An Overview of Mathematical Issues Arising in the Geometric Complexity Theory Approach to $\mathbf{VP}\neq\mathbf{VNP}$

Abstract

We discuss the geometry of orbit closures and the asymptotic behavior of Kronecker coefficients in the context of the geometric complexity theory program to prove a variant of Valiant's algebraic analogue of the $\mathbf{P}\neq\mathbf{NP}$ conjecture. We also describe the precise separation of complexity classes that their program proposes to demonstrate.

MSC codes

  1. 68Q17
  2. 20B30
  3. 14L24

Keywords

  1. geometric complexity theory
  2. $\mathbf{P}$ vs. $\mathbf{NP}$
  3. geometric invariant theory
  4. orbit closure
  5. Kronecker coefficient
  6. determinant
  7. permanent

Get full access to this article

View all available purchase options and get full access to this article.

References

1.
C. M. Ballantine and R. C. Orellana, A combinatorial interpretation for the coefficients in the Kronecker product $s_ {(n-p,p)}\ast s_\lambda$, Sém. Lothar. Combin., 54A (2005/07), B54Af.
2.
A. Berenstein and R. Sjamaar, Coadjoint orbits, moment polytopes, and the Hilbert-Mumford criterion, J. Amer. Math. Soc., 13 (2000), pp. 433–466.
3.
S. J. Berkowitz, On computing the determinant in small parallel time using a small number of processors, Inform. Process. Lett., 18 (1984), pp. 147–150.
4.
P. Botta, Linear transformations that preserve the permanent, Proc. Amer. Math. Soc., 18 (1967), pp. 566–569.
5.
R. P. Brent, The complexity of multiprecision arithmetic, in Proceedings of the Seminar on Complexity of Computational Problem Solving, Brisbane, Australia, 1975, pp. 126–165.
6.
E. Briand, R. Orellana, and M. Rosas, Reduced Kronecker coefficients and counter-examples to Mulmuley's strong saturation conjecture SH, Comput. Complexity, 18 (2008), pp. 577–600.
7.
M. Brion, Stable properties of plethysm: On two conjectures of Foulkes, Manuscripta Math., 80 (1993), pp. 347–371.
8.
P. Bürgisser, Completeness and Reduction in Algebraic Complexity Theory, Algorithms Comput. Math. 7, Springer-Verlag, Berlin, 2000.
9.
P. Bürgisser, Cook's versus Valiant's hypothesis, Theoret. Comput. Sci., 235 (2000), pp. 71–88.
10.
P. Bürgisser, The complexity of factors of multivariate polynomials, Found. Comput. Math., 4 (2004), pp. 369–396.
11.
P. Bürgisser, M. Christandl, and C. Ikenmeyer, Nonvanishing of Kronecker coefficients for rectangular shapes, Adv. Math., 227 (2011), pp. 2082–2091.
12.
P. Bürgisser, M. Christandl, and C. Ikenmeyer, Even partitions in plethysms, J. Algebra, 328 (2011), pp. 322–329.
13.
P. Bürgisser and C. Ikenmeyer, Geometric complexity theory and tensor rank, in Proceedings of the Forty-Third Annual ACM Symposium on Theory of Computing, 2011, pp. 509–518.
14.
M. Christandl, A. W. Harrow, and G. Mitchison, Nonzero Kronecker coefficients and what they tell us about spectra, Comm. Math. Phys., 270 (2007), pp. 575–585.
15.
D. Eisenbud, Commutative Algebra: With a View toward Algebraic Geometry, Grad. Texts in Math. 150, Springer-Verlag, New York, 1995.
16.
D. Eisenbud and J. Harris, Vector spaces of matrices of low rank, Adv. in Math., 70 (1988), pp. 135–155.
17.
M. Franz, Moment polytopes of projective G-varieties and tensor products of symmetric group representations, J. Lie Theory, 12 (2002), pp. 539–549.
18.
G. Frobenius, Über die Darstellung der endlichen Gruppen durch lineare Substitutionen, Sitzungsber Deutsch. Akad. Wiss. Berlin, (1897), pp. 994–1015.
19.
W. Fulton and J. Harris, Representation Theory: A First Course, Grad. Texts in Math. 129, Springer-Verlag, New York, 1991.
20.
J. von zur Gathen, Feasible arithmetic computations: Valiant's hypothesis, J. Symbolic Comput., 4 (1987), pp. 137–172.
21.
D. A. Gay, Characters of the Weyl group of $SU(n)$ on zero weight spaces and centralizers of permutation representations, Rocky Mountain J. Math., 6 (1976), pp. 449–455.
