The maximization of submodular functions have found widespread application in areas such as machine learning, combinatorial optimization, and economics, where practitioners often wish to enforce various constraints; the matroid constraint has been investigated extensively due to its algorithmic properties and expressive power. Recent progress has focused on fast algorithms for important classes of matroids given in explicit form. Currently, nearly-linear time algorithms only exist for graphic and partition matroids [ICALP '19]. In this work, we develop algorithms for monotone submodular maximization constrained by graphic, transversal matroids, or laminar matroids in time near-linear in the size of their representation. Our algorithms achieve an optimal approximation of $1-1/e-\epsilon$ and both generalize and accelerate the results of Ene and Nguyen [ICALP '19]. In fact, the running time of our algorithm cannot be improved within the fast continuous greedy framework of Badanidiyuru and Vondr\'ak [SODA '14]. To achieve near-linear running time, we make use of dynamic data structures that maintain bases with approximate maximum cardinality and weight under certain element updates. These data structures need to support a weight decrease operation and a novel FREEZE operation that allows the algorithm to freeze elements (i.e. force to be contained) in its basis regardless of future data structure operations. For the laminar matroid, we present a new dynamic data structure using the top tree interface of Alstrup, Holm, de Lichtenberg, and Thorup [TALG '05] that maintains the maximum weight basis under insertions and deletions of elements in $O(\log n)$ time. For the transversal matroid the FREEZE operation corresponds to requiring the data structure to keep a certain set $S$ of vertices matched, a property that we call $S$-stability.

翻译：子模函数的最大化在机器学习、组合优化和经济学等领域具有广泛应用，实践者往往希望施加各种约束；拟阵约束因其算法性质和表达能力而受到广泛研究。最新进展聚焦于显式形式的重要拟阵类的快速算法。目前，近线性时间算法仅存在于图和划分拟阵 [ICALP '19]。在本工作中，我们针对由图拟阵、横贯拟阵或分层拟阵约束的单调子模最大化问题，开发出与其表示规模近线性时间的算法。我们的算法实现了最优近似比 $1-1/e-\epsilon$，并同时推广和加速了Ene和Nguyen [ICALP '19] 的结果。事实上，在Badanidiyuru和Vondrák [SODA '14] 的快速连续贪婪框架内，我们的算法运行时间无法进一步改进。为实现近线性运行时间，我们利用动态数据结构维护在特定元素更新下具有近似最大基数和权重的基。这些数据结构需支持权重递减操作以及一种新颖的FREEZE操作，使算法能够在其基中冻结元素（即强制包含），而不受后续数据结构操作影响。对于分层拟阵，我们利用Alstrup、Holm、de Lichtenberg和Thorup [TALG '05] 的顶层树接口，提出一种新的动态数据结构，在元素插入和删除时以 $O(\log n)$ 时间维护最大权重基。对于横贯拟阵，FREEZE操作对应于要求数据结构保持某顶点集 $S$ 的匹配状态，我们称此性质为 $S$-稳定性。