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Computing and proving with pivots

Published online by Cambridge University Press:  09 October 2013

Frédéric Meunier
Frédéric Meunier*
Affiliation:
UniversitéParis Est, CERMICS (ENPC), 77455 Marne-la-Vallée, France. frederic.meunier@enpc.fr
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Abstract

A simple idea used in many combinatorial algorithms is the idea of pivoting. Originally, it comes from the method proposed by Gauss in the 19th century for solving systems of linear equations. This method had been extended in 1947 by Dantzig for the famous simplex algorithm used for solving linear programs. From since, a pivoting algorithm is a method exploring subsets of a ground set and going from one subset σ to a new one σ′ by deleting an element inside σ and adding an element outside σ: σ′ = σ\ { v}  ∪  {u}, with v ∈ σ and u ∉ σ. This simple principle combined with other ideas appears to be quite powerful for many problems. This present paper is a survey on algorithms in operations research and discrete mathematics using pivots. We give also examples where this principle allows not only to compute but also to prove some theorems in a constructive way. A formalisation is described, mainly based on ideas by Michael J. Todd.

Keywords

Combinatorial topologycomplementarity problemsconstructive proofspivoting algorithms
Type
Research Article
Information
RAIRO - Operations Research , Volume 47 , Issue 4 , September 2013 , pp. 331 - 360
Copyright
© EDP Sciences, ROADEF, SMAI, 2013

