Computational Approaches for Zero Forcing and Related Problems

In this paper, we propose computational approaches for the zero forcing problem, the connected zero forcing problem, and the problem of forcing a graph within a specified number of timesteps. Our approaches are based on a combination of integer programming models and combinatorial algorithms, and include formulations for zero forcing as a dynamic process, and as a set-covering problem. We explore several solution strategies for these models, test them on various types of graphs, and show that they are competitive with the state-of-the-art algorithm for zero forcing. Our proposed algorithms for connected zero forcing and for controlling the number of zero forcing timesteps are the first general-purpose computational methods for these problems, and are superior to brute force computation.

[1]  Eyal Ackerman,et al.  Combinatorial model and bounds for target set selection , 2010, Theor. Comput. Sci..

[2]  Michael A. Henning,et al.  Domination in Graphs Applied to Electric Power Networks , 2002, SIAM J. Discret. Math..

[3]  Seth A. Meyer.  Zero forcing sets and bipartite circulants , 2010 .

[4]  Shaun M. Fallat,et al.  The minimum rank of symmetric matrices described by a graph: A survey☆ , 2007 .

[5]  Eduardo L. Pasiliao,et al.  An Integer Programming Approach for Fault-Tolerant Connected Dominating Sets , 2015, INFORMS J. Comput..

[6]  Yiming Wang,et al.  On imposing connectivity constraints in integer programs , 2017, Mathematical Programming.

[7]  David E. Roberson,et al.  Fractional zero forcing via three-color forcing games , 2016, Discret. Appl. Math..

[8]  Leslie Hogben,et al.  Vertex and edge spread of zero forcing number, maximum nullity, and minimum rank of a graph , 2012 .

[9]  Baoyindureng Wu,et al.  Proof of a conjecture on the zero forcing number of a graph , 2016, Discret. Appl. Math..

[10]  Steve Butler,et al.  Using variants of zero forcing to bound the inertia set of a graph , 2015 .

[11]  A. Berman,et al.  Zero forcing for sign patterns , 2013, 1307.2198.

[12]  Pauline van den Driessche,et al.  Parameters Related to Tree‐Width, Zero Forcing, and Maximum Nullity of a Graph , 2013, J. Graph Theory.

[13]  D. West.  Introduction to Graph Theory , 1995 .

[14]  Fabrizio Grandoni,et al.  Solving Connected Dominating Set Faster than 2 n , 2006, Algorithmica.

[15]  Matteo Fischetti,et al.  Thinning out Steiner trees: a node-based model for uniform edge costs , 2017, Math. Program. Comput..

[16]  Nathaniel Dean,et al.  Iteration Index of a Zero Forcing Set in a Graph , 2011, 1105.1492.

[17]  Boris Brimkov,et al.  Characterizations of the Connected Forcing Number of a Graph , 2016, ArXiv.

[18]  Boting Yang.  Fast-mixed searching and related problems on graphs , 2013, Theor. Comput. Sci..

[19]  Simone Severini,et al.  Logic circuits from zero forcing , 2011, Natural Computing.

[20]  Pasquale Avella,et al.  Computational experience with general cutting planes for the Set Covering problem , 2009, Oper. Res. Lett..

[21]  Laurence A. Wolsey,et al.  Integer and Combinatorial Optimization , 1988 .

[22]  Daniela Ferrero,et al.  Power domination and zero forcing , 2015 .

[23]  Ashkan Aazami,et al.  Hardness results and approximation algorithms for some problems on graphs , 2008 .

[24]  Matteo Fischetti,et al.  Optimizing over the first Chvátal closure , 2005, Math. Program..

[25]  David Avis,et al.  Reverse Search for Enumeration , 1996, Discret. Appl. Math..

[26]  M. T. Hagh,et al.  Minimizing the number of PMUs and their optimal placement in power systems , 2012 .

[27]  Michael Young,et al.  Zero forcing propagation time on oriented graphs , 2017, Discret. Appl. Math..

