Maximum coverage problem with group budget constraints

Abstract

We study the maximum coverage problem with group budget constraints (MCG). The input consists of a ground set X , a collection ψ of subsets of X each of which is associated with a combinatorial structure such that for every set S j ∈ ψ, a cost c(S j ) can be calculated based on the combinatorial structure associated with S j , a partition G 1 , G 2 , . . . , G l of ψ, and budgets B 1 , B 2 , . . . , B l , and B. A solution to the problem consists of a subset H of ψ such that S j ∈H c(S j ) ≤ B and for each i ∈ 1, 2, . . . , l, S j ∈H ∩G i c(S j ) ≤ B i . The objective is to maximize | S j ∈H S j |. In our work we use a new and improved analysis of the greedy algorithm to prove that it is a ( α 3+2α )-approximation algorithm, where α is the approximation ratio of a given oracle which takes as an input a subset X new ⊆ X and a group G i and returns a set S j ∈ G i which approximates the optimal solution for max D∈G i |D∩X new | c(D) . This analysis that is shown here to be tight for the greedy algorithm, improves by a factor larger than 2 the analysis of the best known approximation algorithm for MCG.

Maximum Coverage with Cluster Constraints: An LP-Based Approximation Technique

“…For this problem, they obtain a 1 α+1 -approximation algorithm that uses an oracle that, given a current solution, returns a set which contribution to the current solution is within a factor α of the optimal contribution of a single set. For the cost-restricted version, Farbstein and Levin [9] obtain a 1 5 -approximation. Guo et al [12] present a pseudo-polynomial algorithm for CMCG whose approximation ratio is (1 − 1 e ).…”

“…If the solution for a single cluster is seen as a new set, then the MCPC with multiple clusters can be seen as the cardinality restricted version in which each cluster correspond to a group to which at most one set can be assigned. Combining the approaches of Farbstein and Levin [9] and Chekuri and Kumar [5] for the cost-restricted and cardinalty-restricted version, respectively, we obtain a 1 6 -approximation algorithm. In this paper, we improve this ratio to…”

Maximum Coverage with Cluster Constraints: An LP-Based Approximation Technique

“…If the solution for a single cluster is seen as a new set, then the MCPC with multiple clusters can be seen as the cardinality restricted version in which each cluster correspond to a group to which at most one set can be assigned. Combining the approaches of Farbstein and Levin [9] and Chekuri and Kumar [4] for the cost-restricted and cardinalty-restricted version, respectively, we obtain a 1 6 -approximation algorithm. In this paper, we improve this ratio to…”

“…k=1 B k is called the critical knapsack of cluster l. 9 We first show that the z-variables of an optimal LP-solution admit a specific structure. Intuitively, the lemma states that the cluster capacity U l of each cluster l is shared maximally among the first κ(l) − 1 knapsacks in K(l) and the remaining capacity is assigned to the critical knapsack κ(l).…”