The b-Exact Multicover problem takes a universe U of n elements, a family \(\mathcal {F}\) of m subsets of U, a function \({\textsf {dem}}: U \rightarrow \{1,\ldots ,b\}\) and a positive integer k, and decides whether there exists a subfamily(set cover) \(\mathcal {F}^{\prime }\) of size at most k such that each element u ∈ U is covered by exactly dem(u) sets of \(\mathcal {F}^{\prime }\). The b-Exact Coverage problem also takes the same input and decides whether there is a subfamily \(\mathcal {F}^{\prime } \subseteq \mathcal {F}\) such that there are at least k elements that satisfy the following property: u ∈ U is covered by exactly dem(u) sets of \(\mathcal {F}^{\prime }\). Both these problems are known to be NP-complete. In the parameterized setting, when parameterized by k, b-Exact Multicover is W[1]-hard even when b = 1. While b-Exact Coverage is FPT under the same parameter, it is known to not admit a polynomial kernel under standard complexity-theoretic assumptions, even when b = 1. In this paper, we investigate these two problems under the assumption that every set satisfies a given geometric property π. Specifically, we consider the universe to be a set of n points in a real space \(\mathbb {R}^{d}\), d being a positive integer. When d = 2 we consider the problem when π requires all sets to be unit squares or lines. When d > 2, we consider the problem where π requires all sets to be hyperplanes in \(\mathbb {R}^{d}\). These special versions of the problems are also known to be NP-complete. When parameterized by k, the b-Exact Coverage problem has a polynomial size kernel for all the above geometric versions. The b-Exact Multicover problem turns out to be W[1]-hard for squares even when b = 1, but FPT for lines and hyperplanes. Further, we also consider the b-Exact Max. Multicover problem, which takes the same input and decides whether there is a set cover \(\mathcal {F}^{\prime }\) such that every element u ∈ U is covered by at least dem(u) sets and at least k elements satisfy the following property: u ∈ U is covered by exactly dem(u) sets of \(\mathcal {F}^{\prime }\). To the best of our knowledge, this problem has not been studied before, and we show that it is NP-complete (even for the case of lines). In fact, the problem turns out to be W[1]-hard in the general setting, when parameterized by k. However, when we restrict the sets to lines and hyperplanes, we obtain FPT algorithms.

的b -精确Multicover问题需要宇宙ü的Ñ元件，一个家庭\（\ mathcal {F} \）的米的子集ü，函数\（{\ textsf {DEM}}：U \ RIGHTARROW \ {1， \ ldots，b \} \）和一个正整数ķ，并决定是否存在一个亚科（机盖）\（\ mathcal {F} ^ {\素} \）至多大小的ķ这样每个元素ü ∈ U正好被dem ( u ) 组\(\mathcal {F}^{\prime }\)覆盖。在b -精确覆盖问题也采用相同的输入并决定是否存在子族\(\mathcal {F}^{\prime } \subseteq \mathcal {F}\)使得至少有k 个元素满足以下属性：u ∈ U正好被dem ( u ) 组\(\mathcal {F}^{\prime }\)覆盖。已知这两个问题都是 NP 完全问题。在参数化设置中，当由k参数化时，即使b = 1 ，b - Exact Multicover也是 W[1]-hard 。而b - Exact Coverage是相同参数下的 FPT，已知在标准复杂性理论假设下不允许多项式核，即使b = 1。 在本文中，我们在假设每个集合都满足给定的几何性质π 的情况下研究这两个问题. 具体来说，我们认为宇宙是真实空间\(\mathbb {R}^{d}\) 中的一组n个点，d是一个正整数。当d = 2 时，我们考虑π要求所有集合都是单位正方形或直线的问题。当d > 2 时，我们考虑π要求所有集合都是\(\mathbb {R}^{d}\) 中的超平面的问题. 这些问题的特殊版本也被称为 NP 完全问题。当由k参数化时，b -精确覆盖问题具有适用于所有上述几何版本的多项式大小内核。的b -精确Multicover问题被证明是W [1]为-hard正方形即使当b = 1，但为FPT线和超平面。此外，我们还考虑了b - Exact Max。多重覆盖问题，它采用相同的输入并决定是否存在集合覆盖\(\mathcal {F}^{\prime }\)使得每个元素u ∈ U至少被dem (u ) 集合且至少k 个元素满足以下属性：u ∈ U正好被dem ( u ) 集合\(\mathcal {F}^{\prime }\)覆盖。据我们所知，这个问题以前没有被研究过，我们证明它是 NP 完全的（即使是线的情况）。事实上，当由k参数化时，问题在一般设置中变成了 W[1]-hard 。然而，当我们将集合限制为线和超平面时，我们获得了 FPT 算法。