On the Minimum Chordal Completion Polytope

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References

  • Agrawal A, Klein P, Ravi R (1993) Cutting down on fill using nested dissection: Provably good elimination orderings. George A, Gilber JR, Liu JWH, eds. Graph Theory and Sparse Matrix Computation (Springer, New York), 31–55.CrossrefGoogle Scholar
  • Beeri C, Fagin R, Maier D, Yannakakis M (1983) On the desirability of acyclic database schemes. J. ACM. 30(3):479–513.CrossrefGoogle Scholar
  • Bergman D, Raghunathan AU (2015) A Benders approach to the minimum chordal completion problem. Pesant G, ed. Principles and Practice of Constraint Programming—CP 2015, Lecture Notes in Computer Science, vol. 6308 (Springer, Berlin), 47–64.CrossrefGoogle Scholar
  • Berry A, Bordat JP, Heggernes P, Simonet G, Villanger Y (2006) A wide-range algorithm for minimal triangulation from an arbitrary ordering. J. Algorithms 58(1):33–66.CrossrefGoogle Scholar
  • Berry A, Heggernes P, Simonet G (2003) The minimum degree heuristic and the minimal triangulation process. Bodlaender H, ed. Graph-Theoretic Concepts in Computer Science, Lecture Notes in Computer Science, vol. 2880 (Springer, Berlin), 58–70.CrossrefGoogle Scholar
  • Bliznets I, Cygan M, Komosa P, Mach L, Pilipczuk M (2016) Lower bounds for the parameterized complexity of minimum fill-in and other completion problems. Proc. 27th Annual ACM-SIAM Sympos. Discrete Algorithms (Society for Industrial and Applied Mathematics, Philadelphia), 1132–1151.CrossrefGoogle Scholar
  • Bouchitté V, Todinca I (2001) Treewidth and minimum fill-in: Grouping the minimal separators. SIAM J. Comput. 31(1):212–232.CrossrefGoogle Scholar
  • Cao Y, Marx D (2016) Chordal editing is fixed-parameter tractable. Algorithmica 75(1):118–137.CrossrefGoogle Scholar
  • Cao Y, Sandeep RB (2016) Minimum fill-in: Inapproximability and almost tight lower bounds. Proc. 28th Annual ACM-SIAM Sympos. Discrete Algorithms (Society for Industrial and Applied Mathematics, Philadelphia), 875–880.Google Scholar
  • Chang MS (1996) Algorithms for maximum matching and minimum fill-in on chordal bipartite graphs. Asano T, Igarashi Y, Nagamochi H, Miyano S, Suri S, eds. Algorithms and Computation, Lecture Notes in Computer Science, vol. 1178 (Springer, Berlin), 146–155.CrossrefGoogle Scholar
  • Chung F, Mumford D (1994) Chordal completions of planar graphs. J. Combin. Theory Ser. B. 62(1):96–106.CrossrefGoogle Scholar
  • Codato G, Fischetti M (2006) Combinatorial Benders’ cuts for mixed-integer linear programming. Oper. Res. 54(4):756–766.LinkGoogle Scholar
  • Feremans C, Oswald M, Reinelt G (2002) A y-formulation for the treewidth. Technical report, Heidelberg University, Heidelberg, Germany.Google Scholar
  • Fomin FV, Philip G, Villanger Y (2013) Minimum fill-in of sparse graphs: Kernelization and approximation. Algorithmica 71(1):1–20.CrossrefGoogle Scholar
  • Fomin FV, Villanger Y (2012) Subexponential parameterized algorithm for minimum fill-in. Proc. 23rd Annual ACM-SIAM Sympos. Discrete Algorithms (Society for Industrial and Applied Mathematics, Philadelphia), 1737–1746.CrossrefGoogle Scholar
  • Fulkerson DR, Gross OA (1965) Incidence matrices and interval graphs. Pacific J. Math. 15(3):835–855.CrossrefGoogle Scholar
