Deterministic Truncation of Linear Matroids

Abstract

Let $M=(E,\mathcal{I})$ be a matroid. A $k$ -truncation of $M$ is a matroid $M'=(E,\mathcal{I}')$ such that for any $A\subseteq E$, $A\in \mathcal{I}'$ if and only if $|A|\le k$ and $A\in \mathcal {I}$. Given a linear representation of $M$ we consider the problem of finding a linear representation of the $k$-truncation of this matroid. This problem can be expressed as the following problem on matrices. Let $M$ be a $n\times m$ matrix over a field $\mathbb {F}$. A rank $k$ -truncation of the matrix $M$ is a $k\times m$ matrix $M_k$ (over ${\mathbb F}$ or a related field) such that for every subset $I\subseteq \{1,\ldots ,m\}$ of size at most $k$, the set of columns corresponding to $I$ in $M$ has rank $|I|$ if and only if the corresponding set of columns in $M_k$ has rank $|I|$. A common way to compute a rank $k$-truncation of a $n \times m$ matrix is to multiply the matrix with a random $k\times n$ matrix (with the entries from a field of an exponential size), yielding a simple randomized algorithm. So a natural question is whether it possible to obtain a rank $k$-truncation of a matrix, deterministically. In this paper we settle this question for matrices over any field in which the field operations can be done efficiently. This includes any finite field and the field of rationals ($\mathbb Q$).

Our algorithms are based on the properties of the classical Wronskian determinant, and the folded Wronskian determinant, which was recently introduced by Guruswami and Kopparty [ FOCS, 2013 ], and was implicitly present in the work of Forbes and Shpilka [ STOC, 2012 ]. These were used in the context of subspace designs, and reducing randomness for polynomial identity testing and other related problems. Our main conceptual contribution in this paper is to show that the Wronskian determinant can also be used to obtain a representation of the truncation of a linear matroid in deterministic polynomial time. Finally, we use our results to derandomize several parameterized algorithms, including an algorithm for computing $\ell$ -Matroid Parity, to which several problems like $\ell$ -Matroid Intersection can be reduced.

D. Lokshtanov is supported by the “BeHard” grant under the recruitment programme of the of Bergen Research Foundation. F. Panolan is supported by the European Research Council under the European Unions Seventh Framework Programme (FP/2007-2013) / ERC Grant Agreement no. 267959. S. Saurabh is supported by “PARAPPROX" ERC starting grant no. 306992.