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International Conference on Integer Programming and Combinatorial Optimization

IPCO 2007: Integer Programming and Combinatorial Optimization pp 397–409Cite as

Sign-Solvable Linear Complementarity Problems

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 4513))

Abstract

This paper presents a connection between qualitative matrix theory and linear complementarity problems (LCPs). An LCP is said to be sign-solvable if the set of the sign patterns of the solutions is uniquely determined by the sign patterns of the given coefficients. We provide a characterization for sign-solvable LCPs such that the coefficient matrix has nonzero diagonals, which can be tested in polynomial time. This characterization leads to an efficient combinatorial algorithm to find the sign pattern of a solution for these LCPs. The algorithm runs in O(γ) time, where γ is the number of the nonzero coefficients.

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References

  1. Brualdi, R.A., Shader, B.L.: Matrices of Sign-solvable Linear Systems. Cambridge University Press, Cambridge (1995)

    MATH  Google Scholar 

  2. Chandrasekaran, R., Kabadi, S.N., Murty, K.G.: Some NP-complete problems in linear programming. Operations Research Letters 1, 101–104 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  3. Chung, S.J.: NP-completeness of the linear complementarity problem. Journal of Optimization Theory and Applications 60, 393–399 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  4. Cottle, R.W.: The principal pivoting method of quadratic programming. In: Dantzig, G.B., Veinott, A.F. (eds.) Mathematics of Decision Sciences, Part 1, pp. 142–162. American Mathematical Society, Providence (1968)

    Google Scholar 

  5. Cottle, R.W., Dantzig, G.B.: Complementary pivot theory of mathematical programming. Linear Algebra and Its Applications 1, 103–125 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  6. Cottle, R.W., Pang, J.S., Stone, R.E.: The Linear Complementarity Problem. Academic Press, London (1992)

    MATH  Google Scholar 

  7. Coxson, G.E.: The P-matrix problem is co-NP-complete. Mathematical Programming 64, 173–178 (1994)

    Article  MathSciNet  Google Scholar 

  8. Eijndhoven, J.T.J.V.: Solving the linear complementarity problem in circuit simulation. SIAM Journal on Control and Optimization 24, 1050–1062 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  9. Fischer, K.G., Morris, W., Shapiro, J.: Mixed dominating matrices. Linear Algebra and Its Applications 270, 191–214 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  10. Fischer, K.G., Shapiro, J.: Mixed matrices and binomial ideals. Journal of Pure and Applied Algebra 113, 39–54 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  11. Iwata, S., Kakimura, N.: Solving linear programs from sign patterns. Mathematical Programming (to appear)

    Google Scholar 

  12. Kim, S.J., Shader, B.L.: Linear systems with signed solutions. Linear Algebra and Its Applications 313, 21–40 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  13. Kim, S.J., Shader, B.L.: On matrices which have signed null-spaces. Linear Algebra and Its Applications 353, 245–255 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  14. Klee, V.: Recursive structure of S-matrices and O(m 2) algorithm for recognizing strong sign-solvability. Linear Algebra and Its Applications 96, 233–247 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  15. Klee, V., Ladner, R., Manber, R.: Sign-solvability revisited. Linear Algebra and Its Applications 59, 131–158 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  16. Lemke, C.E.: Bimatrix equilibrium points and mathematical programming. Management Science 11, 681–689 (1965)

    Article  MathSciNet  Google Scholar 

  17. Lovász, L., Plummer, M.D.: Matching Theory. Annals of Discrete Mathematics, vol. 29. North-Holland, Amsterdam (1986)

    MATH  Google Scholar 

  18. McCuaig, W.: Brace decomposition. Journal of Graph Theory 38, 124–169 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  19. McCuaig, W.: Pólya’s permanent problem. The Electronic Journal of Combinatorics 11, R79 (2004)

    Google Scholar 

  20. Murty, K.G.: Linear Complementarity, Linear and Nonlinear Programming. Internet Edition (1997)

    Google Scholar 

  21. Pólya, G.: Aufgabe 424. Archiv der Mathematik und Physik 20, 271 (1913)

    Google Scholar 

  22. Robertson, N., Seymour, P.D., Thomas, R.: Permanents, Pfaffian orientations, and even directed circuits. Annals of Mathematics 150, 929–975 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  23. Rohn, J.: Systems of linear interval equations. Linear Algebra and Its Applications 126, 39–78 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  24. Samuelson, P.A.: Foundations of Economics Analysis. Harvard University Press, Cambridge (1947), Atheneum, New York (1971)

    Google Scholar 

  25. Seymour, P., Thomassen, C.: Characterization of even directed graphs. Journal of Combinatorial Theory, Series B 42, 36–45 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  26. Shao, J.Y., Ren, L.Z.: Some properties of matrices with signed null spaces. Discrete Mathematics 279, 423–435 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  27. Vazirani, V.V., Yannakakis, M.: Pfaffian orientations, 0-1 permanents, and even cycles in directed graphs. Discrete Applied Mathematics 25, 179–190 (1989)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematical Informatics, Graduate School of Information Science and Technology, The University of Tokyo, Tokyo 113-8656, Japan

    Naonori Kakimura

Authors
  1. Naonori Kakimura
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Matteo Fischetti David P. Williamson

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Kakimura, N. (2007). Sign-Solvable Linear Complementarity Problems. In: Fischetti, M., Williamson, D.P. (eds) Integer Programming and Combinatorial Optimization. IPCO 2007. Lecture Notes in Computer Science, vol 4513. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-72792-7_30

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  • DOI: https://doi.org/10.1007/978-3-540-72792-7_30

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-72791-0

  • Online ISBN: 978-3-540-72792-7

  • eBook Packages: Computer ScienceComputer Science (R0)

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