Finding Small Multi-Demand Set Covers with Ubiquitous Elements and Large Sets is Fixed-Parameter Tractable

We study a variant of Set Cover where each element of the universe has some demand that determines how many times the element needs to be covered. Moreover, we examine two generalizations of this problem where a set can be included multiple times and where sets have different prices. We prove that all three problems are fixed-parameter tractable with respect to the combined parameter budget, the maximum number of elements missing in one of the sets, and the maximum number of sets in which one of the elements does not occur. Lastly, we point out how our fixed-parameter tractability algorithm can be applied in the context of bribery for the (collective-decision) group identification problem. We consider the following variant of the traditional Set Cover problem: Set Cover with Demands Input: Universe U = [n], a list of demands d1, . . . , dn ∈ [m], a family of subsets F = {F1, . . . , Fm} over U , and an integer k ∈ N. Question: Does there exist a subset S ⊆ F with |S| = k such that each element i ∈ U is included in at least di sets from S? We start by introducing some notation. A family of subsets over some universe U is called covering system. Given some covering system F and an element i ∈ U , we denote by F(i) the subfamily of sets containing i. Analogously, for a subset of elements F ⊆ U , we denote the subfamily of sets from F containing any element from F as F(F ), i.e., F(F ) = ⋃ i∈F F(i). Let smin = mini∈m |Fi| be the minimum size of a set from the covering system and omin = mini∈n |F(i)| the minimum number of occurrences of an element in sets from the covering system. Symmetrically, let smax = maxi∈m |Fi| be the maximum size of a set from the covering system and omax = maxi∈n |F(i)| the maximum number of occurrences of an element in sets from the covering system. We consider Set Cover with Demands parameterized by the number of sets to be selected, the maximum number of elements missing in a set, and the maximum number of sets in which an element does not appear. This means that we develop