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 $O(n^\omega)$-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 $O(n^\omega)$. 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 $O(n^{\omega})$-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 $O(n^{\omega+1})$ time. This is the first polynomial-time algorithm for the problem, solving an open question of Kakimura and Takamatsu.

翻译：我们展示了线性德尔塔拟阵上的新算法与构造。我们观察到线性德尔塔拟阵的一种替代表示——基于斜对称矩阵的收缩表示。该表示与更标准的"扭曲表示"在 $O(n^\omega)$ 时间变换下等价，但在算法任务中更为便利。例如，寻找最大权重可行集的问题可直接简化为线性拟阵中寻找最大权重基的问题。在该表示的支持下，我们给出了线性德尔塔拟阵上的新算法与构造。我们证明了线性德尔塔拟阵的并运算与德尔塔和运算仍定义线性德尔塔拟阵，且可在随机化时间 $O(n^\omega)$ 内构造出结果德尔塔拟阵的表示。此前仅知这两种运算能定义德尔塔拟阵。我们还注意到，每个投影线性德尔塔拟阵可表示为初等投影。这意味着，投影线性德尔塔拟阵上的若干优化问题（包括覆盖、德尔塔覆盖及奇偶性问题）在判定版本中均可归约为单次 $O(n^\omega)$ 时间矩阵秩计算。利用 Harvey 的方法（此前被 Cheung、Lao 和 Leung 用于线性拟阵奇偶性问题），我们进一步展示了如何在相同时间内解决搜索版本。这改进了 Geelen、Iwata 和 Murota 提出的 $O(n^4)$ 时间增广路径算法。最后，我们研究了最大基数德尔塔拟阵交问题。利用 Storjohann 的符号行列式算法，我们证明了该问题的解可在 $O(n^{\omega+1})$ 时间内找到。这是该问题的首个多项式时间算法，解决了 Kakimura 与 Takamatsu 提出的开放问题。