  • Garey MR, Johnson DS (1979) Computers and Intractability: A Guide to the Theory of NP-Completeness (W. H. Freeman & Co., New York).Google Scholar
  • George A, Liu WH (1989) The evolution of the minimum degree ordering algorithm. SIAM Rev. 31(1):1–19.CrossrefGoogle Scholar
  • Grone R, Johnson CR, Sá EM, Wolkowicz H (1984) Positive definite completions of partial hermitian matrices. Linear Algebra Appl. 58(0):109–124.CrossrefGoogle Scholar
  • Heggernes P (2006) Minimal triangulations of graphs: A survey. Discrete Math. 306(3):297–317.CrossrefGoogle Scholar
  • IBM ILOG (2017) CPLEX Optimization Studio 12.7.1 User Manual (IBM ILOG, New York).Google Scholar
  • Judd S, Kearns M, Vorobeychik Y (2011) Behavioral conflict and fairness in social networks. Chen N, Elkind E, Koutsoupias E, eds. Internet and Network Economics, Lecture Notes in Computer Science, vol. 7090 (Springer, New York), 242–253.CrossrefGoogle Scholar
  • Kaplan H, Shamir R, Tarjan RE (1999) Tractability of parameterized completion problems on chordal, strongly chordal, and proper interval graphs. SIAM J. Comput. 28(5):1906–1922.CrossrefGoogle Scholar
  • Kim S, Kojima M, Mevissen M, Yamashita M (2011) Exploiting sparsity in linear and nonlinear matrix inequalities via positive semidefinite matrix completion. Math. Programming 129(1):33–68.CrossrefGoogle Scholar
  • Kloks T, Bodlaender H, Müller H, Kratsch D (1993) Computing Treewidth and Minimum Fill-In: All You Need Are the Minimal Separators (Springer, Berlin), 260–271.Google Scholar
  • Kloks T, Kratsch D, Müller H (1999) Approximating the bandwidth for asteroidal triple-free graphs. J. Algorithms 32(1):41–57.CrossrefGoogle Scholar
  • Kloks T, Kratsch D, Wong C (1998) Minimum fill-in on circle and circular-arc graphs. J. Algorithms 28(2):272–289.CrossrefGoogle Scholar
  • Kratsch D, Müller H (2009) On a property of minimal triangulations. Discrete Math 309(6):1724–1729.CrossrefGoogle Scholar
  • Lauritzen SL, Spiegelhalter DJ (1990) Local computations with probabilities on graphical structures and their application to expert systems. Shafer G, Pearl J, eds. Readings in Uncertain Reasoning (Morgan Kaufmann Publishers Inc., San Francisco), 415–448.Google Scholar
  • Mezzini M, Moscarini M (2010) Simple algorithms for minimal triangulation of a graph and backward selection of a decomposable Markov network. Theoret. Comput. Sci. 411(7–9):958–966.CrossrefGoogle Scholar
  • Nakata K, Fujisawa K, Fukuda M, Kojima M, Murota K (2003) Exploiting sparsity in semidefinite programming via matrix completion II: Implementation and numerical results. Math. Programming 95(2):303–327.CrossrefGoogle Scholar
  • Natanzon A, Shamir R, Sharan R (2000) A polynomial approximation algorithm for the minimum fill-in problem. SIAM J. Comput. 30(4):1067–1079.CrossrefGoogle Scholar
  • Nikolaou C, Koubarakis M (2014) Fast consistency checking of very large real-world RCC-8 constraint networks using graph partitioning. Proc. 28th AAAI Conf. A.I. (Association for the Advancement of Artificial Intelligence, Menlo Park, CA), 2724–2730.Google Scholar
  • Nikolopoulos SD, Palios L (2007) Detecting holes and antiholes in graphs. Algorithmica 47(2):119–138.CrossrefGoogle Scholar
  • Ohtsuki T (1976) A fast algorithm for finding an optimal ordering for vertex elimination on a graph. SIAM J. Comput. 5(1):133–145.CrossrefGoogle Scholar
