Empirical Evaluation of Approximation Algorithms for Generalized Graph Coloring and Uniform Quasi-wideness

The notions of bounded expansion and nowhere denseness not only offer robust and general definitions of uniform sparseness of graphs, they also describe the tractability boundary for several important algorithmic questions. In this paper we study two structural properties of these graph classes that are of particular importance in this context, namely the property of having bounded generalized coloring numbers and the property of being uniformly quasi-wide. We provide experimental evaluations of several algorithms that approximate these parameters on real-world graphs. On the theoretical side, we provide a new algorithm for uniform quasi-wideness with polynomial size guarantees in graph classes of bounded expansion and show a lower bound indicating that the guarantees of this algorithm are close to optimal in graph classes with fixed excluded minor.

[1]  Jure Leskovec,et al.  {SNAP Datasets}: {Stanford} Large Network Dataset Collection , 2014 .

[2]  Duncan J. Watts,et al.  Collective dynamics of ‘small-world’ networks , 1998, Nature.

[3]  Xuding Zhu,et al.  Colouring graphs with bounded generalized colouring number , 2009, Discret. Math..

[4]  Ken-ichi Kawarabayashi,et al.  Contraction decomposition in h-minor-free graphs and algorithmic applications , 2011, STOC '11.

[5]  Stephan Kreutzer,et al.  Neighborhood complexity and kernelization for nowhere dense classes of graphs , 2016, ICALP.

[6]  Patrice Ossona de Mendez,et al.  Distributed Domination on Graph Classes of Bounded Expansion , 2017, SPAA.

[7]  Blair D. Sullivan,et al.  Treedepth Bounds in Linear Colorings , 2018, WG.

[8]  Luc Segoufin,et al.  First-order queries on classes of structures with bounded expansion , 2018, Log. Methods Comput. Sci..

[9]  Christos Faloutsos,et al.  Graphs over time: densification laws, shrinking diameters and possible explanations , 2005, KDD '05.

[10]  Zdenek Dvorak,et al.  Constant-factor approximation of domination number in sparse graphs , 2011, ArXiv.

[11]  Bruce A. Reed,et al.  Excluding any graph as a minor allows a low tree-width 2-coloring , 2004, J. Comb. Theory, Ser. B.

[12]  F. Chung,et al.  Connected Components in Random Graphs with Given Expected Degree Sequences , 2002 .

[13]  Anuj Dawar,et al.  Homomorphism Preservation on Quasi-Wide Classes , 2008, J. Comput. Syst. Sci..

[14]  M. Newman,et al.  Finding community structure in networks using the eigenvectors of matrices. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.

[15]  J. Nesetril,et al.  Grad and classes with bounded expansion III. restricted dualities , 2005, math/0508325.

[16]  Stephan Kreutzer,et al.  Model-checking for successor-invariant first-order formulas on graph classes of bounded expansion , 2017, 2017 32nd Annual ACM/IEEE Symposium on Logic in Computer Science (LICS).

[17]  Anne Berry,et al.  Generating All the Minimal Separators of a Graph , 2000, Int. J. Found. Comput. Sci..

[18]  M E J Newman,et al.  Community structure in social and biological networks , 2001, Proceedings of the National Academy of Sciences of the United States of America.

[19]  Erik D. Demaine,et al.  Approximation algorithms via contraction decomposition , 2007, SODA '07.

[20]  W. Zachary,et al.  An Information Flow Model for Conflict and Fission in Small Groups , 1977, Journal of Anthropological Research.

[21]  Michal Pilipczuk,et al.  Kernelization and approximation of distance-r independent sets on nowhere dense graphs , 2018, ArXiv.

[22]  Ken-ichi Kawarabayashi,et al.  Algorithmic Graph Minor Theory: Improved Grid Minor Bounds and Wagner’s Contraction , 2009, Algorithmica.

[23]  Jérôme Kunegis,et al.  KONECT: the Koblenz network collection , 2013, WWW.

[24]  Felix Reidl,et al.  Characterising Bounded Expansion by Neighbourhood Complexity , 2016, Eur. J. Comb..

[25]  Patrice Ossona de Mendez,et al.  On the generalised colouring numbers of graphs that exclude a fixed minor , 2015, Eur. J. Comb..

[26]  Jure Leskovec,et al.  Friendship and mobility: user movement in location-based social networks , 2011, KDD.

[27]  Michal Pilipczuk,et al.  Polynomial bounds for centered colorings on proper minor-closed graph classes , 2018, SODA.

[28]  Stephan Kreutzer,et al.  Deciding first-order properties of nowhere dense graphs , 2013, STOC.

[29]  Christian Schulz,et al.  Exactly Solving the Maximum Weight Independent Set Problem on Large Real-World Graphs , 2018, ALENEX.

[30]  M. DePamphilis,et al.  HUMAN DISEASE , 1957, The Ulster Medical Journal.

[31]  S. Brenner,et al.  The structure of the nervous system of the nematode Caenorhabditis elegans. , 1986, Philosophical transactions of the Royal Society of London. Series B, Biological sciences.

[32]  Fahad Panolan,et al.  Reconfiguration on sparse graphs , 2018, J. Comput. Syst. Sci..

[33]  Stephan Kreutzer,et al.  Domination Problems in Nowhere-Dense Classes , 2009, FSTTCS.

[34]  Wojciech Nadara.  Experimental evaluation of kernelization algorithms to Dominating Set , 2018, ArXiv.

[35]  A. Barabasi,et al.  The human disease network , 2007, Proceedings of the National Academy of Sciences.

[36]  Stephan Kreutzer,et al.  The Generalised Colouring Numbers on Classes of Bounded Expansion , 2016, MFCS.

[37]  Michal Pilipczuk,et al.  Parameterized Algorithms , 2015, Springer International Publishing.