22.
A. È. Guterman and A. V. Mikhalëv, General algebra and linear mappings that preserve matrix invariants, Fundam. Prikl. Mat., 9 (2003), pp. 83–101.
23.
R. Hartshorne, Ample Subvarieties of Algebraic Varieties, Notes written in collaboration with C. Musili, Lecture Notes in Math. 156, Springer-Verlag, Berlin, 1970.
24.
D. Hilbert, Über die vollen Invariantensysteme, Math. Ann., 42 (1893), pp. 313–373.
25.
H. Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero. I, II, Ann. of Math. (2), 79 (1964), pp. 109–203; ibid., 79 (1964), pp. 205–326.
26.
R. Howe, Perspectives on invariant theory: Schur duality, multiplicity-free actions and beyond, in The Schur Lectures (1992) (Tel Aviv), Israel Math. Conf. Proc. 8, Bar-Ilan University, Ramat Gan, Israel, 1995, pp. 1–182.
27.
E. Kaltofen and P. Koiran, Expressing a fraction of two determinants as a determinant, in Proceedings of ISSAC '08, ACM, New York, 2008, pp. 141–146.
28.
G. R. Kempf, Instability in invariant theory, Ann. of Math. (2), 108 (1978), pp. 299–316.
29.
A. Klyachko, Quantum marginal problem and representations of the symmetric group, arXiv:quant-ph/0409113v1, 2004.
30.
A. W. Knapp, Lie Groups Beyond an Introduction, 2nd ed., Progr. Math. 140, Birkhäuser Boston, Boston, MA, 2002.
31.
B. Kostant, Lie group representations on polynomial rings, Amer. J. Math., 85 (1963), pp. 327–404.
32.
H. Kraft, Geometrische Methoden in der Invariantentheorie, Aspects Math. D1, Friedr. Vieweg & Sohn, Braunschweig, 1984.
33.
S. Kumar, Geometry of orbits of permanents and determinants, arXiv:1007.1695v1, 2010.
34.
S. Kumar, Kac-Moody Groups, Their Flag Varieties and Representation Theory, Progr. Math. 204, Birkhäuser Boston, Boston, MA, 2002.
35.
J. M. Landsberg, L. Manivel, and N. Ressayre, Hypersurfaces with degenerate duals and the geometric complexity theory program, Comment. Math. Helv., to appear.
36.
I. G. Macdonald, Symmetric Functions and Hall Polynomials, 2nd ed., Oxford Math. Monogr., The Clarendon Press, Oxford University Press, New York, 1995.
37.
K. Magaard and G. Malle, Irreducibility of alternating and symmetric squares, Manuscripta Math., 95 (1998), pp. 169–180.
38.
G. Malod and N. Portier, Characterizing Valiant's algebraic complexity classes, J. Complexity, 24 (2008), pp. 16–38.
39.
L. Manivel, Applications de Gauss et pléthysme, Ann. Inst. Fourier (Grenoble), 47 (1997), pp. 715–773.
40.
L. Manivel, Gaussian maps and plethysm, in Algebraic Geometry (Catania, 1993/Barcelona, 1994), Lecture Notes in Pure and Appl. Math. 200, Dekker, New York, 1998, pp. 91–117.
41.
L. Manivel, A note on certain Kronecker coefficients, Proc. Amer. Math. Soc., 138 (2010), pp. 1–7.
42.
M. Marcus and F. C. May, The permanent function, Canad. J. Math., 14 (1962), pp. 177–189.
43.
Y. Matsushima, Espaces homogènes de Stein des groupes de Lie complexes, Nagoya Math. J., 16 (1960), pp. 205–218.
44.
H. Minc, Permanents, in Encyclopedia of Mathematics and its Applications, Vol. 6, Encyclopedia Math. Appl. 9999, Addison–Wesley, Reading, MA, 1978.
45.
K. D. Mulmuley and M. Sohoni, Geometric complexity theory I: An approach to the P vs. NP and related problems, SIAM J. Comput., 31 (2001), pp. 496–526.
46.
K. D. Mulmuley and M. Sohoni, Geometric complexity theory II: Towards explicit obstructions for embeddings among class varieties, SIAM J. Comput., 38 (2008), pp. 1175–1206.
47.
K. D. Mulmuley and M. Sohoni, Geometric complexity theory III: On deciding positivity of Littlewood-Richardson coefficients, arXiv:cs/0501076v1, 2005.
48.
K. D. Mulmuley and M. Sohoni, Geometric Complexity Theory IV: Quantum Group for the Kronecker Problem, preprint, Department of Computer Science, The University of Chicago, Chicago, IL.
49.
