Covering Graphs with Few Complete Bipartite Subgraphs *
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Nearly Tight Approximability Results for Minimum Biclique Cover and Partition
Nearly Tight Approximability Results for Minimum Biclique Cover and Partition Parinya Chalermsook1, Sandy Heydrich1;3, Eugenia Holm2, and Andreas Karrenbauer1∗ 1 Max Planck Institute for Informatics, Saarbr¨ucken, Germany 2 Dept. for Computer and Information Science, University of Konstanz, Germany 3 Saarbr¨ucken Graduate School of Computer Science {andreas.karrenbauer,parinya.chalermsook,sandy.heydrich}@mpi-inf.mpg.de [email protected] Abstract. In this paper, we consider the minimum biclique cover and minimum biclique partition problems on bipartite graphs. In the min- imum biclique cover problem, we are given an input bipartite graph G = (V; E), and our goal is to compute the minimum number of complete bipartite subgraphs that cover all edges of G. This problem, besides its correspondence to a well-studied notion of bipartite dimension in graph theory, has applications in many other research areas such as artificial intelligence, computer security, automata theory, and biology. Since it is NP-hard, past research has focused on approximation algorithms, fixed parameter tractability, and special graph classes that admit polynomial time exact algorithms. For the minimum biclique partition problem, we are interested in a biclique cover that covers each edge exactly once. We revisit the problems from approximation algorithms' perspectives and give nearly tight lower and upper bound results. We first show that both problems are NP-hard to approximate to within a factor of n1−" (where n is the number of vertices in the input graph). Using a stronger complexity assumption, the hardness becomes Ω~(n), where Ω~(·) hides lower order terms. Then we show that approximation factors of the form n=(log n)γ for some γ > 0 can be obtained. -
On Covering Numbers of Different Kinds
On Covering Numbers of Different Kinds Bachelor Thesis of Peter Stumpf At the Department of Informatics Institute of Theoretical Computer Science Reviewers: Prof. Dr. Dorothea Wagner Prof. Maria Axenovich Ph.D. Advisors: Torsten Ueckerdt Thomas Blaesius Time Period: 17.4.2015 – 16.8.2015 KIT – University of the State of Baden-Wuerttemberg and National Laboratory of the Helmholtz Association www.kit.edu Statement of Authorship I hereby declare that this document has been composed by myself and describes my own work, unless otherwise acknowledged in the text. Karlsruhe, 11th May 2015 iii Abstract A covering number measures how “difficult” it is to cover all edges of a host graph with guest graphs of a given guest class. E.g., the global covering number, which has received the most attention, is the smallest number k such that the host graph is the union of k guest graphs from the guest class. In this thesis, we consider the global, the local and the folded covering number and compare them for same pairs of host and guest classes. We investigate by how much global, local and folded covering number can differ. We give an example of a hereditary union-closed guest class where the folded covering number is at most 2, whereas the local covering number can be arbitrarily large. In contrast we show that within every host class the global covering number with regards to a topological minor-closed union-closed guest class is bounded if and only if its corresponding folded covering number is bounded. In the context of computational complexity, we construct a host class within which computing the local covering number with regards to interval graphs is NP-hard, whereas computing the global covering number is possible in constant time. -
On the Fine-Grained Complexity of Rainbow Coloring∗
On the Fine-Grained Complexity of Rainbow Coloring∗ Łukasz Kowalik1, Juho Lauri2, and Arkadiusz Socała3 1 University of Warsaw, Warsaw, Poland [email protected] 2 Tampere University of Technology, Tampere, Finland [email protected] 3 University of Warsaw, Warsaw, Poland [email protected] Abstract The Rainbow k-Coloring problem asks whether the edges of a given graph can be colored in k colors so that every pair of vertices is connected by a rainbow path, i.e., a path with all edges of different colors. Our main result states that for any k ≥ 2, there is no algorithm for 3/2 Rainbow k-Coloring running in time 2o(n ), unless ETH fails. Motivated by this negative result we consider two parameterized variants of the problem. In the Subset Rainbow k- Coloring problem, introduced by Chakraborty et al. [STACS 2009, J. Comb. Opt. 2009], we are additionally given a set S of pairs of vertices and we ask if there is a coloring in which all the pairs in S are connected by rainbow paths. We show that Subset Rainbow k-Coloring is FPT when parameterized by |S|. We also study Maximum Rainbow k-Coloring problem, where we are additionally given an integer q and we ask if there is a coloring in which at least q anti-edges are connected by rainbow paths. We show that the problem is FPT when parameterized by q and has a kernel of size O(q) for every k ≥ 2, extending the result of Ananth et al. [FSTTCS 2011]. We believe that our techniques used for the lower bounds may shed some light on the complexity of the classical Edge Coloring problem, where it is a major open question if a 2O(n)-time algorithm exists. -
