Summary: Determinant Versus Permanent
Abstract. We study the problem of expressing permanent of matrices as determinant
of (possibly larger) matrices. This problem has close connections with complexity of
arithmetic computations: complexities of computing permanent and determinant roughly
correspond to arithmetic versions of the classes NP and P respectively. We survey known
results about their relative complexity and describe two recently developed approaches
that might lead to a proof of the conjecture that permanent can only be expressed as
determinant of exponential-sized matrices.
Mathematics Subject Classification (2000). Primary 68Q17; Secondary 68W30.
Keywords. Arithmetic computation, complexity classes, determinant, permanent.
Determinant of square matrices plays a fundamental role in linear algebra. It
is a linear function on rows (and columns) of the matrix, and has several nice
interpretations. Geometrically, it is the volume of the parallelopied defined by rows
(or columns) of the matrix, and algebraically, it is the product of all eigenvalues,
with multiplicity, of the matrix. It also satisfies a number of other interesting
properties, e.g., it is multiplicative, invariant under linear combinations of rows
(and columns) etc. Permanent of a square matrix is a number that is defined in a
way similar to the determinant. For matrix X = [xi,j]1i,jn,