메뉴 건너뛰기




Volumn 27, Issue 2, 1998, Pages 356-400

On the complexity of computing mixed volumes

Author keywords

Approximation; Computation; Computational complexity; Computational convexity; Convex body; Deterministic algorithm; Mixed volumes; Parallelotope; Polynomial time algorithm; Polytope; Randomized algorithm; Volume; Zonotope; hardness

Indexed keywords


EID: 0001599636     PISSN: 00975397     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0097539794278384     Document Type: Article
Times cited : (57)

References (82)
  • 1
    • 0011332819 scopus 로고
    • On the theory of mixed volumes of convex bodies, II. New inequalities between mixed volumes and their applications
    • [Al37] in Russian
    • [Al37] A.D. ALEKSANDROV, On the theory of mixed volumes of convex bodies, II. New inequalities between mixed volumes and their applications, Math. Sb. N.S., 2 (1937), pp. 1205-1238 (in Russian).
    • (1937) Math. Sb. N.S. , vol.2 , pp. 1205-1238
    • Aleksandrov, A.D.1
  • 2
    • 0000864732 scopus 로고
    • On the theory of mixed volumes of convex bodies, IV. Mixed discriminants and mixed volumes
    • [Al38] in Russian
    • [Al38] A.D. ALEKSANDROV, On the theory of mixed volumes of convex bodies, IV. Mixed discriminants and mixed volumes, Math. Sb. N.S., 3 (1938), pp. 227-251 (in Russian).
    • (1938) Math. Sb. N.S. , vol.3 , pp. 227-251
    • Aleksandrov, A.D.1
  • 3
    • 84966253125 scopus 로고
    • Computing volumes of polyhedra
    • [AS86]
    • [AS86] E.L. ALLGOWER AND P.M. SCHMIDT, Computing volumes of polyhedra, Math. of Comp., 46 (1986), pp. 171-174.
    • (1986) Math. of Comp. , vol.46 , pp. 171-174
    • Allgower, E.L.1    Schmidt, P.M.2
  • 4
    • 85025262253 scopus 로고
    • Sampling and integration of near log-concave functions
    • [AK90] ACM, New York
    • [AK90] D. APPLEGATE AND R. KANNAN, Sampling and integration of near log-concave functions, in Proc. 23rd ACM Symp. on Theory of Computing, ACM, New York, 1990, pp. 156-163.
    • (1990) Proc. 23rd ACM Symp. on Theory of Computing , pp. 156-163
    • Applegate, D.1    Kannan, R.2
  • 5
    • 0000075710 scopus 로고
    • Computing the volume is difficult
    • [BF86] ACM, New York
    • [BF86] I. BÁRÁNY AND Z. FÜREDI, Computing the volume is difficult, in Proc. 18th ACM Symp. on Theory of Computing, ACM, New York, 1986, pp. 442-447. Reprinted in Discrete Comput. Geom., 2 (1987), pp. 319-326.
    • (1986) Proc. 18th ACM Symp. on Theory of Computing , pp. 442-447
    • Bárány, I.1    Füredi, Z.2
  • 6
    • 0000075710 scopus 로고
    • [BF86] I. BÁRÁNY AND Z. FÜREDI, Computing the volume is difficult, in Proc. 18th ACM Symp. on Theory of Computing, ACM, New York, 1986, pp. 442-447. Reprinted in Discrete Comput. Geom., 2 (1987), pp. 319-326.
    • (1987) Discrete Comput. Geom. , vol.2 , pp. 319-326
  • 7
    • 0000975439 scopus 로고
    • A polynomial time algorithm for counting integral points in polyhedra when the dimension is fixed
    • [Ba94]
    • [Ba94] A.I. BARVINOK, A polynomial time algorithm for counting integral points in polyhedra when the dimension is fixed, Math. Oper. Res., 19 (1994), pp. 769-779.
    • (1994) Math. Oper. Res. , vol.19 , pp. 769-779
    • Barvinok, A.I.1
  • 9
    • 34250406508 scopus 로고
    • The number of roots of a system of equations
    • [Be75]
    • [Be75] D.N BERNSHTEIN, The number of roots of a system of equations, Funct. Anal. Appl., 9 (1975), pp. 183-185.
    • (1975) Funct. Anal. Appl. , vol.9 , pp. 183-185
    • Bernshtein, D.N.1
  • 10
    • 0012589965 scopus 로고
    • Mixed volumes of polytopes
    • [Be92]
    • [Be92] U. BETKE, Mixed volumes of polytopes, Arch. Math., 58 (1992), pp. 388-391.