[28]  Vittorio Giovannetti,et al.  Full control by locally induced relaxation. , 2007, Physical review letters.

[29]  Min Zhao,et al.  Power domination in graphs , 2006, Discret. Math..

[30]  Ilan Newman,et al.  Treewidth governs the complexity of target set selection , 2011, Discret. Optim..

[31]  W. Haemers.  Zero forcing sets and minimum rank of graphs , 2008 .

[32]  Hong-Gwa Yeh,et al.  Some results on the target set selection problem , 2011, Journal of Combinatorial Optimization.

[33]  Alexandre Salles da Cunha,et al.  The k-Cardinality Tree Problem: Reformulations and Lagrangian Relaxation , 2010, Discret. Appl. Math..

[34]  Nathan Warnberg,et al.  Positive semidefinite propagation time , 2016, Discret. Appl. Math..

[35]  Gevork B. Gharehpetian,et al.  Integrated Model Considering Effects of Zero Injection Buses and Conventional Measurements on Optimal PMU Placement , 2017, IEEE Transactions on Smart Grid.

[36]  Boris Brimkov,et al.  Graphs with Extremal Connected Forcing Numbers , 2017, ArXiv.

[37]  Michael Young,et al.  Throttling zero force propagation speed on graphs , 2013, Australas. J Comb..

[38]  Hong-Gwa Yeh,et al.  On minimum rank and zero forcing sets of a graph , 2010 .

[39]  Neng Fan,et al.  Solving the Connected Dominating Set Problem and Power Dominating Set Problem by Integer Programming , 2012, COCOA.

[40]  Boris Brimkov,et al.  Connected power domination in graphs , 2017, Journal of Combinatorial Optimization.

[41]  Fabrizio Grandoni,et al.  Solving Connected Dominating Set Faster than 2n , 2007, Algorithmica.

[42]  Darren D. Row.  A technique for computing the zero forcing number of a graph with a cut-vertex , 2012 .

[43]  Jean-Charles Delvenne,et al.  Zero forcing number, constraint matchings and strong structural controllability , 2014, ArXiv.

[44]  Michael Young,et al.  Positive semidefinite zero forcing , 2013, Linear Algebra and its Applications.

[45]  Sarah Meyer,et al.  Propagation time for zero forcing on a graph , 2012, Discret. Appl. Math..

[46]  Linda Eroh,et al.  Metric dimension and zero forcing number of two families of line graphs , 2012, 1207.6127.

[47]  R. A. Zemlin,et al.  Integer Programming Formulation of Traveling Salesman Problems , 1960, JACM.

[48]  William J. Cook,et al.  Solution of a Large-Scale Traveling-Salesman Problem , 1954, 50 Years of Integer Programming.

[49]  Boris Brimkov,et al.  Complexity and computation of connected zero forcing , 2017, Discret. Appl. Math..

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

[51]  Egon Balas,et al.  On the set covering polytope: I. All the facets with coefficients in {0, 1, 2} , 1986, Math. Program..

[52]  Miguel Constantino,et al.  Imposing Connectivity Constraints in Forest Planning Models , 2013, Oper. Res..

[53]  Shaun M. Fallat,et al.  Zero forcing parameters and minimum rank problems , 2010, 1003.2028.

[54]  Michael A. Henning,et al.  Bounds on the connected domination number of a graph , 2013, Discret. Appl. Math..

[55]  Yair Caro,et al.  Upper bounds on the k-forcing number of a graph , 2014, Discret. Appl. Math..

[56]  George B. Dantzig,et al.  Solution of a Large-Scale Traveling-Salesman Problem , 1954, Oper. Res..

[57]  Ashkan Aazami,et al.  Domination in graphs with bounded propagation: algorithms, formulations and hardness results , 2008, J. Comb. Optim..

[58]  Raphael Yuster,et al.  Connected Domination and Spanning Trees with Many Leaves , 2000, SIAM J. Discret. Math..

[59]  David A. Bader,et al.  Benchmarking for Graph Clustering and Partitioning , 2014, Encyclopedia of Social Network Analysis and Mining.