Approximation algorithm for the partial set multi-cover problem

Abstract

Partial set cover problem and set multi-cover problem are two generalizations of set cover problem. In this paper, we consider the partial set multi-cover problem which is a combination of them: given an element set E, a collection of sets S ⊆ 2 E , a total covering ratio q which is a constant between 0 and 1, each set S ∈ S is associated with a cost c S , each element e ∈ E is associated with a covering requirement r e , the goal is to find a minimum cost sub-collection S ′ ⊆ S to fully cover at least q|E| elements, where element e is fully covered if it belongs to at least r e sets of S ′ . Denote by r max = max{r e : e ∈ E} the maximum covering requirement. We present an (O( rmax log 2 n ε ), 1 − ε)-bicriteria approximation algorithm, that is, the output of our algorithm has cost at most O( rmax log 2 n ε ) times of the optimal value while the number of fully covered elements is at least (1 − ε)q|E|.

Majority Opinion Diffusion in Social Networks: An Adversarial Approach

“…Furthermore, assume that in the MI problem, the seed set is given as a part of the input and the attacker's task is only to decide on the color of the seed nodes. Then, the problem can be reduced to the partial set multi-cover problem [45], which has a constant factor approximation algorithm. Therefore, if the attacker has an effective heuristic method to select the seed set, then it can find a reasonably efficient solution.…”

On Cost-Effective, Reliable Coverage for LoS Communications in Urban Areas

“…This is called a partial set multi-cover problem and is less studied than the previous one. Ran et al address the difficulty of the problem [14], while recent works [15], [16] provide heuristics with bounded error that still require solving an integer linear problem at each step, which is computationally impossible in our context.…”