[38]  Sebastian Siebertz,et al.  Reconfiguration on nowhere dense graph classes , 2017, Electron. J. Comb..

[39]  Jakub Gajarský,et al.  Kernelization Using Structural Parameters on Sparse Graph Classes , 2013, ESA.

[40]  Hal A. Kierstead,et al.  Orderings on Graphs and Game Coloring Number , 2003, Order.

[41]  Ryan A. Rossi,et al.  The Network Data Repository with Interactive Graph Analytics and Visualization , 2015, AAAI.

[42]  Felix Reidl,et al.  Structural sparseness and complex networks , 2016 .

[43]  Azadeh Iranmehr,et al.  Trust Management for Semantic Web , 2009, 2009 Second International Conference on Computer and Electrical Engineering.

[44]  Stephan Kreutzer,et al.  Colouring and Covering Nowhere Dense Graphs , 2015, WG.

[45]  Zdenek Dvorak.  On distance r-dominating and 2r-independent sets in sparse graphs , 2019, J. Graph Theory.

[46]  Lada A. Adamic,et al.  The political blogosphere and the 2004 U.S. election: divided they blog , 2005, LinkKDD '05.

[47]  Stephan Kreutzer,et al.  Polynomial Kernels and Wideness Properties of Nowhere Dense Graph Classes , 2016, SODA.

[48]  D. Bu,et al.  Topological structure analysis of the protein-protein interaction network in budding yeast. , 2003, Nucleic acids research.

[49]  M. Newman,et al.  The structure of scientific collaboration networks. , 2000, Proceedings of the National Academy of Sciences of the United States of America.

[50]  Hans L. Bodlaender,et al.  A linear time algorithm for finding tree-decompositions of small treewidth , 1993, STOC.

[51]  Blair D. Sullivan,et al.  Experimental Evaluation of Counting Subgraph Isomorphisms in Classes of Bounded Expansion , 2017, ArXiv.

[52]  Kathryn B. Laskey,et al.  Stochastic blockmodels: First steps , 1983 .

[53]  Stephan Kreutzer,et al.  Kernelization and Sparseness: the case of Dominating Set , 2014, STACS.

[54]  Michal Pilipczuk,et al.  On the number of types in sparse graphs , 2017, LICS.

[55]  D. Lusseau,et al.  The bottlenose dolphin community of Doubtful Sound features a large proportion of long-lasting associations , 2003, Behavioral Ecology and Sociobiology.

[56]  Erik D. Demaine,et al.  The Bidimensionality Theory and Its Algorithmic Applications , 2008, Comput. J..

[57]  Jaroslav Nesetril,et al.  On nowhere dense graphs , 2011, Eur. J. Comb..

[58]  Sampo Niskanen,et al.  Cliquer user's guide, version 1.0 , 2003 .

[59]  Jaroslav Nesetril,et al.  Grad and classes with bounded expansion II. Algorithmic aspects , 2008, Eur. J. Comb..

[60]  Robin Thomas,et al.  Testing first-order properties for subclasses of sparse graphs , 2011, JACM.

[61]  Jaroslav Nesetril,et al.  Grad and classes with bounded expansion III. Restricted graph homomorphism dualities , 2008, Eur. J. Comb..

[62]  Rolf Niedermeier,et al.  Polynomial-time data reduction for dominating set , 2002, JACM.

[63]  Yiming Yang,et al.  Introducing the Enron Corpus , 2004, CEAS.

[64]  Ciro Cattuto,et al.  High-Resolution Measurements of Face-to-Face Contact Patterns in a Primary School , 2011, PloS one.

[65]  Ian T. Foster,et al.  Mapping the Gnutella Network , 2002, IEEE Internet Comput..

[66]  Marcin Pilipczuk,et al.  Empirical Evaluation of Approximation Algorithms for Generalized Graph Coloring and Uniform Quasi-wideness , 2019, ACM J. Exp. Algorithmics.

[67]  Jaroslav Nesetril,et al.  Grad and classes with bounded expansion I. Decompositions , 2008, Eur. J. Comb..

[68]  Jaroslav Nesetril,et al.  First order properties on nowhere dense structures , 2010, The Journal of Symbolic Logic.

[69]  Kedar Nath Das,et al.  Heuristics to Find Maximum Independent Set: An Overview , 2011, SocProS.

[70]  Arie M. C. A. Koster,et al.  Treewidth computations I. Upper bounds , 2010, Inf. Comput..

[71]  Fahad Panolan,et al.  Lossy Kernels for Connected Dominating Set on Sparse Graphs , 2017, STACS.

[72]  Donald E. Knuth,et al.  The Stanford GraphBase - a platform for combinatorial computing , 1993 .

[73]  Fan Chung Graham,et al.  The Average Distance in a Random Graph with Given Expected Degrees , 2004, Internet Math..

[74]  F. Chung,et al.  The average distances in random graphs with given expected degrees , 2002, Proceedings of the National Academy of Sciences of the United States of America.

[75]  Ken-ichi Kawarabayashi,et al.  Algorithmic graph minor theory: Decomposition, approximation, and coloring , 2005, 46th Annual IEEE Symposium on Foundations of Computer Science (FOCS'05).

[76]  Jaroslav Nesetril,et al.  Tree-depth, subgraph coloring and homomorphism bounds , 2006, Eur. J. Comb..

[77]  Christian Komusiewicz,et al.  The First Parameterized Algorithms and Computational Experiments Challenge , 2017, IPEC.

[78]  Blair D. Sullivan,et al.  Structural Sparsity of Complex Networks: Random Graph Models and Linear Algorithms , 2014, ArXiv.

[79]  Jure Leskovec,et al.  Signed networks in social media , 2010, CHI.