    • (1992) Arch. Math. , vol.58 , pp. 388-391
    • Betke, U.1
  • 11
    • 51249167675 scopus 로고
    • Approximating the volume of convex bodies
    • [BH93]
    • [BH93] U. BETKE AND M. HENK, Approximating the volume of convex bodies, Discrete Comput. Geom., 10 (1993), pp. 15-21.
    • (1993) Discrete Comput. Geom. , vol.10 , pp. 15-21
    • Betke, U.1    Henk, M.2
  • 12
    • 0004261538 scopus 로고
    • [BF34] Springer-Verlag, Berlin, 1934 (in German); reprinted: Chelsea, New York
    • [BF34] T. BONNESEN AND W. FENCHEL, Theorie der konvexen Körper, Springer-Verlag, Berlin, 1934 (in German); (reprinted: Chelsea, New York, 1948); Theory of Convex Bodies, BCS Associates, Moscow, Idaho, 1987, (in English).
    • (1948) Theorie der Konvexen Körper
    • Bonnesen, T.1    Fenchel, W.2
  • 13
    • 84971273934 scopus 로고    scopus 로고
    • BCS Associates, Moscow, Idaho, 1987, (in English)
    • [BF34] T. BONNESEN AND W. FENCHEL, Theorie der konvexen Körper, Springer-Verlag, Berlin, 1934 (in German); (reprinted: Chelsea, New York, 1948); Theory of Convex Bodies, BCS Associates, Moscow, Idaho, 1987, (in English).
    • Theory of Convex Bodies
  • 14
    • 0013157009 scopus 로고
    • Counting linear extensions is #P-complete
    • [BW92]
    • [BW92] G. BRIGHTWELL AND P. WINKLER, Counting linear extensions is #P-complete, Order, 8 (1992), pp. 225-242.
    • (1992) Order , vol.8 , pp. 225-242
    • Brightwell, G.1    Winkler, P.2
  • 15
    • 0002370076 scopus 로고
    • How hard is it to marry at random? (On the approximation of the permanent)
    • [Br86] ACM, New York
    • [Br86] A. Z. BRODER, How hard is it to marry at random? (On the approximation of the permanent), in Proc. 18th ACM Symp. on Theory of Computing, ACM, New York, 1986, pp. 50-58.
    • (1986) Proc. 18th ACM Symp. on Theory of Computing , pp. 50-58
    • Broder, A.Z.1
  • 16
    • 0005388856 scopus 로고
    • Determinantal identities: Gauss, Schur, Cauchy, Sylvester, Kronecker, Jacobi, Binet, Laplace, Muir, and Cayley
    • [BS83]
    • [BS83] R. A. BRUALDI AND H. SCHNEIDER, Determinantal identities: Gauss, Schur, Cauchy, Sylvester, Kronecker, Jacobi, Binet, Laplace, Muir, and Cayley, Linear Algebra Appl., 52-53 (1983), pp. 769-791.
    • (1983) Linear Algebra Appl. , vol.52-53 , pp. 769-791
    • Brualdi, R.A.1    Schneider, H.2
  • 18
    • 85010519644 scopus 로고
    • An efficient algorithm for the sparse mixed resultant
    • [CE93] Proc. 10th Intl. Symp. Appl. Algebra, Algebraic Alg., Error-Corr. Codes, Springer-Verlag, New York
    • [CE93] J. CANNY AND I.Z. EMIRIS, An efficient algorithm for the sparse mixed resultant, in Proc. 10th Intl. Symp. Appl. Algebra, Algebraic Alg., Error-Corr. Codes, Lecture Notes in Comput. Sci. 263, Springer-Verlag, New York, 1993, pp. 89-104.
    • (1993) Lecture Notes in Comput. Sci. , vol.263 , pp. 89-104
    • Canny, J.1    Emiris, I.Z.2
  • 19
    • 84968511777 scopus 로고
    • An optimality condition for determining the exact number of roots of a polynomial system
    • [CR91] Bonn, Germany, ACM, New York
    • [CR91] J. CANNY AND J.M. ROJAS, An optimality condition for determining the exact number of roots of a polynomial system, in Proc. ACM Intl. Symp. Algebraic Symbolic Comput., Bonn, Germany, ACM, New York, 1991, pp. 96-102.
    • (1991) Proc. ACM Intl. Symp. Algebraic Symbolic Comput. , pp. 96-102
    • Canny, J.1    Rojas, J.M.2
  • 20
    • 0018491818 scopus 로고
    • Two algorithms for determining volumes of convex polyhedra
    • [CH79]
    • [CH79] J. COHEN AND T. HICKEY, Two algorithms for determining volumes of convex polyhedra, J. Assoc. Comput. Mach., 26 (1979), pp. 401-414.
