Polynomial instances of the positive semidefinite and Euclidean distance matrix completion problems

SIAM Journal on Matrix Analysis and Applications

Volumn 22, Issue 3, 2000, Pages 874-894

  Laurent, Monique ^a  

^a CWI   (Netherlands)

Author keywords

Bit model; Chordal graph; Euclidean distance matrix; Matrix completion; Minimum fill in; Order of a graph; Polynomial algorithm; Positive semidefinite matrix; Real number model

Indexed keywords

EID: 0035219189     PISSN: 08954798     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0895479899352689     Document Type: Article

References (44)

1. J. AGLER, J. W. HELTON, S. MCCULLOUGH, AND L. RODMAN, Positive semidefinite matrices with a given sparsity pattern, Linear Algebra Appl., 107 (1988), pp. 101-149.
2. A. Y. ALFAKIH, A. KHANDANI, AND H. WOLKOWICZ, Solving Euclidean distance matrix completion problems via semidefinite programming, Comput. Optim. Appl., 12 (1998), pp. 13-30.
3. F. ALIZADEH, Interior point methods in semidefinite programming with applications in combinatorial optimization, SIAM J. Optim., 5 (1995), pp. 13-51.
4. M. BAKONYI AND C. R. JOHNSON, The Euclidean distance matrix completion problem, SIAM J. Matrix Anal. Appl., 16 (1995), pp. 646-654.
5. M. BAKONYI AND G. NAEVDAL, On the matrix completion method for multidimensional moment problems, Acta Sci. Math. (Szeged), 64 (1998), pp. 547-558.
6. F. BARAHONA AND A. R. MAHJOUB, On the cut polytope, Math. Programming, 36 (1986), pp. 157-173.
7. W. BARRETT, C. R. JOHNSON, AND P. TARAZAGA, The real positive definite completion problem for a simple cycle, Linear Algebra Appl., 192 (1993), pp. 3-31.
8. A. I. BARVINOK, Feasibility testing for systems of real quadratic equations, Discrete Comput. Geom., 10 (1993), pp. 1-13.
9. A. I. BARVINOK, personal communication, 1998.
10. L. BLUM, M. SHUB, AND S. SMALE, On a theory of computation and complexity over the real numbers: NP-completeness, recursive functions and universal machines, Bull. Amer. Math. Soc., 21 (1989), pp. 1-46.
11. G. M. CRIPPEN AND T. F. HAVEL, Distance Geometry and Molecular Conformation, Research Studies Press, Taunton, Somerset, England, 1988.
12. H. DYM AND I. GOHBERG, Extensions of band matrices with band inverses, Linear Algebra Appl., 36 (1981), pp. 1-24.
13. M. R. GAREY AND D. S. JOHNSON, Computers and Intractability: A Guide to the Theory of NP-Completeness, W. H. Freeman and Company, New York, 1979.
14. M. C. GOLUMBIC, Algorithmic Theory and Perfect Graphs, Academic Press, New York, 1980.
15. R. GRONE, C. R. JOHNSON, E. M. SÁ, AND H. WOLKOWICZ, Positive definite completions of partial Hermitian matrices, Linear Algebra Appl., 58 (1984), pp. 109-124.
16. M. GRÖTSCHEL, L. LOVÁSZ, AND A. SCHRIJVER, Geometric Algorithms and Combinatorial Optimization, Springer-Verlag, Berlin, 1988.
17. C. HELMBERG, F. RENDL, R. J. VANDERBEI, AND H. WOLKOWICZ, An interior-point method for semidefinite programming, SIAM J. Optim., 6 (1996), pp. 342-361.
18. B. HENDRICKSON, The molecule problem: Exploiting structure in global optimization, SIAM J. Optim., 5 (1995), pp. 835-857.
19. C. R. JOHNSON, Matrix completion problems: A survey, in Matrix Theory and Applications 40, Proc. Sympos. Appl. Math., C. R. Johnson, ed., AMS, Providence, RI, 1990, pp. 171-198.