On Different Graph Covering Numbers Structural, Extremal and Algorithmic Results
More on Different Graph Covering Numbers Structural, Extremal and Algorithmic Results Master Thesis of Peter Stumpf At the Department of Informatics Institute of Theoretical Computer Science Reviewers: Prof. Dr. Dorothea Wagner Prof. Dr. Peter Sanders Prof. Maria Axenovich Ph.D. Advisors: Torsten Ueckerdt Marcel Radermacher Time Period: 23.02.2017 – 22.08.2017 KIT – University of the State of Baden-Wuerttemberg and National Laboratory of the Helmholtz Association www.kit.edu Statement of Authorship I hereby declare that this document has been composed by myself and describes my own work, unless otherwise acknowledged in the text. Karlsruhe, 22nd August 2017 iii Abstract A covering number measures how “difficult” it is to cover all edges of a host graph with guest graphs of a given guest class. E.g., the global covering number, which has received the most attention, is the smallest number k such that the host graph is the union of k guest graphs from the guest class. In this thesis, we consider the recent framework of the global, the local and the folded covering number (Knauer and Ueckerdt, Discrete Mathematics 339 (2016)). The local covering number relaxes the global covering number by counting the guest graphs only locally at every vertex. And the folded covering number relaxes the local covering number even further, by allowing several vertices in the same guest graph to be identified, at the expense of counting these with multiplicities. More precisely, we consider a cover of a host graph H with regards to a guest class G to be a finite set of guest graphs S = {G1,...,Gm} ⊂ G paired with an edge-surjective homomorphism φ: V (G1 ∪· .. -
On the Fine-Grained Complexity of Rainbow Coloring
On the fine-grained complexity of rainbow coloring∗ Łukasz Kowalik† Juho Lauri‡ Arkadiusz Socała§ Abstract The Rainbow k-Coloring problem asks whether the edges of a given graph can be colored in k colors so that every pair of vertices is connected by a rainbow path, i.e., a path with all edges of different colors. Our main result states that for any k ≥ 2, there 3/2 is no algorithm for Rainbow k-Coloring running in time 2o(n ), unless ETH fails. Motivated by this negative result we consider two parameterized variants of the problem. In Subset Rainbow k-Coloring problem, introduced by Chakraborty et al. [STACS 2009, J. Comb. Opt. 2009], we are additionally given a set S of pairs of vertices and we ask if there is a coloring in which all the pairs in S are connected by rainbow paths. We show that Subset Rainbow k-Coloring is FPT when parameterized by |S|. We also study Maximum Rainbow k-Coloring problem, where we are additionally given an integer q and we ask if there is a coloring in which at least q anti-edges are connected by rainbow paths. We show that the problem is FPT when parameterized by q and has a kernel of size O(q) for every k ≥ 2 (thus proving that the problem is FPT), extending the result of Ananth et al. [FSTTCS 2011]. 1 Introduction The Rainbow k-Coloring problem asks whether the edges of a given graph can be colored in k colors so that every pair of vertices is connected by a rainbow path, i.e., a path with all edges of different colors. -
Technical Program
TECHNICAL PROGRAM Wednesday, 9:00-10:30 Wednesday, 10:50-12:20 WA-01 WB-02 Wednesday, 9:00-10:30 - Fo1 Wednesday, 10:50-12:20 - Fo2 Opening Ceremony Mixed Integer Linear Programming Stream: Invited Presentations and Ceremonies Stream: Discrete and Combinatorial Optimization, Plenary session Graphs and Networks Chair: Marco Lübbecke Invited session Chair: Alexander Martin 1 - Sports Scheduling meets Business Analytics Michael Trick 1 - Error bounds for mixed integer linear optimization Faster computers and algorithms have transformed how sports sched- problems ules have been created in practice in a wide range of sports. Techniques Oliver Stein such as Combinatorial Benders Decomposition, Large Scale Neighbor- hood Search, and Brand-and-Price have greatly increased the range of We introduce computable a-priori and a-posteriori error bounds for op- sports leagues that can use operations research methods to create their timality and feasibility of a point generated as the rounding of an op- schedules. With this increase in computational and algorithmic power timal point of the LP relaxation of a mixed integer linear optimization comes the opportunity to create not just playable schedules but more problem. Treating the mesh size of integer vectors as a parameter al- profitable schedules. Using data mining and other predictive analytics lows us to study the effect of different ‘granularities’ in the discrete techniques, it is possible to model attendance and other revenue effects variables on the error bounds. Our analysis mainly bases on the con- of the schedule. Combining these models with advanced schedule cre- struction of a so-called grid relaxation retract. -
