Tensors: Geometry and Applications J.M. Landsberg - Texas A&M ...

20 1. Introduction
Given a bipartite graph on (n,n) vertices one can check if the graph has
a complete matching in polynomial time [153]. However there is no known
polynomial time algorithm to count the number of perfect matchings.
Problems such as the marriage problem appear to require a number of
arithmetic operations that grows exponentially with the size of the data in
order to solve them, however a proposed solution can be verified by performing
a number of arithmetic operations that grows polynomialy with the
size of the data. Such problems are said to be of class NP. (See Chapter
13 for precise definitions.)
Form an incidence matrix X = (xi j ) for a bipartite graph by letting
the upper index correspond to one set of nodes and the lower index the
other. One then places a 1 in the (i,j)-th slot if there is an edge joining the
corresponding nodes and a zero if there is not.
Define the permanent of an n × n matrix X = (xi j ) by
(1.4.2) permn(X) := �
σ∈Sn
and observe the similarities with (1.4.1).
x 1 σ(1) x2 σ(2) · · · xn σ(n) ,
Exercise 1.4.2.1: Verify directly that the permanent of the incidence matrix
for the following graph indeed equals its number of perfect matchings.
Exercise 1.4.2.2: Show that if X is an incidence matrix for an (n,n)bipartite
graph Γ, that the number of perfect matchings of Γ is given by the
permanent.
1.4.3. Algebraic variants of P v. NP. Matrix multiplication, and thus
computing the determinant of a matrix, can be computed by performing a
number of arithmetic operations that is polynomial in the size of the data.
(If the data size is of order m = n 2 then one needs roughly m 3
2 = n 3
operations, or roughly n 4 if one wants an algorithm without divisions or
decisions, see §13.4.2.) Roughly speaking, such problems are said to be of
class P, or are computable in polynomial time.
L. Valiant [307] had the following idea: Let P(x 1 ,... ,x v ) be a homogeneous
polynomial of degree m in v variables. We say P is an affine
projection of a determinant of size n if there exists an affine linear function
f : C v → Matn(C) such that P = det ◦ f. Write dc(P) for the smallest n