    • (1979) J. Assoc. Comput. Mach. , vol.26 , pp. 401-414
    • Cohen, J.1    Hickey, T.2
  • 21
    • 0000296729 scopus 로고
    • On lattices admitting unique Lagrange interpolation
    • [CY77]
    • [CY77] K.C. CHUNG AND T.H. YAO, On lattices admitting unique Lagrange interpolation, SIAM J. Numer. Anal., 14 (1977), pp. 735-743.
    • (1977) SIAM J. Numer. Anal. , vol.14 , pp. 735-743
    • Chung, K.C.1    Yao, T.H.2
  • 22
    • 0020155051 scopus 로고
    • Some NP-complete problems in linear programming
    • [CKM82]
    • [CKM82] R. CHANDRASEKARAN, S.N. KABADI, AND K.G. MURTY, Some NP-complete problems in linear programming, Oper. Res. Lett., 1 (1982), pp. 101-104.
    • (1982) Oper. Res. Lett. , vol.1 , pp. 101-104
    • Chandrasekaran, R.1    Kabadi, S.N.2    Murty, K.G.3
  • 23
    • 0024091845 scopus 로고
    • The complexity of computing the volume of a polyhedron
    • [DF88]
    • [DF88] M.E. DYER AND A.M. FRIEZE, The complexity of computing the volume of a polyhedron, SIAM J. Comput., 17 (1988), pp. 967-974.
    • (1988) SIAM J. Comput. , vol.17 , pp. 967-974
    • Dyer, M.E.1    Frieze, A.M.2
  • 24
    • 0003672936 scopus 로고
    • Computing the volume of convex bodies: A case where randomness provably helps
    • [DF91] Probabilistic Combinatorics and its Applications, Béla Bollobás, ed., American Mathematical Society, Providence, RI
    • [DF91] M.E. DYER AND A.M. FRIEZE, Computing the volume of convex bodies: a case where randomness provably helps, in Probabilistic Combinatorics and its Applications, Proceedings of Symposia in Applied Mathematics, Vol. 44, Béla Bollobás, ed., American Mathematical Society, Providence, RI, 1991, pp. 123-169.
    • (1991) Proceedings of Symposia in Applied Mathematics , vol.44 , pp. 123-169
    • Dyer, M.E.1    Frieze, A.M.2
  • 25
    • 0025720957 scopus 로고
    • A random polynomial time algorithm for approximating the volume of a convex body
    • [DFK91]
    • [DFK91] M.E. DYER, A.M. FRIEZE, AND R. KANNAN, A random polynomial time algorithm for approximating the volume of a convex body, J. Assoc. Comput. Mach., 38 (1991), pp. 1-17.
    • (1991) J. Assoc. Comput. Mach. , vol.38 , pp. 1-17
    • Dyer, M.E.1    Frieze, A.M.2    Kannan, R.3
  • 26
    • 0031210630 scopus 로고    scopus 로고
    • On Barvinok's algorithm for counting lattice points in fixed dimension
    • [DK97] to appear
    • [DK97] M.E. DYER AND R. KANNAN, On Barvinok's algorithm for counting lattice points in fixed dimension, Math. Oper. Res., 22 (1997), to appear.
    • (1997) Math. Oper. Res. , vol.22
    • Dyer, M.E.1    Kannan, R.2
  • 27
    • 0002167079 scopus 로고
    • Submodular functions, matroids, and certain polyhedra
    • [Ed70] R. Guy, H. Hanani, N. Sauer, and J. Schönheim, eds., Gordon and Breach, New York
    • [Ed70] J. EDMONDS, Submodular functions, matroids, and certain polyhedra, in Combinatorial Structures and their Applications, R. Guy, H. Hanani, N. Sauer, and J. Schönheim, eds., Gordon and Breach, New York, 1970, pp. 69-87.
    • (1970) Combinatorial Structures and Their Applications , pp. 69-87
    • Edmonds, J.1
  • 28
    • 84941843652 scopus 로고
    • Sur un probléme de géometrie diophantienne linéaire
    • [Eh67]
    • [Eh67] E. EHRHART, Sur un probléme de géometrie diophantienne linéaire, J. Reine Angew. Math., 226 (1967), pp. 1-29; 227 (1967), pp. 25-49.
    • (1967) J. Reine Angew. Math. , vol.226 , pp. 1-29
    • Ehrhart, E.1
  • 29
    • 84941843652 scopus 로고
    • [Eh67] E. EHRHART, Sur un probléme de géometrie diophantienne linéaire, J. Reine Angew. Math., 226 (1967), pp. 1-29; 227 (1967), pp. 25-49.