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References

Aharoni, R. and Fleiner, T., On a lemma of Scarf. J. Comb. Theor. Ser. B 87 (2003) 72–80. Google Scholar
Aharoni, R. and Haxell, P., Hall’s theorem for hypergraphs. J. Graph Theory 35 (2000) 83–88. Google Scholar
Aharoni, R. and Holzman, R., Fractional kernels in digraphs. J. Comb. Theor. Ser. B 73 (1998) 1–6. Google Scholar
C. Berge and P. Duchet, Séminaire MSH. Paris (1983).
Boros, E. and Gurvich, V., Perfect graphs are kernel solvable. Discrete Math. 159 (1996) 35–55. Google Scholar
Cheng, X. and Deng, X., On the complexity of 2d discrete fixed point problem. Theor. Comput. Sci. 410 (2009) 448–4456. Google Scholar
V. Chvátal, Linear programming. W.H. Freeman; 1st edn. (1983).
Cloutier, J., Nyman, K.L. and Su, F.E., Two-player envy-free multi-cake division. Math. Soc. Sci. 59 (2010) 26–37. Google Scholar
R. Cottle, J. Pang and R. Stone, The linear complementarity problem. Academic Press, Boston (1992).
Cottle, R.W. and Dantzig, G.B., A generalization of the linear complementary problem. J. Comb. Theor. Ser. B 8 (1970) 79–90. Google Scholar
G.B. Dantzig, Maximization of a linear function of variables subject to linear inequalities, in Activity analysis of production and allocation, edited by T.C. Koopmans. Wiley and Chapman-Hall (1947) 339–347.
M. De Longueville, A course in topological combinatorics. Springer (2012).
De Longueville, M. and Živaljevic, R., The Borsuk-Ulam-property, Tucker-property and constructive proofs in combinatorics. J. Comb. Theor. Ser. A 113 (2006) 839–850. Google Scholar
Antoine, Deza, Sui, Huang, Tamon Stephen and Tamás Terlaky, The colourful feasibility problem. Discrete Appl. Math. 156 (2008) 2166–2177. Google Scholar
B.C. Eaves, The linear complementary problem in mathematical programming. Tech. report, Department of Operations Research, Standford University, Standford, California (1969).
Eaves, B.C., Homotopies for the computation of fixed points. Math. Programm. 3 (1972) 1–22. Google Scholar
Eaves, B.C. and Saigal, R., Homotopies for computation of fixed points on unbounded regions. Math. Programm. 3 (1972) 225–237. Google Scholar
J. Edmonds, Euler complexes, Research trends in combinatorial optimization. Springer (2009) 65–68.
Edmonds, J. and Sanità, L., On finding another room-partitioning of the vertices, Electron. Notes in Discrete Math. 36 (2010) 1257–1264. Google Scholar
Fan, K., A generalization of Tucker’s combinatorial lemma with topological applications. Ann. Math. 56 (1952) 128–140. Google Scholar
Fan, K., Combinatorial properties of certain simplicial and cubical vertex maps. Arch. Mathematiks 11 (1960) 368–377. Google Scholar
Freud, R.M. and Todd, J., A constructive proof of Tucker’s combinatorial lemma. J. Comb. Theor. Ser. A 30 (1981) 321–325. Google Scholar
O. Friedmann, A subexponential lower bound for Zadeh’s pivoting rule for solving linear programs and games, in Proc. of the 15th Conference on Integer Programming and Combinatorial Optimization, IPCO’11. New York, NY, USA (2011).
Garcia, C.B., A fixed point theorem including the last theorem of Poincaré. Math. Programm. 8 (1975) 227–239. Google Scholar
Garcia, C.B. and Zangwill, W.I., An approach to homotopy and degree theory. Math. Oper. Res. 4 (1979) 390–405. Google Scholar
C.B. Garcia and W.I. Zangwill, Pathways to solutions, fixed points and equilibria. Prentice-Hall, Englewood Cliffs (1981).
Grigni, M., A Sperner lemma complete for PPA. Inform. Process. Lett. 77 (1995) 255–259. Google Scholar
Hanke, B., Sanyal, R., Schultz, C. and Ziegler, G., Combinatorial Stokes formulas via minimal resolutions. J. Comb. Theor. Ser. A 116 (2009) 404–420. Google Scholar
Herings, P.J.-J. and van den Elzen, A., Computation of the Nash equilibrium selected by the tracing procedure in n-person games. Games and Economic Behavior 38 (2002) 89–117. Google Scholar
Jeroslow, R., The simplex algorithm with the pivot rule of maximizing criterion improvement. Discrete Math. 4 (1973) 367–377. Google Scholar
S. Kintali, L.J. Poplawski, R. Rajaraman, R. Sundaram and S.-H. Teng, Reducibility among fractional stability problems. IEEE Symposium Found. Comput. Sci. FOCS (2009).
V. Klee and G.J. Minty, How good is the simplex method?, Inequalities III, in Proc. of Third Sympos. (New York), Univ. California, CA, 1969. Academic Press (1972) 159–175.
