Faster algorithms on linear delta-matroids
Tomohiro Koana, Magnus Wahlström
TL;DR
The paper studies faster algorithms for linear delta-matroids by introducing a contraction representation over skew-symmetric matrices, which is equivalent to the traditional twist representation up to $O(n^{\omega})$-time transformations. It proves that unions and delta-sums of linear delta-matroids remain linear and can be represented in randomized time $O(n^{\omega})$, extending to projected linear delta-matroids. Algorithmically, max-weight feasible sets reduce to max-weight column bases, and many decision problems reduce to rank computations, yielding $O(n^{\omega})$-time (high-probability) algorithms for Linear DM Parity and related problems, plus $O(n^{\omega+1})$ for maximum cardinality DM Intersection; weighted variants are also addressed. The work connects delta-matroid theory to factor and path-packing problems via Ishikawa–Wakayama formulas, providing a broad algebraic toolkit and resolving open questions about polynomial-time solvability in this domain.
Abstract
We show new algorithms and constructions over linear delta-matroids. We observe an alternative representation for linear delta-matroids, as a contraction representation over a skew-symmetric matrix. This is equivalent to the more standard "twist representation" up to $O(n^ω)$-time transformations, but is much more convenient for algorithmic tasks. For instance, the problem of finding a max-weight feasible set now reduces directly to the problem of finding a max-weight basis in a linear matroid. Supported by this representation, we provide new algorithms and constructions over linear delta-matroids. We show that the union and delta-sum of linear delta-matroids define linear delta-matroids, and a representation for the resulting delta-matroid can be constructed in randomized time $O(n^ω)$. Previously, it was only known that these operations define delta-matroids. We also note that every projected linear delta-matroid can be represented as an elementary projection. This implies that several optimization problems over (projected) linear delta-matroids, including the coverage, delta-coverage, and parity problems, reduce (in their decision versions) to a single $O(n^ω)$-time matrix rank computation. Using the methods of Harvey, previously used by Cheung, Lao and Leung for linear matroid parity, we furthermore show how to solve the search versions in the same time. This improves on the $O(n^4)$-time augmenting path algorithm of Geelen, Iwata and Murota. Finally, we consider the maximum-cardinality delta-matroid intersection problem. Using Storjohann's algorithms for symbolic determinants, we show that such a solution can be found in $O(n^{ω+1})$ time. This is the first polynomial-time algorithm for the problem, solving an open question of Kakimura and Takamatsu.