  In this paper we address the classification problem for purely infinite
simple Leavitt path algebras of finite graphs over a field $\ell$. Each graph
$E$ has associated a Leavitt path $\ell$-algebra $L(E)$. There is an open
question which asks whether the pair $(K_0(L(E)), [1_{L(E)}])$, consisting of
the Grothendieck group together with the class $[1_{L(E)}]$ of the identity, is
a complete invariant for the classification, up to algebra isomorphism, of
those Leavitt path algebras of finite graphs which are purely infinite simple.
We show that $(K_0(L(E)), [1_{L(E)}])$ is a complete invariant for the
classification of such algebras up to polynomial homotopy equivalence. To prove
this we develop the bivariant algebraic $K$-theory of Leavitt path algebras and
obtain several results of independent interest.