    • (1967) J. Reine Angew. Math. , vol.227 , pp. 25-49
  • 31
    • 0000348667 scopus 로고
    • Efficient incremental algorithms for the sparse resultant and the mixed volume
    • [EC95]
    • [EC95] I.Z. EMIRIS AND J.F. CANNY, Efficient incremental algorithms for the sparse resultant and the mixed volume, J. Symbolic Comput., 20 (1995), pp. 117-149.
    • (1995) J. Symbolic Comput. , vol.20 , pp. 117-149
    • Emiris, I.Z.1    Canny, J.F.2
  • 32
    • 0001782882 scopus 로고
    • Monomial bases and polynomial system solving
    • [ER94] Oxford, UK, ACM, New York
    • [ER94] I.Z. EMIRIS AND A. REGE, Monomial bases and polynomial system solving, in Proc. 8th ACM Intl. Symp. Algebraic Symbolic Comput. '94, Oxford, UK, ACM, New York, 1994, pp. 114-122.
    • (1994) Proc. 8th ACM Intl. Symp. Algebraic Symbolic Comput , vol.94 , pp. 114-122
    • Emiris, I.Z.1    Rege, A.2
  • 33
    • 0011484987 scopus 로고
    • Inégalités quadratique entre les volumes mixtes des corps convexes
    • [Fe36]
    • [Fe36] W. FENCHEL, Inégalités quadratique entre les volumes mixtes des corps convexes, C.R. Acad. Sci. Paris, 203 (1936), pp. 647-650.
    • (1936) C.R. Acad. Sci. Paris , vol.203 , pp. 647-650
    • Fenchel, W.1
  • 35
    • 0010135519 scopus 로고
    • Newton polytopes and the classical resultant and discriminant
    • [GKZ90]
    • [GKZ90] I.M. GELFAND, M.M. KAPRANOV, AND A.V. ZELEVINSKY, Newton polytopes and the classical resultant and discriminant, Adv. Math., 84 (1990), pp. 237-254.
    • (1990) Adv. Math. , vol.84 , pp. 237-254
    • Gelfand, I.M.1    Kapranov, M.M.2    Zelevinsky, A.V.3
  • 36
    • 85034485987 scopus 로고
    • Zonotopes and mixture management
    • [GV89] J.P. Penot, ed., ISNM87, Birkhäuser, Basel
    • [GV89] D. GIRAD AND P. VALENTIN, Zonotopes and mixture management, in New Methods in Optimization and their Industrial Uses, J.P. Penot, ed., ISNM87, Birkhäuser, Basel, 1989, pp. 57-71.
    • (1989) New Methods in Optimization and Their Industrial Uses , pp. 57-71
    • Girad, D.1    Valentin, P.2
  • 37
    • 0011428975 scopus 로고
    • On the complexity of some basic problems in computational convexity: II. Volume and mixed volumes
    • [GK94] T. Bisztriczky, P. McMullen, R. Schneider, and A. Ivic Weiss, eds., Kluwer, Boston
    • [GK94] P. GRITZMANN AND V. KLEE, On the complexity of some basic problems in computational convexity: II. Volume and mixed volumes, in Polytopes: Abstract, Convex and Computational, T. Bisztriczky, P. McMullen, R. Schneider, and A. Ivic Weiss, eds., Kluwer, Boston, 1994, pp. 373-466.
    • (1994) Polytopes: Abstract, Convex and Computational , pp. 373-466
    • Gritzmann, P.1    Klee, V.2
  • 38
    • 51249169029 scopus 로고
    • Largest k-simplices in d-polytopes
    • [GKL95]
    • [GKL95] P. GRITZMANN, V. KLEE, AND D. LARMAN, Largest k-simplices in d-polytopes, Discrete Comput. Geom., 13 (1995), pp. 477-515.
    • (1995) Discrete Comput. Geom. , vol.13 , pp. 477-515
    • Gritzmann, P.1    Klee, V.2    Larman, D.3
  • 39
    • 0001734192 scopus 로고
    • Minkowski addition of polytopes: Computational complexity and applications to Gröbner bases
    • [GS93]
    • [GS93] P. GRITZMANN AND B. STURMFELS, Minkowski addition of polytopes: computational complexity and applications to Gröbner bases, SIAM J. Discrete Math., 6 (1993), pp. 246-269.
