices onto a line and edges to simple pairwise non-crossing curves drawn into "pages", which are half-planes bounded by that line. Two-page book embeddings, i.e., book embeddings into 2 pages, are of special importance as they are both NP-hard to compute and have specific applications. We obtain a 2O(√n) algorithm for computing a book embedding of an n-vertex graph on two pages – a result which is asymptotically tight under the Exponential Time Hypothesis. As a key tool in our approach, we obtain a single-exponential fixed-parameter algorithm for the same problem when parameterized by the treewidth of the input graph. We conclude by establishing the fixed-parameter tractability of computing minimum-page book embeddings when parameterized by the feedback edge number, settling an open question arising from previous work on the problem. © Robert Ganian, Haiko Müller, Sebastian Ordyniak, Giacomo Paesani, and Mateusz Rychlicki.

关键词： Parameter estimation

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Lossy Kernelization of Same-Size Clustering 1

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17th International Computer Science Symposium in Russia, CSR 2022

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

ISBN: (数字)9783031095740

ISBN: (纸本)9783031095733

In this work, we study the k-median clustering problem with an additional equal-size constraint on the clusters, from the perspective of parameterized preprocessing. Our main result is the first lossy (2-approximate) polynomial kernel for this problem, parameterized by the cost of clustering. We complement this result by establishing lower bounds for the problem that eliminate the existences of an (exact) kernel of polynomial size and a PTAS. © 2022, Springer Nature Switzerland AG.

关键词： Parameterization

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Path-contractions, edge deletions and connectivity preservation 25

Path-contractions, edge deletions and connectivity preservat...

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25th European Symposium on algorithms, ESA 2017

作者： Gutin, Gregory Ramanujan, M.S. Reidl, Felix Wahlström, Magnus Algorithms and Complexity Group TU Wien Vienna Austria Royal Holloway University Egham United Kingdom

ISBN: (纸本)9783959770491

We study several problems related to graph modification problems under connectivity constraints from the perspective of parameterized complexity: (Weighted) Biconnectivity Deletion, where we are tasked with deleting k edges while preserving biconnectivity in an undirected graph, Vertexdeletion Preserving Strong Connectivity, where we want to maintain strong connectivity of a digraph while deleting exactly k vertices, and Path-contraction Preserving Strong Connectivity, in which the operation of path contraction on arcs is used instead. The parameterized tractability of this last problem was posed in [Bang-Jensen and Yeo, Discrete Applied Math 2008] as an open question and we answer it here in the negative: both variants of preserving strong connectivity are W[1]-hard. Preserving biconnectivity, on the other hand, turns out to be fixed parameter tractable (FPT) and we provide an FPT algorithm that solves Weighted Biconnectivity Deletion. Further, we show that the unweighted case even admits a randomized polynomial kernel. All our results provide further interesting data points for the systematic study of connectivitypreservation constraints in the parameterized setting.

关键词： Undirected graphs

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Parameterised Partially-Predrawn Crossing Number 38

Parameterised Partially-Predrawn Crossing Number

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38th International Symposium on Computational Geometry, SoCG 2022

作者： Hamm, Thekla Hlinený, Petr Algorithms and Complexity Group TU Wien Austria Faculty of Informatics Masaryk University Brno Czech Republic

ISBN: (纸本)9783959772273

Inspired by the increasingly popular research on extending partial graph drawings, we propose a new perspective on the traditional and arguably most important geometric graph parameter, the crossing number. Specifically, we define the partially predrawn crossing number to be the smallest number of crossings in any drawing of a graph, part of which is prescribed on the input (not counting the prescribed crossings). Our main result - an FPT-algorithm to compute the partially predrawn crossing number - combines advanced ideas from research on the classical crossing number and so called partial planarity in a very natural but intricate way. Not only do our techniques generalise the known FPT-algorithm by Grohe for computing the standard crossing number, they also allow us to substantially improve a number of recent parameterised results for various drawing extension problems. © Thekla Hamm and Petr Hlinn;licensed under Creative Commons License CC-BY 4.0

关键词： Parameterization

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Boundary Labeling in a Circular Orbit 32

Boundary Labeling in a Circular Orbit

引用

32nd International Symposium on Graph Drawing and Network Visualization, GD 2024

作者： Bonerath, Annika Nöllenburg, Martin Terziadis, Soeren Wallinger, Markus Wulms, Jules University of Bonn Germany Algorithms and Complexity Group TU Wien Austria TU Eindhoven Netherlands Chair for Efficient Algorithms TU Munich Germany

ISBN: (纸本)9783959773430

Boundary labeling is a well-known method for displaying short textual labels for a set of point features in a figure alongside the boundary of that figure. Labels and their corresponding points are connected via crossing-free leaders. We propose orbital boundary labeling as a new variant of the problem, in which (i) the figure is enclosed by a circular contour and (ii) the labels are placed as disjoint circular arcs in an annulus-shaped orbit around the contour. The algorithmic objective is to compute an orbital boundary labeling with the minimum total leader length. We identify several parameters that define the corresponding problem space: two leader types (straight or orbital-radial), label size and order, presence of candidate label positions, and constraints on where a leader attaches to its label. Our results provide polynomial-time algorithms for many variants and NP-hardness for others, using a variety of geometric and combinatorial insights. © Annika Bonerath, Martin Nöllenburg, Soeren Terziadis, Markus Wallinger, and Jules Wulms.

关键词： Polynomial approximation

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Discovering Archipelagos of Tractability for Constraint Satisfaction and Counting 16

Discovering Archipelagos of Tractability for Constraint Sati...

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Annual ACM-Society for Industrial and Applied Mathmatics Symposium on Discrete algorithms

作者： Robert Ganian M. S. Ramanujan Stefan Szeider Algorithms and Complexity Group

ISBN: (纸本)9781510819672

The Constraint Satisfaction Problem (CSP) is a central and generic computational problem which provides a common framework for many theoretical and practical applications. A central line of research is concerned with the identification of classes of instances for which CSP can be solved in polynomial time; such classes are often called "islands of tractability." A prominent way of defining islands of tractability for CSP is to restrict the relations that may occur in the constraints to a fixed set, called a constraint language, whereas a constraint language is conservative if it contains all unary relations. Schaefer's famous Dichotomy Theorem (STOC 1978) identifies all islands of tractability in terms of tractable constraint languages over a Boolean domain of values. Since then many extensions and generalizations of this result have been obtained. Recently, Bulatov (TOCL 2011, JACM 2013) gave a full characterization of all islands of tractability for CSP and the counting version #CSP that are defined in terms of conservative constraint languages. This paper addresses the general limit of the mentioned tractability results for CSP and #CSP, that they only apply to instances where all constraints belong to a single tractable language (in general, the union of two tractable languages isn't tractable). We show that we can overcome this limitation as long as we keep some control of how constraints over the various considered tractable languages interact with each other. For this purpose we utilize the notion of a strong backdoor of a CSP instance, as introduced by Williams et al. (IJCAI 2003), which is a set of variables that when instantiated moves the instance to an island of tractability, i.e., to a tractable class of instances. We consider strong backdoors into scattered classes, consisting of CSP instances where each connected component belongs entirely to some class from a list of tractable classes. Figuratively speaking, a scattered class constitutes an archipela

关键词： chip scale packaging communicating sequential processes phosphocellulose Language islands constraint satisfaction problems archipelagoes Constraint Polynomial time

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