A practical volume algorithm

Mathematical Programming Computation

Volumn 8, Issue 2, 2016, Pages 133-160

  Cousins, Ben ^a   Vempala, Santosh ^a  

^a Georgia Institute of Technology   (United States)

Author keywords

Convex geometry; Hit and run; Random walks; Simulated annealing; Volume computation

Indexed keywords

SIMULATED ANNEALING;

COMMODITY HARDWARE; CONVEX GEOMETRY; HIT-AND-RUN; IMPROVE PERFORMANCE; LINEAR INEQUALITIES; RANDOM WALK; ROUNDING ALGORITHM; VOLUME ALGORITHMS;

ALGORITHMS;

EID: 84971278966     PISSN: 18672949     EISSN: 18672957     Source Type: Journal    
DOI: 10.1007/s12532-015-0097-z     Document Type: Article

References (31)

1. Genz, A., Bretz, F.: Computation of Multivariate Normal and t Probabilities. Springer, New York (2009)
2. Iyengar, S.: Evaluation of normal probabilities of symmetric regions. SIAM J. Sci. Stat. Comput. 9, 812–837 (1988)
3. Kannan, R., Li, G.: Sampling according to the multivariate normal density. In: FOCS ’96: Proceedings of the 37th Annual Symposium on Foundations of Computer Science, Washington, DC, USA, p. 204. IEEE Computer Society (1996)
4. Martynov, G.: Evaluation of the normal distribution function. J. Soviet Math 77, 1857–1875 (1980)
5. Schellenberger, J., Palsson, B.: Use of randomized sampling for analysis of metabolic networks. J. Biol. Chem. 284(9), 5457–5461 (2009)
6. Somerville, P.N.: Numerical computation of multivariate normal and multivariate-t probabilities over convex regions. J. Comput. Graph. Stat. 7, 529–545 (1998)
7. Dyer, M.E., Frieze, A.M., Kannan, R.: A random polynomial time algorithm for approximating the volume of convex bodies. In: STOC, pp. 375–381 (1989)
8. Dyer, M.E., Frieze, A.M., Kannan, R.: A random polynomial-time algorithm for approximating the volume of convex bodies. J. ACM 38(1), 1–17 (1991)
9. 4) volume algorithm. J. Comput. Syst. Sci. 72(2), 392–417 (2006)
10. Cousins, B., Vempala, S.: A cubic algorithm for computing Gaussian volume. In: SODA, pp. 1215–1228 (2014)
11. 3) volume algorithm. In: STOC, pp. 539–548 (2015)
12. 4) volume algorithm. Eur. J. Oper. Res. 216 (2012)
13. Cousins, B., Vempala, S.: Volume computation of convex bodies. MATLAB File Exchange. http://www.mathworks.com/matlabcentral/fileexchange/43596-volume-computation-of-convex-bodies (2013)
14. Emiris, I., Fisikopoulos, V.: Efficient random-walk methods for approximating polytope volume. In: Proceedings of the 30th Annual Symposium on Computational Geometry, p. 318. ACM (2014)
15. Lovász, L., Vempala, S.: Hit-and-run from a corner. SIAM J. Comput. 35, 985–1005 (2006)
16. Bourgain, J.: Random points in isotropic convex sets. Convex Geom. Anal. 34, 53–58 (1996)
17. Rudelson, M.: Random vectors in the isotropic position. J. Funct. Anal. 164, 60–72 (1999)
18. Adamczak, R., Litvak, A., Pajor, A., Tomczak-Jaegermann, N.: Quantitative estimates of the convergence of the empirical covariance matrix in log-concave ensembles. J. Am. Math. Soc. 23, 535–561 (2010)
19. Štefankovič, D., Vempala, S., Vigoda, E.: Adaptive simulated annealing: a near-optimal connection between sampling and counting. J. ACM 56(3), 18–36 (2009)
20. 5) volume algorithm for convex bodies. Random Struct. Algorithms 11, 1–50 (1997)
21. Gillman, D.: A Chernoff bound for random walks on expander graphs. In: FOCS, pp. 680–691. IEEE Comput. Soc. Press, Los Alamitos (1993)
22. Kannan, R., Lovász, L., Simonovits, M.: Isoperimetric problems for convex bodies and a localization lemama. Discrete Comput. Geom. 13, 541–559 (1995)
23. Canfield, E.R., McKay, B.: The asymptotic volume of the Birkhoff polytope. Online J. Anal. Comb. 4, 4 (2009)
24. Chan, C., Robbins, D., Yuen, D.: On the volume of a certain polytope. Exp. Math. 9(1), 91–99 (2000)
25. De Loera, J.A., Liu, F., Yoshida, R.: A generating function for all semi-magic squares and the volume of the Birkhoff polytope. J. Algebraic Comb. 30(1), 113–139 (2009)
26. Pak, I.: Four questions on Birkhoff polytope. Ann. Comb. 4(1), 83–90 (2000)
27. Zeilberger, D.: Proof of a conjecture of Chan, Robbins, and Yuen. Electron. Trans. Numer. Anal. 9, 147–148 (electronic) (1999) [Orthogonal polynomials: numerical and symbolic algorithms (Leganés, 1998)]
28. Beck, M., Pixton, D.: The Ehrhart polynomial of the Birkhoff polytope. Discrete Comput. Geom. 30(4), 623–637 (2003)
29. Chan, C., Robbins, D.: On the volume of the polytope of doubly stochastic matrices. Exp. Math. 8(3), 291–300 (1999)
30. Dyer, M., Gritzmann, P., Hufnagel, A.: On the complexity of computing mixed volumes. SIAM J. Comput. 27(2), 356–400 (1998)
31. Cousins, B., Vempala, S.: Volume computation and sampling. http://www.cc.gatech.edu/~bcousins/volume.html (2013)