Abstract
We introduce two new combinatorial optimization problems: the Maximum Spider Problem and the Spider Cover Problem; we study their approximability and illustrate their applications. In these problems we are given a directed graph , a distinguished vertex , and a family D of subsets of vertices. A spider centered at vertex s is a collection of arc-disjoint paths all starting at s but ending into pairwise distinct vertices. We say that a spider covers a subset of vertices X if at least one of the endpoints of the paths constituting the spider other than s belongs to X. In the Maximum Spider Problem the goal is to find a spider centered at s that covers the maximum number of elements of the family D. Conversely, the Spider Cover Problem consists of finding the minimum number of spiders centered at s that covers all subsets in D. We motivate the study of the Maximum Spider and Spider Cover Problems by pointing out a variety of applications. We show that a natural greedy algorithm gives a 2-approximation algorithm for the Maximum Spider Problem and a -approximation algorithm for the Spider Cover Problem.
Highlights
Given a digraph G V, E and a vertex s ∈ V, a spider centered at s is a subgraph S of G consisting of arc-disjoint paths sharing the initial vertex s and ending into pairwise distinct vertices
We show that a natural greedy algorithm gives a 2-approximation algorithm for the Maximum Spider Problem and a log |D| 1 -approximation algorithm for the Spider Cover Problem
To the best of our knowledge, the Maximum Spider and the Spider Cover Problems have not been considered before, apart from the different special cases mentioned in the previous section
Summary
Given a digraph G V, E and a vertex s ∈ V , a spider centered at s is a subgraph S of G consisting of arc-disjoint paths sharing the initial vertex s and ending into pairwise distinct vertices. The endpoints of the paths composing the spider S—other than the center s—are called the terminals of the spider. Given a spider S, we say that S reaches a vertex x ∈ V if x is a terminal of S; we say that the spider S covers a subset of vertices D ⊆ V if S reaches at least a vertex in D. We are given a digraph G V, E , a distinguished node s, and a family D ⊆ 2V \{s} of subsets of vertices. We are given a digraph G V, E , a distinguished vertex s ∈ V , and a family D ∈ 2V \{s} of subsets of vertices. The goal is to find a minimum cardinality collection of spiders centered at s such that each subset D ∈ D is covered by at least a spider in the collection
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