    • (1993) SIAM J. Discrete Math. , vol.6 , pp. 246-269
    • Gritzmann, P.1    Sturmfels, B.2
  • 40
    • 0000838163 scopus 로고
    • Lattice points
    • [GW93] P.M. Gruber and J.M. Wills, eds., Elsevier, Amsterdam
    • [GW93] P. GRITZMANN AND J.M. WILLS, Lattice points, in Handbook of Convex Geometry, P.M. Gruber and J.M. Wills, eds., Elsevier, Amsterdam, 1993, pp. 765-798.
    • (1993) Handbook of Convex Geometry , pp. 765-798
    • Gritzmann, P.1    Wills, J.M.2
  • 42
    • 84968520678 scopus 로고
    • A polyhedral method for solving sparse polynomial systems
    • [HS95]
    • [HS95] B. HUBER AND B. STURMFELS, A polyhedral method for solving sparse polynomial systems, Math. Comput., 64 (1995), pp. 1541-1555.
    • (1995) Math. Comput. , vol.64 , pp. 1541-1555
    • Huber, B.1    Sturmfels, B.2
  • 43
    • 0024904712 scopus 로고
    • Approximating the permanent
    • [JS89]
    • [JS89] M. R. JERRUM AND A. J. SINCLAIR, Approximating the permanent, SIAM J. Comput., 18 (1989), pp. 1149-1178.
    • (1989) SIAM J. Comput. , vol.18 , pp. 1149-1178
    • Jerrum, M.R.1    Sinclair, A.J.2
  • 44
    • 38249043088 scopus 로고
    • Random generation of combinatorical structures from a uniform distribution
    • [JVV86]
    • [JVV86] M. R. JERRUM, L. G. VALIANT, AND V.V. VAZIRANI, Random generation of combinatorical structures from a uniform distribution, Theoret. Comput. Sci., 43 (1986), pp. 169-188.
    • (1986) Theoret. Comput. Sci. , vol.43 , pp. 169-188
    • Jerrum, M.R.1    Valiant, L.G.2    Vazirani, V.V.3
  • 45
    • 0001951331 scopus 로고
    • A catalog of complexity classes
    • [Jo90] J. van Leeuwen, ed., Elsevier and M.I.T. Press, Amsterdam and Cambridge, MA
    • [Jo90] D.S. JOHNSON, A catalog of complexity classes, in Handbook of Theoretical Computer Science, vol. A: Algorithms and Complexity, J. van Leeuwen, ed., Elsevier and M.I.T. Press, Amsterdam and Cambridge, MA, 1990 , pp. 67-161.
    • (1990) Handbook of Theoretical Computer Science, vol. A: Algorithms and Complexity , vol.A , pp. 67-161
    • Johnson, D.S.1
  • 47
    • 0027574032 scopus 로고
    • A Monte-Carlo algorithm for estimationg the permanent
    • [KKLLL93]
    • [KKLLL93] N. KARMARKAR, R. KARP, R. LIPTON, L. LOVÁSZ, AND M. LUBY, A Monte-Carlo algorithm for estimationg the permanent, SIAM J. Comput., 22 (1993), pp. 284-293.
    • (1993) SIAM J. Comput. , vol.22 , pp. 284-293
    • Karmarkar, N.1    Karp, R.2    Lipton, R.3    Lovász, L.4    Luby, M.5
  • 48
    • 0040174181 scopus 로고
    • Complexity of polytope volume computation
    • [Kh93] J. Pach, ed., Springer-Verlag, Berlin
    • [Kh93] L.G. KHACHIYAN, Complexity of polytope volume computation, in New Trends in Discrete and Computational Geometry, J. Pach, ed., Springer-Verlag, Berlin, 1993, pp. 91-101.
    • (1993) New Trends in Discrete and Computational Geometry , pp. 91-101
    • Khachiyan, L.G.1
  • 49
    • 0042778011 scopus 로고
    • On the complexity of approximating the maximal inscribed ellipsoid for a polytope
    • [KT93]
    • [KT93] L.G. KHACHIYAN AND M.J. TODD, On the complexity of approximating the maximal inscribed ellipsoid for a polytope, Math. Programming, 61 (1993), pp. 137-159.
    • (1993) Math. Programming , vol.61 , pp. 137-159
    • Khachiyan, L.G.1    Todd, M.J.2
  • 50
    • 84966243783 scopus 로고
    • Polytope volume computation
    • [La91]
    • [La91] J. LAWRENCE, Polytope volume computation, Math. Comput., 57 (1991), pp. 259-271.
    • (1991) Math. Comput. , vol.57 , pp. 259-271
    • Lawrence, J.1
  • 51
    • 0030163190 scopus 로고    scopus 로고
    • Counting affine roots of polynomial systems via pointed Newton polytopes
    • [LRW96]
    • [LRW96] T.Y. Li, J.M. ROJAS, AND X. WANG, Counting affine roots of polynomial systems via pointed Newton polytopes, J. Complexity, 12 (1996), pp. 116-133.
