a b s t r a c tWe consider a variety of vehicle routing problems. The input consists of an undirected graph and edge lengths. Customers located at the nodes have to be visited by a set of vehicles. Two important parameters are k the number of vehicles, and the longest distance traveled by a vehicle denoted by λ. Here, we consider k to be a given bound on the maximum number of vehicles, and thus the decision maker cannot increase its value. Therefore, the goal will be to minimize λ. We study different variations of this problem, where for instance instead of servicing the customers using paths, we can serve them using spanning trees or cycles. For all these variations, we present new approximation algorithms with FPT time (where k is the parameter) which improve the known approximation guarantees for these problems.
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.
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