Krawczyk, A., The complexity of finding a second Hamiltonian cycle in cubic graphs. J. Comput. System Sci. 58 (1999) 641–647. Google Scholar
Kuhn, H.W., Some combinatorial lemmas in topology. IBM J. 4 (1960) 518–524. Google Scholar
H.W. Kuhn, Approximate search for fixed points, in Computing methods in optimization problems 2. Academic Press, New York (1969).
van der Laan, G. and Talman, A.J.J., A restart algorithm for computing fixed points without an extra dimension. Math. Programm. 17 (1979) 74–84. Google Scholar
G. van der Laan and A.J.J. Talman, A restart algorithm without an artificial level for computing fixed points on unbounded regions, in Functional differential equations and approximation of fixed points, edited by H.O. Peitgen and M.O. Walther. Springer-Verlag, Berlin (1979) 247–256.
Lemke, C.E., Bimatrix equilibrium points and equilibrium programming. Manage. Sci. 11 (1965) 681–689. Google Scholar
Lemke, C.E. and Howson, J.T., Equilibrium points of bimatrix games. J. Soc. Industr. Appl. Math. 12 (1964) 413–423. Google Scholar
J. Matoušek, Using the Borsuk-Ulam theorem. Springer (2003).
J. Matoušek and B. Gärtner, Understanding and using linear programming. Springer (2006).
F. Meunier, Configurations équilibrées, Ph.D. thesis, Université Joseph Fourier. Grenoble (2006).
F. Meunier, A Z q-Fan formula. Tech. report, Laboratoire Leibniz, INPG, Grenoble (2006).
Meunier, F., Discrete splittings of the necklace. Math. Oper. Res. 33 (2008) 678–688. Google Scholar
Frédéric Meunier and Antoine Deza, A further generalization of the colourful Carathéodory theorem, Discrete Geometry and Optimization. Fields Institute Communications 69 (2013).
Monsky, P., On dividing a square into triangles. Am. Math. Monthly 77 (1970) 161–164. Google Scholar
Morris, D.M., Lemke paths on simple polytopes. Math. Oper. Res. 19 (1994) 780–789. Google Scholar
J.M. Munkres, Elements of algebraic topology. Perseus Books (1995).
Nash, J.F., Equilibrium points in n-person games. Proc. Natl. Acad. Sci. USA 36 (1950) 48–49. Google Scholar
Neyman, J., Un théorème d’existence. Compt. R. Math. Acad. Sci. de Paris 222 (1946) 843–845. Google Scholar
M.J. Osborne and A. Rubinstein, A course in game theory. MIT Press (1994).
Pálvölgyi, D., 2D-TUCKER is PPAD-complete. WINE, Lect. Note Comput. Sci. 5929 (2009) 569–574. Google Scholar
Papadimitriou, C., On the complexity of the parity argument and other inefficient proofs of existence. J. Comput. Syst. Sci. 48 (1994) 498–532. Google Scholar
Prescott, T. and Su, F.E., A constructive proof of Ky Fan’s generalization of Tucker’s lemma. J. Combi. Theor. Ser. A 111 (2005) 257–265. Google Scholar
Savani, R. and von Stengel, B., Hard-to-solve bimatrix games. Econometrica 74 (2006) 397–429. Google Scholar
Scarf, H., The approximation of fixed points of a continuous mapping. SIAM J. Appl. Math. 15 (1967) 1328–1343. Google Scholar
Scarf, H., The core of an n person game. Econometrica 35 (1967) 50–69. Google Scholar
H. Scarf, The computation of equilibrium prices: an exposition. in Handbook of mathematical economics, vol II, edited by K. Arrow and A. Kirman (1982).
Shapley, L.S., A note on the Lemke-Howson algorithm. Math. Programm. Study 1 (1974) 175–189. Google Scholar
Sperner, E., Neuer Beweis für die Invarianz der Dimensionszahl und des Gebietes, Abh. Math. Sem. Univ. Hambourg 6 (1928) 265–272. Google Scholar
Terlaky, T. and Zhang, S., Pivot rules for linear programming: A survey on recent theoretical developments. Annal. Operat. Res. 46 (1993) 203–233. Google Scholar
Thomason, A.G., Hamiltonian cycles and uniquely edge colourable graphs. Annal. Discrete Math. 3 (1978) 259–268. Google Scholar
Todd, M.J., A generalized complementary pivoting algorithm. Math. Programm. 6 (1974) 243–263. Google Scholar
Todd, M.J., Orientations in complementary pivot algorithms. Math. Oper. Res. 1 (1976) 54–66. Google Scholar
A.W. Tucker, Some topological properties of disk and sphere, in Proc. of the First Canadian Mathematical Congress, University of Toronto Press (1946).
Tutte, W.T., On Hamiltonian circuits. J. London Math. Soc. 21 (1946) 98–101. Google Scholar
L.A. Végh and B. von Stengel, Oriented Euler complexes and signed perfect matchings. Tech. report (2012).
Živaljević, R., Oriented matroids and Ky Fan’s theorem. Combinatorica 30 (2010) 471–484. Google Scholar
Wolsey, L.A., Cubical Sperner lemmas as applications of generalized complementary pivoting. J. Comb. Theor. Ser. A 23 (1977) 78–87. Google Scholar