    • (1996) J. Complexity , vol.12 , pp. 116-133
    • Li, T.Y.1    Rojas, J.M.2    Wang, X.3
  • 52
    • 0006217743 scopus 로고
    • Random walks on graphs: A survey
    • [Lo95] D. Miklós, V. T. Sós and T. Szönyi, eds., Bolyai Soc. Math. Stud. 2, János Bolyai Math. Soc., Budapest
    • [Lo95] L. LOVÁSZ, Random walks on graphs: A survey, in Combinatorics: Paul Erdös is 80, Vol. 2, D. Miklós, V. T. Sós and T. Szönyi, eds., Bolyai Soc. Math. Stud. 2, János Bolyai Math. Soc., Budapest, 1995, pp. 353-397.
    • (1995) Combinatorics: Paul Erdös Is 80 , vol.2 , pp. 353-397
    • Lovász, L.1
  • 53
    • 0040174183 scopus 로고
    • The mixing rate of Markov chains, an isoperimetric inequality and computing the volume
    • [LS90] IEEE Computer Society Press, Los Alamitos, CA
    • [LS90] L. LOVÁSZ AND M. SIMONIVITS, The mixing rate of Markov chains, an isoperimetric inequality and computing the volume, in Proc. IEEE 31st Annual Symp. Found. Comput. Sci., IEEE Computer Society Press, Los Alamitos, CA, 1990, pp. 364-355.
    • (1990) Proc. IEEE 31st Annual Symp. Found. Comput. Sci. , pp. 364-1355
    • Lovász, L.1    Simonivits, M.2
  • 54
    • 84990634606 scopus 로고
    • Random walks in a convex body and an improved volume algorithm
    • [LS93]
    • [LS93] L. LOVÁSZ AND M. SIMONIVITS, Random walks in a convex body and an improved volume algorithm, Random Structures Algorithms, 4 (1993), pp. 359-412.
    • (1993) Random Structures Algorithms , vol.4 , pp. 359-412
    • Lovász, L.1    Simonivits, M.2
  • 55
    • 84976163829 scopus 로고
    • The maximum number of faces of a convex polytope
    • [Mc70]
    • [Mc70] P. MCMULLEN, The maximum number of faces of a convex polytope, Mathematika, 17 (1970), pp. 179-184.
    • (1970) Mathematika , vol.17 , pp. 179-184
    • Mcmullen, P.1
  • 56
    • 1542451806 scopus 로고
    • On the condition number of Lagrangian numerical differentiation
    • [MM85]
    • [MM85] G. MIEL AND R. MOONEY, On the condition number of Lagrangian numerical differentiation, Appl. Math. Comput., 16 (1985), pp. 241-252.
    • (1985) Appl. Math. Comput. , vol.16 , pp. 241-252
    • Miel, G.1    Mooney, R.2
  • 57
    • 0039480862 scopus 로고
    • Theorie der konvexen Körper, insbesondere Begründung ihres Oberflächenbegriffs
    • [Mi11] Leipzig, Berlin
    • [Mi11] H. MINKOWSKI, Theorie der konvexen Körper, insbesondere Begründung ihres Oberflächenbegriffs, in Collected Works Vol. II, Leipzig, Berlin, 1911, pp. 131-229.
    • (1911) Collected Works , vol.2 , pp. 131-229
    • Minkowski, H.1
  • 58
    • 1542341905 scopus 로고
    • Computing the volume of a zonotope
    • [Mo89]
    • [Mo89] H. L. MONTGOMERY, Computing the volume of a zonotope, Amer. Math. Monthly, 96 (1989), p. 431.
    • (1989) Amer. Math. Monthly , vol.96 , pp. 431
    • Montgomery, H.L.1
  • 59
    • 84966255056 scopus 로고
    • Two formulas for the general multivariate polynomial which interpolates a regular grid on a simplex
    • [Ol86]
    • [Ol86] C. OLMSTED, Two formulas for the general multivariate polynomial which interpolates a regular grid on a simplex, Math. Comput., 47 (1986), pp. 275-284.
    • (1986) Math. Comput. , vol.47 , pp. 275-284
    • Olmsted, C.1
  • 60
    • 0342542805 scopus 로고
    • On recognizing integer polyhedra
    • [PY90]
    • [PY90] C.H. PAPADIMITRIOU AND M. YANNAKAKIS, On recognizing integer polyhedra, Combinatorica, 10 (1990), pp. 107-109.