P4-Free Partition and Cover Numbers and Application
P4-free Partition and Cover Numbers and Application Alexander R. Block Simina Br^anzei Hemanta K. Maji Himanshi Mehta Tamalika Mukherjee Hai H. Nguyen Abstract P4-free graphs{ also known as cographs, complement-reducible graphs, or hereditary Dacey graphs{have been well studied in graph theory. Motivated by computer science and information theory applications, our work encodes (flat) joint probability distributions and Boolean functions as bipartite graphs and studies bipartite P4-free graphs. For these applications, the graph properties of edge partitioning and covering a bipartite graph using the minimum number of these graphs are particularly relevant. Previously, such graph properties have appeared in leakage-resilient cryptography and (variants of) coloring problems. Interestingly, our covering problem is closely related to the well-studied problem of prod- uct/Prague dimension of loopless undirected graphs, which allows us to employ algebraic lower- bounding techniques for the product/Prague dimension. We prove that computing these num- bers is NP-complete, even for bipartite graphs. We establish a connection to the (unsolved) Zarankiewicz problem to show that there are bipartite graphs with size-N partite sets such that these numbers are at least " · N 1−2", for " 2 f1=3; 1=4; 1=5;::: g. Finally, we accurately esti- mate these numbers for bipartite graphs encoding well-studied Boolean functions from circuit complexity, such as set intersection, set disjointness, and inequality. For applications in information theory and communication & cryptographic complexity, we consider a system where a setup samples from a (flat) joint distribution and gives the partic- ipants, Alice and Bob, their portion from this joint sample. -
The Biclique Covering Number of Grids
THE BICLIQUE COVERING NUMBER OF GRIDS KRYSTAL GUO1, TONY HUYNH2, AND MARCO MACCHIA3 Abstract. We determine the exact value of the biclique covering number for all grid graphs. 1. Introduction Let G be a graph. A biclique of G is a complete bipartite subgraph. The biclique covering number of G, denoted bc(G), is the minimum number of bicliques of G required to cover the edges of G. The biclique covering number is studied in fields as diverse as polyhedral combinatorics [10,2], biology [12], and communication complexity [13], where it is also known as bipartite dimension or rectangle covering number. Computing the biclique covering number is a classic NP-hard problem. Indeed, deciding if bc(G) 6 k appears as problem GT18 in Garey and Johnson [7]. It is also NP-hard to approximate within a factor of n1−", where n is the number of vertices of G (see [4]). As such, there are very few classes of graphs for which we know the biclique covering number exactly. For example, it is well-known that the biclique covering number of the complete graph Kn is dlog2 ne (see [5] for a proof). Note that the minimum number of bicliques that partition E(Kn) is n − 1 by the Graham-Pollak theorem [9]. This result was later extended to biclique coverings C of Kn such that each edge is in at most k bicliques of C. Alon [1] showed that the minimum number of bicliques in such coverings is Θ(kn1=k). Two − − more classes of graphs for which we know the biclique covering number exactly are K2n and Kn;n, which are the graphs obtained from K2n and Kn;n by deleting the edges of a perfect matching. -
Enumeration Algorithms and Graph Theoretical Models to Address Biological Problems Related to Symbiosis
Università Italo-Francese / Université Franco-Italienne, AGREEMENT FOR THE CO-DIRECTION OF THE Ph.D THESIS: Sapienza, Università di Roma: Université Claude Bernard, Lyon 1: Dept. of Computer Science Ecole doctorale E2M2 Supervisor Tiziana Calamoneri Supervisor Marie-France Sagot Author: Mattia Gastaldello Enumeration Algorithms and Graph Theoretical Models to Address Biological Problems Related To Symbiosis Contents Introduction 5 0.1 Enumeration algorithms ............................ 10 0.1.1 Complexity Classes for Enumeration Algorithms ........... 10 0.1.2 Techniques to Design Enumeration Algorithms ............ 12 Cytoplasmic Incompatibility and Chain Graphs 17 1.1 Introduction and Motivations ......................... 17 1.2 Models for the Cytoplasmic Incompatibilty .................. 19 1.3 Preliminary Notation .............................. 25 1.4 Toxin and Antitoxin: The graph theoretic interpretation .......... 27 1.5 Chian Graphs and Bipartite Edge Covers ................... 30 1.5.1 Enumerating Maximal Chain Subgraphs ............... 30 1.5.2 Minimum Chain Subgraph Cover ................... 37 1.5.3 Enumeration of Minimal Chain Subgraph Covers .......... 38 1.6 Chain Graphs and Interval Orders ....................... 42 1.6.1 Preliminaries on Poset Theory ..................... 43 1.6.2 Computation of the Interval Order Dimension ............ 43 1.6.3 Enumeration of Minimal Extensions and Maximal Reductions of Interval Order .............................. 44 1.7 Computing the Poset Dimension ........................ 44 1.7.1 -
