Permanent versus Determinant conjecture

Permanent versus Determinant conjecture
Name	Permanent versus Determinant conjecture
Field	Theoretical computer science; Algebraic complexity
Creators	Leslie Valiant
Date	1979
Status	Open problem

Permanent versus Determinant conjecture The Permanent versus Determinant conjecture posits a separation between the algebraic complexity of the permanent and the determinant functions, asserting that expressing the permanent as a small-size determinant of a matrix whose entries are linear forms is impossible within polynomial blowup. The conjecture influenced research in Leslie Valiant's program, relates to classifications in complexity theory and has guided work by researchers associated with institutions such as Institute for Advanced Study, Massachusetts Institute of Technology, Princeton University, and Stanford University.

Statement of the conjecture

The conjecture asserts that there is no polynomial-size family of matrices with entries given by linear forms over variables such that the determinant of those matrices equals the permanent of an arbitrary n×n matrix, implying a superpolynomial lower bound on the determinantal complexity of the permanent. Statements and formulations appear in foundational papers by Leslie Valiant and are discussed in surveys by researchers at Carnegie Mellon University, University of California, Berkeley, University of Toronto, University of Cambridge, and University of Oxford.

Historical background and motivation

The conjecture originated in the late 1970s and early 1980s as part of efforts to formalize algebraic analogs of classical separations like those sought in Cook–Levin theorem-era discussions and the P versus NP problem program promoted by researchers including Richard Karp, Stephen Cook, and Alan Turing-era influences. Early impetus came from Leslie Valiant's 1979 and 1982 works that introduced algebraic complexity classes analogous to NP and P, prompting further study at workshops hosted by American Mathematical Society, International Congress of Mathematicians, and labs at Bell Laboratories. Motivation also connects to historical algebraic results such as those by Arthur Cayley, James Joseph Sylvester, and later developments in linear algebra at École Normale Supérieure and University of Göttingen.

Known results and partial progress

Proven lower bounds and conditional separations include exponential or superpolynomial lower bounds in restricted models studied by teams at École Polytechnique Fédérale de Lausanne, École Normale Supérieure, University of Chicago, and University of Washington. Results by researchers affiliated with Princeton University, Columbia University, University of California, San Diego, University of Illinois Urbana-Champaign, and Microsoft Research established barriers such as the natural proofs framework analogues and arithmetic circuit depth reductions. Geometric and representation-theoretic partial results using techniques from groups like GL_n and tools developed in collaborations involving Institute for Advanced Study and Max Planck Institute have produced complexity lower bounds for special classes of matrices, with contributions by scholars connected to Harvard University, Yale University, Cornell University, and ETH Zurich.

Relations to complexity theory and Valiant's conjectures

The conjecture lies at the heart of Leslie Valiant's algebraic complexity program and is intimately connected to the separation of algebraic classes such as VP and VNP, analogous to P versus NP conjecture in classical complexity. Work by investigators at Clay Mathematics Institute, National Science Foundation, Simons Foundation, and research groups at University of California, Los Angeles and Duke University related the conjecture to complete problems, hardness reductions, and completeness results; these include characterizations that echo themes from the Cook–Levin theorem and structural frameworks explored at Bell Labs and IBM Research.

Approaches and techniques

Approaches employ algebraic geometry, representation theory, and invariant theory developed at centers such as Institut des Hautes Études Scientifiques, Max Planck Society, Centre National de la Recherche Scientifique, and universities including University of Paris, Princeton University, University of Michigan, and Brown University. Techniques include studying orbit closures under actions of groups like GL_n, leveraging tools from the Geometric Complexity Theory initiative spearheaded by collaborations among researchers at University of Toronto, Rutgers University, and University of California, Berkeley, and exploiting combinatorial and matrix-analytic methods traceable to work by John von Neumann, Paul Erdős, and Alfréd Rényi. Circuit lower-bound strategies draw on depth-reduction results from groups at Carnegie Mellon University and Massachusetts Institute of Technology, while barriers and no-go theorems have been developed in papers involving scholars at Columbia University and University of California, Santa Barbara.

Open problems and future directions

Key open problems include proving superpolynomial lower bounds on determinantal complexity for the permanent, establishing unconditional separations between VP and VNP, and resolving whether geometric approaches in Geometric Complexity Theory can circumvent known barriers. Directions for future work emphasize cross-disciplinary collaboration among institutions such as Institute for Advanced Study, Simons Institute for the Theory of Computing, Royal Society, and funding bodies like European Research Council and National Science Foundation to push algebraic, geometric, and combinatorial techniques further. Progress will likely involve new insights from scholars at Princeton University, Harvard University, Stanford University, University of Cambridge, and emergent research centers worldwide.

Category:Algebraic complexity theory