We consider the problem of optimizing a coverage function under a $\ell$-matchoid of rank $k$. We design fixed-parameter algorithms as well as streaming algorithms to compute an exact solution. Unlike previous work that presumes linear representativity of matroids, we consider the general oracle model. For the special case where the coverage function is linear, we give a deterministic fixed-parameter algorithm parameterized by $\ell$ and $k$. This result, combined with the lower bounds of Lovasz~\cite{Lovasz1981}, and Jensen and Korte~\cite{Jensen82}, demonstrates a separation between the $\ell$-matchoid and the matroid $\ell$-parity problems in the setting of fixed-parameter tractability. For a general coverage function, we give both deterministic and randomized fixed-parameter algorithms, parameterized by $\ell$ and $z$, where $z$ is the number of points covered in an optimal solution. The resulting algorithms can be directly translated into streaming algorithms. For unweighted coverage functions, we show that we can find an exact solution even when the function is given in the form of a value oracle (and so we do not have access to an explicit representation of the set system). Our result can be implemented in the streaming setting and stores a number of elements depending only on $\ell$ and $z$, but completely indpendent of the total size $n$ of the ground set. This shows that it is possible to circumvent the recent space lower bound of Feldman et al~\cite{feldman2020}, by parameterizing the solution value. This result, combined with existing lower bounds, also provides a new separation between the space and time complexity of maximizing an arbitrary submodular function and a coverage function in the value oracle model.
翻译:我们考虑的是以美元/ ell$- unatch massy 来优化一个覆盖函数的问题。 我们设计固定参数算法以及流算算法来计算准确的解决方案。 与先前假定机器人线性代表率的工作不同, 我们考虑的是一般标准模型。 对于覆盖函数线性的特例, 我们给出了确定性固定参数算法参数参数, 以美元/ ell$和 美元计算。 这个结果, 加上Lovaszzácite{ Lovasz181} 和Jentsen和 Korte ⁇ cite{ Jensen82} 的下限。 我们设计固定参数可移动性代表率和配方 。 对于一般覆盖函数, 我们给出了确定性固定值和随机化固定值的固定参数, 以美元/ 美元和 美元为参数的参数, 这是最佳解决方案中可能覆盖的点数。 由此产生的算法可以直接转换成 $- 美元- 美元- 元- 和 直线值表示我们设定一个固定值的计算结果。