Polytope volume computation
Author:
Jim Lawrence
Journal:
Math. Comp. 57 (1991), 259271
MSC:
Primary 52B55
MathSciNet review:
1079024
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Abstract: A combinatorial form of Gram's relation for convex polytopes can be adapted for use in computing polytope volume. We present an algorithm for volume computation based on this observation. This algorithm is useful in finding the volume of a polytope given as the solution set of a system of linear inequalities, . As an illustration we compute a formula for the volume of a projective image of the ncube. From this formula we deduce that, when A and b have rational entries (so that the volume of P is also a rational number), the number of binary digits in the denominator of the volume cannot be bounded by a polynomial in the total number of digits in the numerators and denominators of entries of A and b . This settles a question posed by Dyer and Frieze.
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 [1]
 E. L. Allgower and P. M. Schmidt, Computing volumes of polyhedra, Math. Comp. 46 (1986), 171174. MR 815838 (87d:51036)
 [2]
 M. L. Balinski, An algorithm for finding all vertices of convex polyhedral sets, SIAM J. Appl. Math. 9 (1961), 7288. MR 0142057 (25:5451)
 [3]
 J. Bárány and Z. Füredi, Computing the volume is difficult, Proc. 18th Annual ACM Sympos. on Theory of Computing, 1986, pp. 442447.
 [4]
 D. L. Barrow and P. W. Smith, Spline notation applied to a volume problem, Amer. Math. Monthly 86 (1979), 5051. MR 1538918
 [5]
 J. Cohen and T. Hickey, Two algorithms for determining volumes of convex polyhedra, J. Assoc. Comput. Mach. 26 (1979), 401414. MR 535261 (80g:52006)
 [6]
 M. E. Dyer, The complexity of vertex enumeration methods, Math. Oper. Res. 8 (1983), 381402. MR 716120 (85j:68102)
 [7]
 M. E. Dyer and A. M. Frieze, On the complexity of computing the volume of a polyhedron, SIAM J. Comput. 17 (1988), 967974. MR 961051 (90f:68077)
 [8]
 H. G. Eggleston, Convexity, Cambridge Univ. Press, 1958. MR 0124813 (23:A2123)
 [9]
 G. Elekes, A geometric inequality and the complexity of computing volume, Discrete Comput. Geom. 1 (1986), 289292. MR 866364 (87k:68138)
 [10]
 S. I. Gass, Linear programming, McGrawHill, 1958. MR 0373586 (51:9786)
 [11]
 B. Grünbaum, Convex polytopes, Interscience, 1967.
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 G. H. Hardy and E. M. Wright, An introduction to the theory of numbers, fourth ed., Clarendon Press, 1968. MR 2445243 (2009i:11001)
 [13]
 J. B. Lasserre, An analytical expression and an algorithm for the volume of a convex polyhedron in , J. Optim. Theory Appl. 39 (1983), 363377. MR 703477 (84m:52018)
 [14]
 J. Lawrence, Valuations and polarity, Discrete Comput. Geom. 3 (1988), 307324. MR 947219 (90b:52001)
 [15]
 Y. T. Lee and A. A. G. Requicha, Algorithms for computing the volume and other integral properties of solids. I. Known methods and open issues, Comm. ACM 25 (1982), 635641. MR 680293 (84a:52003a)
 [16]
 , Algorithms for computing the volume and other integral properties of solids. II. A family of algorithms based on representation conversion and cellular approximation, Comm. ACM 25 (1982), 642650. MR 680294 (84a:52003b)
 [17]
 L. Lovász, An algorithmic theory of numbers, graphs and convexity, SIAM, 1986. MR 861822 (87m:68066)
 [18]
 P. McMullen, Anglesum relations for polyhedral sets, Mathematika 33 (1986), 173188. MR 882490 (88f:52009)
 [19]
 P. McMullen and G. C. Shephard, Convex polytopes and the upper bound conjecture, Cambridge Univ. Press, 1971. MR 0301635 (46:791)
 [20]
 M. Manas and J. Nedoma, Finding all vertices of a convex polyhedron, Numer. Math. 12 (1968), 226229. MR 0235705 (38:4008)
 [21]
 T. H. Mattheiss, An algorithm for determining irrelevant constraints and all vertices in systems of linear inequalities, Oper. Res. 21 (1973), 247260. MR 0437087 (55:10020)
 [22]
 T. H. Mattheis and D. S. Rubin, A survey and comparison of methods for finding all vertices of convex polyhedral sets, Math. Oper. Res. 5 (1980), 167185. MR 571811 (81e:90059)
 [23]
 G. C. Rota, The valuation ring, Studies in Pure Mathematics (L. Mirsky, ed.), Academic Press, New York, 1971.
 [24]
 A. Schrijver, Theory of linear and integer programming, Wiley, Chichester, 1986. MR 874114 (88m:90090)
 [25]
 G. C. Shephard, An elementary proof of Gram's theorem for convex polytopes, Canad. J. Math. 19 (1967), 12141217. MR 0225228 (37:822)
 [26]
 D. P. Shoemaker and T. C. Huang, A systematic method for calculating volumes of polyhedra corresponding to Brillouin zones, Acta Cryst. Sect. A 7 (1954), 249259.
 [27]
 T. Speevak, An efficient algorithm for obtaining the volume of a special kind of pyramid and application to convex polyhedra, Math. Comp. 46 (1986), 531536. MR 829623 (87d:52012)
 [28]
 J. Stoer and C. Witzgall, Convexity and optimization in finite dimensions, SpringerVerlag, 1970. MR 0286498 (44:3707)
 [29]
 L. G. Valiant, The complexity of enumeration and reliability problems, SIAM J. Comput. 8 (1979), 410421. MR 539258 (80f:68055)
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 B. Von Hohenbalken, Finding Simplicial subdivisions of polytopes, Math. Programming 21 (1981), 233234. MR 623842 (82h:52001)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718199110790242
PII:
S 00255718(1991)10790242
Article copyright:
© Copyright 1991 American Mathematical Society