K. D. Mulmuley and H. Narayaran, Geometric Complexity Theory V: On Deciding Nonvanishing of a Generalized Littlewood-Richardson Coefficient, Technical report TR-2007-05, Department of Computer Science, The University of Chicago, Chicago, IL, 2007.
50.
K. D. Mulmuley, Geometric Complexity Theory VI: The Flip via Saturated and Positive Integer Programming in Representation Theory and Algebraic Geometry, Technical report TR-2007-04, Department of Computer Science, The University of Chicago, Chicago, IL, 2007.
51.
K. D. Mulmuley, Geometric Complexity Theory VII: Nonstandard Quantum Group for the Plethysm Problem, Technical report TR-2007-14, Department of Computer Science, The University of Chicago, Chicago, IL, 2007.
52.
K. D. Mulmuley, Geometric Complexity Theory: On Canonical Bases for the Nonstandard Quantum Groups, Technical report TR-2007-15, Department of Computer Science, The University of Chicago, Chicago, IL, 2007.
53.
K. D. Mulmuley, On P vs. NP and geometric complexity theory, J. ACM, 58 (2011), article 5.
54.
F. D. Murnaghan, The analysis of the Kronecker product of irreducible representations of the symmetric group, Amer. J. Math., 60 (1938), pp. 761–784.
55.
A. M. Popov, Irreducible simple linear Lie groups with finite standard subgroups in general position, Funkcional. Anal. i Priložen., 9 (1975), pp. 81–82.
56.
V. L. Popov, Two orbits: When is one in the closure of the other?, Tr. Mat. Inst. Steklova, 264 (2009), pp. 152–164.
57.
C. Procesi, Lie Groups: An Approach through Invariants and Representations, Universitext, Springer-Verlag, New York, 2007.
58.
N. Ressayre, Geometric invariant theory and the generalized eigenvalue problem, Invent. Math., 180 (2010), pp. 389–441.
59.
H. J. Ryser, Combinatorial Mathematics, Carus Math. Monogr. 14, Mathematical Association of America, Washington, DC, 1963.
60.
C. E. Shannon, A mathematical theory of communication, Bell System Tech. J., 27 (1948), pp. 379–423, 623–656.
61.
A. Skowroński and J. Weyman, The algebras of semi-invariants of quivers, Transform. Groups, 5 (2000), pp. 361–402.
62.
T. A. Springer, Invariant Theory, Lecture Notes in Math. 585, Springer-Verlag, Berlin, 1977.
63.
R. P. Stanley, Irreducible symmetric group characters of rectangular shape, Sém. Lothar. Combin., 50 (2003/04), B50d.
64.
S. Toda, Classes of arithmetic circuits capturing the complexity of computing the determinant, IEICE Trans. Inf. Syst., E75-D (1992), pp. 116–124.
65.
L. G. Valiant, Reducibility by algebraic projections, in Logic and Algorithmic: An International Symposium Held in Honor of Ernst Specker, Monogr. Enseign. Math. 30, University of Geneva, Geneva, Switzerland, 1982, pp. 365–380.
66.
L. G. Valiant, Completeness classes in algebra, in Proceedings of the 11th Annual ACM Symposium on Theory of Computing, 1979, pp. 249–261.
67.
J. Weyman, Cohomology of Vector Bundles and Syzygies, Cambridge Tracts in Math. 149, Cambridge University Press, Cambridge, UK, 2003.

Information & Authors

Information

Published In

cover image SIAM Journal on Computing
SIAM Journal on Computing
Pages: 1179 - 1209
ISSN (online): 1095-7111

History

Submitted: 17 July 2009
Accepted: 2 May 2011
Published online: 23 August 2011

MSC codes

  1. 68Q17
  2. 20B30
  3. 14L24

Keywords

  1. geometric complexity theory
  2. $\mathbf{P}$ vs. $\mathbf{NP}$
  3. geometric invariant theory
  4. orbit closure
  5. Kronecker coefficient
  6. determinant
  7. permanent

Authors

Affiliations

Metrics & Citations

Metrics

Citations

If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.

Cited By

Media

Figures

Other

Tables

Share

Share

Copy the content Link

Share with email

Email a colleague

Share on social media

The SIAM Publications Library now uses SIAM Single Sign-On for individuals. If you do not have existing SIAM credentials, create your SIAM account https://my.siam.org.