  The Odd Cycle Transversal problem (OCT) asks whether a given graph can be
made bipartite by deleting at most $k$ of its vertices. In a breakthrough
result Reed, Smith, and Vetta (Operations Research Letters, 2004) gave a
$\BigOh(4^kkmn)$ time algorithm for it, the first algorithm with polynomial
runtime of uniform degree for every fixed $k$. It is known that this implies a
polynomial-time compression algorithm that turns OCT instances into equivalent
instances of size at most $\BigOh(4^k)$, a so-called kernelization. Since then
the existence of a polynomial kernel for OCT, i.e., a kernelization with size
bounded polynomially in $k$, has turned into one of the main open questions in
the study of kernelization.
  This work provides the first (randomized) polynomial kernelization for OCT.
We introduce a novel kernelization approach based on matroid theory, where we
encode all relevant information about a problem instance into a matroid with a
representation of size polynomial in $k$. For OCT, the matroid is built to
allow us to simulate the computation of the iterative compression step of the
algorithm of Reed, Smith, and Vetta, applied (for only one round) to an
approximate odd cycle transversal which it is aiming to shrink to size $k$. The
process is randomized with one-sided error exponentially small in $k$, where
the result can contain false positives but no false negatives, and the size
guarantee is cubic in the size of the approximate solution. Combined with an
$\BigOh(\sqrt{\log n})$-approximation (Agarwal et al., STOC 2005), we get a
reduction of the instance to size $\BigOh(k^{4.5})$, implying a randomized
polynomial kernelization.