Graph Factorization from Wikipedia, the Free Encyclopedia Contents
Graph factorization From Wikipedia, the free encyclopedia Contents 1 2 × 2 real matrices 1 1.1 Profile ................................................. 1 1.2 Equi-areal mapping .......................................... 2 1.3 Functions of 2 × 2 real matrices .................................... 2 1.4 2 × 2 real matrices as complex numbers ............................... 3 1.5 References ............................................... 4 2 Abelian group 5 2.1 Definition ............................................... 5 2.2 Facts ................................................. 5 2.2.1 Notation ........................................... 5 2.2.2 Multiplication table ...................................... 6 2.3 Examples ............................................... 6 2.4 Historical remarks .......................................... 6 2.5 Properties ............................................... 6 2.6 Finite abelian groups ......................................... 7 2.6.1 Classification ......................................... 7 2.6.2 Automorphisms ....................................... 7 2.7 Infinite abelian groups ........................................ 8 2.7.1 Torsion groups ........................................ 9 2.7.2 Torsion-free and mixed groups ................................ 9 2.7.3 Invariants and classification .................................. 9 2.7.4 Additive groups of rings ................................... 9 2.8 Relation to other mathematical topics ................................. 10 2.9 A note on the typography ...................................... -
The Biclique Covering Number of Grids
The biclique covering number of grids Krystal Guo1 Tony Huynh2 Marco Macchia3 D´epartement de Math´ematique, Universit´elibre de Bruxelles, Belgium. (1) [email protected] (2) [email protected] (3) [email protected] Submitted: Nov 16, 2018; Accepted: Oct 28, 2019; Published: Nov 8, 2019 c The authors. Released under the CC BY-ND license (International 4.0). Abstract We determine the exact value of the biclique covering number for all grid graphs. Mathematics Subject Classifications: 05C70 1 Introduction Let G be a graph. A biclique of G is a complete bipartite subgraph. The biclique covering number of G, denoted bc(G), is the minimum number of bicliques of G required to cover the edges of G. The biclique covering number is studied in fields as diverse as polyhedral combinatorics [10,2], biology [11], and communication complexity [12], where it is also known as bipartite dimension or rectangle covering number. Computing the biclique covering number is a classic NP-hard problem. Indeed, decid- ing if bc(G) 6 k appears as problem GT18 in Garey and Johnson [7]. It is also NP-hard to approximate within a factor of n1−", where n is the number of vertices of G (see [4]). As such, there are very few classes of graphs for which we know the biclique covering number exactly. For example, it is well-known that the biclique covering number of the complete graph Kn is dlog2 ne (see [5] for a proof). Note that the minimum number of bicliques that partition E(Kn) is n−1 by the Graham-Pollak theorem [9]. -
Juho Lauri Chasing the Rainbow Connection: Hardness, Algorithms, and Bounds
Juho Lauri Chasing the Rainbow Connection: Hardness, Algorithms, and Bounds Julkaisu 1428 • Publication 1428 Tampere 2016 Tampereen teknillinen yliopisto. Julkaisu 1428 Tampere University of Technology. Publication 1428 Juho Lauri Chasing the Rainbow Connection: Hardness, Algorithms, and Bounds Thesis for the degree of Doctor of Science in Technology to be presented with due permission for public examination and criticism in Tietotalo Building, Auditorium TB224, at Tampere University of Technology, on the 3rd of November 2016, at 12 noon. Tampereen teknillinen yliopisto - Tampere University of Technology Tampere 2016 ISBN 978-952-15-3836-0 (printed) ISBN 978-952-15-3842-1 (PDF) ISSN 1459-2045 Abstract We study rainbow connectivity of graphs from the algorithmic and graph-theoretic points of view. The study is divided into three parts. First, we study the complexity of deciding whether a given edge-colored graph is rainbow-connected. That is, we seek to verify whether the graph has a path on which no color repeats between each pair of its vertices. We obtain a comprehensive map of the hardness landscape of the problem. While the problem is NP-complete in general, we identify several structural properties that render the problem tractable. At the same time, we strengthen the known NP-completeness results for the problem. We pinpoint various parameters for which the problem is fixed-parameter tractable, including dichotomy results for popular width parameters, such as treewidth and pathwidth. The study extends to variants of the problem that consider vertex-colored graphs and/or rainbow shortest paths. We also consider upper and lower bounds for exact parameterized algorithms.