  • Ohtsuki T, Cheung LK, Fujisawa T (1976) Minimal triangulation of a graph and optimal pivoting order in a sparse matrix. J. Math. Anal. Appl. 54(3):622–633.CrossrefGoogle Scholar
  • Parra A, Scheffler P (1995) How to use the minimal separators of a graph for its chordal triangulation. Fülöp Z, Gécseg F, eds. Automata, Languages and Programming—ICALP 1995, Lecture Notes in Computer Science, vol. 944 (Springer, Berlin), 123–134.CrossrefGoogle Scholar
  • Parra A, Scheffler P (1997) Characterizations and algorithmic applications of chordal graph embeddings. Discrete Appl. Math. 79(1):171–188.CrossrefGoogle Scholar
  • Parter S (1961) The use of linear graphs in gauss elimination. SIAM Rev. 3(2):119–130.CrossrefGoogle Scholar
  • Renz J (2002) The region connection calculus. Qualitative Spatial Reasoning with Topological Information (Springer, Berlin), 41–50.CrossrefGoogle Scholar
  • Rollon E, Larrosa J (2011) On mini-buckets and the min-fill elimination ordering. Lee J, ed. Principles and Practice of Constraint Programming (CP) (Springer, Berlin), 759–773.CrossrefGoogle Scholar
  • Rose DJ (1972) A graph-theoretic study of the numerical solution of sparse positive definite systems of linear equations. Read RC, ed. Graph Theory and Computing (Academic Press, Cambridge, MA), 183–217.CrossrefGoogle Scholar
  • Rose DJ, Tarjan RE (1978) Algorithmic aspects of vertex elimination on directed graphs. SIAM J. Appl. Math. 34(1):176–197.CrossrefGoogle Scholar
  • Rose DJ, Tarjan RE, Lueker GS (1976) Algorithmic aspects of vertex elimination on graphs. SIAM J. Comput. 5(2):266–283.CrossrefGoogle Scholar
  • Rostami B, Malucelli F, Frey D, Buchheim C (2015) On the quadratic shortest path problem. Internat. Sympos. Experiment. Algorithms (Springer, New York), 379–390.CrossrefGoogle Scholar
  • Sioutis M (2014) Triangulation versus graph partitioning for tackling large real world qualitative spatial networks. 2014 IEEE 26th Internat. Conf. Tools Artificial Intelligence (IEEE, Piscataway, NJ), 194–201.CrossrefGoogle Scholar
  • Sioutis M, Koubarakis M (2012) Consistency of chordal rcc-8 networks. 2012 IEEE 24th Internat. Conf. Tools Artificial Intelligence, vol. 1 (IEEE, Piscataway, NJ), 436–443.CrossrefGoogle Scholar
  • Tarjan RE, Yannakakis M (1984) Simple linear-time algorithms to test chordality of graphs, test acyclicity of hypergraphs, and selectively reduce acyclic hypergraphs. SIAM J. Comput. 13(3):566–579.CrossrefGoogle Scholar
  • Uno T, Satoh H (2014) An efficient algorithm for enumerating chordless cycles and chordless paths. Internat. Conf. Discovery Sci. (Springer, New York), 313–324.CrossrefGoogle Scholar
  • Vandenberghe L, Andersen MS (2015) Chordal graphs and semidefinite optimization. Foundations Trends Optim. 1(4):241–433.CrossrefGoogle Scholar
  • Yannakakis M (1981) Computing the minimum fill-in is NP-Complete. SIAM J. Algebraic Discrete Methods 2(1):77–79.CrossrefGoogle Scholar
  • Yüceoğlu B (2015) Branch-and-cut algorithms for graph problems. Unpublished doctoral thesis, Maastricht University, Mastricht, Netherlands.Google Scholar
  • Zhang C, Zhao T, Li W (2015) Ontology Data Query in Geospatial Semantic Web (Springer International Publishing, Cham, Switzerland), 57–87.CrossrefGoogle Scholar
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