    • (1990) Combinatorica , vol.10 , pp. 107-109
    • Papadimitriou, C.H.1    Yannakakis, M.2
  • 61
    • 51249161636 scopus 로고
    • Product formulas for sparse resultants
    • [PS93]
    • [PS93] P. PEDERSEN AND B. STURMFELS, Product formulas for sparse resultants, Math. Z., 214 (1993), pp. 377-396.
    • (1993) Math. Z , vol.214 , pp. 377-396
    • Pedersen, P.1    Sturmfels, B.2
  • 62
    • 0016568017 scopus 로고
    • Optimally stable Lagrangian numerical differentiation
    • [Ri75]
    • [Ri75] T.J. RIVLIN, Optimally stable Lagrangian numerical differentiation, SIAM J. Numer. Anal., 12 (1975), pp. 712-725.
    • (1975) SIAM J. Numer. Anal. , vol.12 , pp. 712-725
    • Rivlin, T.J.1
  • 64
    • 0028517599 scopus 로고
    • A convex geometric approach to counting the roots of a polynomial system
    • [Ro94]
    • [Ro94] J.M. ROJAS, A convex geometric approach to counting the roots of a polynomial system, Theoret. Comput. Sci., 133 (1994), pp. 105-140.
    • (1994) Theoret. Comput. Sci. , vol.133 , pp. 105-140
    • Rojas, J.M.1
  • 65
    • 85034462593 scopus 로고    scopus 로고
    • Toric intersection for affine root counting
    • [Ro97] to appear
    • [Ro97] J.M. ROJAS, Toric intersection for affine root counting, J. Pure Appl. Math., 133 (1997), to appear.
    • (1997) J. Pure Appl. Math. , vol.133
    • Rojas, J.M.1
  • 66
    • 84966261969 scopus 로고
    • Some problems in optimally stable Lagrangian differentiation
    • [Sa74]
    • [Sa74] H.E. SALZER, Some problems in optimally stable Lagrangian differentiation, Math. Comput., 28 (1974), pp. 1105-1115.
    • (1974) Math. Comput. , vol.28 , pp. 1105-1115
    • Salzer, H.E.1
  • 67
    • 0001855015 scopus 로고
    • Mixed volumes
    • [Sa93] P.M. Gruber and J.M. Wills, eds., Elsevier, Amsterdam
    • [Sa93] J.R. SANGWINE-YAGER, Mixed volumes, in Handbook of Convex Geometry, P.M. Gruber and J.M. Wills, eds., Elsevier, Amsterdam, 1993, pp. 43-72.
    • (1993) Handbook of Convex Geometry , pp. 43-72
    • Sangwine-Yager, J.R.1
  • 68
    • 0003333144 scopus 로고
    • Convex Bodies: The Brunn-Minkowski Theory
    • [Sc93] Cambridge University Press, Cambridge, MA
    • [Sc93] R. SCHNEIDER, Convex Bodies: The Brunn-Minkowski Theory, Encyclopedia of Mathematics and its Applications, Vol. 44, Cambridge University Press, Cambridge, MA, 1993.
    • (1993) Encyclopedia of Mathematics and Its Applications , vol.44
    • Schneider, R.1
  • 69
    • 0013522153 scopus 로고
    • Polytopes and Brunn-Minkowski theory
    • [Sc94] T. Bisztriczky, P. McMullen, R. Schneider, and A. Ivic Weiss, eds., Kluwer, Boston
    • [Sc94] R. SCHNEIDER, Polytopes and Brunn-Minkowski theory, in Polytopes: Abstract, Convex and Computational, T. Bisztriczky, P. McMullen, R. Schneider, and A. Ivic Weiss, eds., Kluwer, Boston, 1994, pp. 273-300.
    • (1994) Polytopes: Abstract, Convex and Computational , pp. 273-300
    • Schneider, R.1
  • 71
    • 0000613625 scopus 로고
    • Combinatorial properties of associated zonotopes
    • [Sh74]
    • [Sh74] G.C. SHEPHARD, Combinatorial properties of associated zonotopes, Canad. J. Math., 26 (1974), pp. 302-321.
    • (1974) Canad. J. Math. , vol.26 , pp. 302-321
    • Shephard, G.C.1
  • 72
    • 0024699930 scopus 로고
    • Approximate counting, uniform generation and rapidly mixing Markov chains
    • [SJ89]
    • [SJ89] A. SINCLAIR AND M. JERRUM, Approximate counting, uniform generation and rapidly mixing Markov chains, inform. Comput., 82 (1989), pp. 93-133.
