On the Threshold of Intractability

We study the computational complexity of the graph modification problems Open image in new window and Open image in new window , adding and deleting as few edges as possible to transform the input into a threshold (or chain) graph. In this article, we show that both problems are Open image in new window -hard, resolving a conjecture by Natanzon, Shamir, and Sharan (2001). On the positive side, we show that these problems admit quadratic vertex kernels. Furthermore, we give a subexponential time parameterized algorithm solving Open image in new window in Open image in new window time, making it one of relatively few natural problems in this complexity class on general graphs. These results are of broader interest to the field of social network analysis, where recent work of Brandes (2014) posits that the minimum edit distance to a threshold graph gives a good measure of consistency for node centralities. Finally, we show that all our positive results extend to Open image in new window , as well as the completion and deletion variants of both problems.

[1]  Fahad Panolan,et al.  Faster Parameterized Algorithms for Deletion to Split Graphs , 2012, Algorithmica.

[2]  Roded Sharan,et al.  Cluster graph modification problems , 2002, Discret. Appl. Math..

[3]  R. Möhring.  Algorithmic graph theory and perfect graphs , 1986 .

[4]  Yixin Cao,et al.  Chordal Editing is Fixed-Parameter Tractable , 2014, Algorithmica.

[5]  N. Mahadev,et al.  Threshold graphs and related topics , 1995 .

[6]  Yun Zhang,et al.  The Cluster Editing Problem: Implementations and Experiments , 2006, IWPEC.

[7]  Federico Mancini,et al.  Graph modification problems related to graph classes , 2008 .

[8]  Heikki Mannila,et al.  Approximating the Minimum Chain Completion problem , 2009, Inf. Process. Lett..

[9]  Yunlong Liu,et al.  Complexity and parameterized algorithms for Cograph Editing , 2012, Theor. Comput. Sci..

[10]  Yunlong Liu,et al.  Edge deletion problems: Branching facilitated by modular decomposition , 2015, Theor. Comput. Sci..

[11]  M. Yannakakis.  Computing the Minimum Fill-in is NP^Complete , 1981 .

[12]  Noga Alon,et al.  Fast Fast , 2009, ICALP.

[13]  A. Brandstädt,et al.  Graph Classes: A Survey , 1987 .

[14]  Fedor V. Fomin,et al.  Subexponential parameterized algorithm for minimum fill-in , 2011, SODA.

[15]  Michal Pilipczuk,et al.  Exploring Subexponential Parameterized Complexity of Completion Problems , 2013, STACS.

[16]  Peter Damaschke.  Parameterized enumeration, transversals, and imperfect phylogeny reconstruction , 2006, Theor. Comput. Sci..

[17]  Jiong Guo,et al.  Problem Kernels for NP-Complete Edge Deletion Problems: Split and Related Graphs , 2007, ISAAC.

[18]  Wang Jian-xin,et al.  An Overview of Kernelization Algorithms for Graph Modification Problems , 2014 .

[19]  Roded Sharan,et al.  Complexity classification of some edge modification problems , 2001, Discret. Appl. Math..

[20]  Peter L. Hammer,et al.  The splittance of a graph , 1981, Comb..

[21]  R. Sharan,et al.  Complexity classication of some edge modication problems , 1999 .

[22]  Flavia Bonomo,et al.  NP-completeness results for edge modification problems , 2006, Discret. Appl. Math..

[23]  Ulrik Brandes,et al.  Stars, neighborhood inclusion and network centrality , 2015 .

[24]  Zhi-Zhong Chen,et al.  Computing Phylogenetic Roots with Bounded Degrees and Errors , 2001, WADS.

[25]  Michal Pilipczuk,et al.  A Polynomial Kernel for Trivially Perfect Editing , 2015, ESA.

[26]  Leizhen Cai,et al.  Fixed-Parameter Tractability of Graph Modification Problems for Hereditary Properties , 1996, Inf. Process. Lett..

[27]  Jörg Flum,et al.  Parameterized Complexity Theory , 2006, Texts in Theoretical Computer Science. An EATCS Series.

[28]  Yong Gao,et al.  Familial groups in social networks , 2013, Soc. Networks.

[29]  Michal Pilipczuk,et al.  Exploring the Subexponential Complexity of Completion Problems , 2015, TOCT.