20. C. R. JOHNSON, B. KROSCHEL, AND H. WOLKOWICZ, An interior-point method for approximate positive semidefinite completions, Comput. Optim. Appl., 9 (1998), pp. 175-190.
21. C. R. JOHNSON AND P. TARAZAGA, Connections between the real positive semidefinite and distance matrix completion problems, Linear Algebra Appl., 223/224 (1995), pp. 375-391.
22. N. KARMARKAR, A new polynomial-time algorithm for linear programming, Combinatorica, 4 (1984), pp. 373-395.
23. L. KHACHIYAN, A polynomial algorithm in linear programming, Soviet Mathematics Doklady, 20 (1979), pp. 191-194.
24. L. KHACHIYAN AND L. PORKOLAB, Computing integral points in convex semi-algebraic sets, in 38th Annual IEEE Symposium on Foundations of Computer Science, Miami, FL, 1997, pp. 162-171.
25. M. LAURENT, The real positive semidefinite completion problem for series-parallel graphs, Linear Algebra Appl., 252 (1997), pp. 347-366.
26. M. LAURENT, Cuts, matrix completions, and graph rigidity, Math. Programming, 79 (1997), pp. 255-283.
27. M. LAURENT, A connection between positive semidefinite and Euclidean distance matrix completion problems, Linear Algebra Appl., 273 (1998), pp. 9-22.
28. M. LAURENT, On the order of a graph and its deficiency in chordality, Combinatorica, to appear.
29. J. DE LEEUW AND W. HEISER, Theory of multidimensional scaling, in Handbook of Statistics, Vol. 2, P. R. Krishnaiah and L. N. Kanal, eds., North Holland, 1982, pp. 285-316.
30. H. W. LENSTRA, JR., Integer programming with a fixed number of variables, Math. Oper. Res., 8 (1983), pp. 538-548.
31. J. J. MORÉ AND Z. WU, Distance geometry optimization for protein structures, J. Global Optim., 15 (1999), pp. 219-234.
32. Y. E. NESTEROV AND A. S. NEMIROVSKY, Interior Point Polynomial Algorithms in Convex Programming: Theory and Algorithms, SIAM, Philadelphia, 1994.
33. L. PORKOLAB AND L. KHACHIYAN, On the complexity of semidefinite programs, J. Global Optim., 10 (1997), pp. 351-365.
34. L. PORKOLAB, private communication, 2000.
35. M. V. RAMANA, An exact duality theory for semidefinite programming and its complexity implications, Math. Programming, 77 (1997), pp. 129-162.
36. D. J. ROSE, R. E. TARJAN, AND G. S. LUEKER, Algorithmic aspects of vertex elimination on graphs, SIAM J. Comput., 5 (1976), pp. 266-283.
37. I. J. SCHOENBERG, Remarks to M. Fréchet's article "Sur la définition axiomatique d'une classe d'espaces vectoriels distanciés applicables vectoriellement sur l'espace de Hilbert," Ann. of Math., 36 (1935), pp. 724-732.
38. A. SCHRIJVER, Theory of Linear and Integer Programming, John Wiley and Sons, New York, 1986.
39. R. E. TARJAN, Decomposition by clique separators, Discrete Math., 55 (1985), pp. 221-232.
40. L. VANDENBERGHE AND S. BOYD, Semidefinite programming, SIAM Rev., 38 (1996), pp. 49-95.
41. J. A. WALD AND C. J. COLBOURN, Steiner trees, partial 2-trees and minimum IFI networks, Networks, 13 (1983), pp. 159-167.
42. H. WOLKOWICZ, R. SAIGAL, AND L. VANDENBERGHE, EDS., Handbook of Semidefinite Programming: Theory, Algorithms, and Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2000.
43. S. J. WRIGHT, Primal-Dual Interior-Point Methods, SIAM, Philadelphia, 1996.
44. M. YANNAKAKIS, Computing the minimum fill-in is NP-complete, SIAM J. Algebraic Discrete Methods, 2 (1981), pp. 77-79.