    • (1989) Inform. Comput. , vol.82 , pp. 93-133
    • Sinclair, A.1    Jerrum, M.2
  • 73
    • 0002876380 scopus 로고
    • Two combinatorial applications of the Aleksandrov-Fenchel inequalities
    • [St81]
    • [St81] R.M. STANLEY, Two combinatorial applications of the Aleksandrov-Fenchel inequalities, J. Combin. Theory Ser. A, 17 (1981), pp. 56-65.
    • (1981) J. Combin. Theory Ser. A , vol.17 , pp. 56-65
    • Stanley, R.M.1
  • 74
    • 0003652674 scopus 로고
    • [St86] Wadsworth & Brooks/Cole, Pacific Grove, CA
    • [St86] R.M. STANLEY, Enumerative Combinatorics, Vol. 1, Wadsworth & Brooks/Cole, Pacific Grove, CA, 1986.
    • (1986) Enumerative Combinatorics , vol.1
    • Stanley, R.M.1
  • 75
    • 0000657549 scopus 로고
    • A zonotope associated with graphical degree sequences
    • [St91] P. Gritzmann and B. Sturmfels, eds., Amer. Math. Soc. and Assoc. Comput. Mach., Providence, RI
    • [St91] R. STANLEY, A zonotope associated with graphical degree sequences, in Applied Geometry and Discrete Mathematics: The Victor Klee Festschrift, P. Gritzmann and B. Sturmfels, eds., Amer. Math. Soc. and Assoc. Comput. Mach., Providence, RI, 1991, pp. 555-570.
    • (1991) Applied Geometry and Discrete Mathematics: The Victor Klee Festschrift , pp. 555-570
    • Stanley, R.1
  • 76
    • 49249151236 scopus 로고
    • The complexity of computing the permanent
    • [Va77]
    • [Va77] L.G. VALIANT, The complexity of computing the permanent, Theoret. Comput. Sci., 8 (1977), pp. 189-201.
    • (1977) Theoret. Comput. Sci. , vol.8 , pp. 189-201
    • Valiant, L.G.1
  • 77
    • 0000142982 scopus 로고
    • The complexity of enumeration and reliability problems
    • [Va79]
    • [Va79] L.G. VALIANT, The complexity of enumeration and reliability problems, SIAM J. Comput., 8 (1979), pp. 410-421.
    • (1979) SIAM J. Comput. , vol.8 , pp. 410-421
    • Valiant, L.G.1
  • 78
    • 21144472893 scopus 로고
    • Nonlinear reduction for solving deficient polynomial systems by continuation methods
    • [VC92]
    • [VC92] J. VERSCHELDE AND R. COOLS, Nonlinear reduction for solving deficient polynomial systems by continuation methods, Numer. Math., 63 (1992), pp. 263-282.
    • (1992) Numer. Math. , vol.63 , pp. 263-282
    • Verschelde, J.1    Cools, R.2
  • 79
    • 0001917438 scopus 로고
    • Symmetric Newton polytopes for solving sparse polynomial systems
    • [VG95]
    • [VG95] J. VERSCHELDE AND K. GATERMANN, Symmetric Newton polytopes for solving sparse polynomial systems, Adv. Appl. Math, 16 (1995), pp. 95-127.
    • (1995) Adv. Appl. Math , vol.16 , pp. 95-127
    • Verschelde, J.1    Gatermann, K.2
  • 80
    • 0028449799 scopus 로고
    • Homotopies exploiting Newton polytopes for solving sparse polynomial systems
    • [VVC94]
    • [VVC94] J. VERSCHELDE, P. VERLINDEN, AND R. COOLS, Homotopies exploiting Newton polytopes for solving sparse polynomial systems, SIAM J. Numer. Anal., 31 (1994), pp. 915-930.
    • (1994) SIAM J. Numer. Anal. , vol.31 , pp. 915-930
    • Verschelde, J.1    Verlinden, P.2    Cools, R.3
  • 81
    • 0004045430 scopus 로고
    • [We76] Academic Press, London
    • [We76] D.J.A. WELSH, Matroid Theory, Academic Press, London, 1976.
    • (1976) Matroid Theory
    • Welsh, D.J.A.1
  • 82
    • 0010992572 scopus 로고
    • Unimodular matroids
    • [Wh87] N. White, eds., Cambridge University Press, Cambridge, MA
    • [Wh87] N. WHITE, Unimodular matroids, in Combinatorial Geometries, N. White, eds., Cambridge University Press, Cambridge, MA, 1987, pp. 40-52.
    • (1987) Combinatorial Geometries , pp. 40-52
    • White, N.1


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.