The minimum degree Group Steiner problem

Abstract

The DB-GST problem is given an undirected graph G(V,E), and a collection of groups S={Si}i=1q,Si⊆V, find a tree that contains at least one vertex from every group Si, so that the maximum degree is minimal. This problem was motivated by On-Line algorithms Hajiaghayi (2016), and has applications in VLSI design and fast Broadcasting. In the WDB-GST problem, every vertex v has individual degree bound dv, and every e∈E has a cost c(e)>0. The goal is, to find a tree that contains at least one terminal from every group, so that for every v, degT(v)≤dv, and among such trees, find the one with minimum cost. We give the first approximation for this problem, an (O(log2n),O(log2n)) bicriteria approximation ratio the WDB-GST problem on trees inputs. This implies an O(log2n) approximation for DB-GST on tree inputs. The previously best known ratio for the WDB-GST problem on trees was a bicriterion (O(log2n),O(log3n)) (the approximation for the degrees is O(log3n)) ratio which is folklore. Getting O(log2n) approximation requires careful case analysis and was not known.Our result for WDB-GST generalizes the classic result of Garg et al. (2016) that approximated the cost within O(log2n), but did not approximate the degree.Our main result is an O(log3n) approximation for BD-GST on Bounded Treewidth graphs.The DB-Steiner k-tree problem is given an undirected graph G(V,E), a collection of terminals S⊆V, and a number k, find a tree T(V′,E′) that contains at least k terminals, of minimum maximum degree. We prove that if the DB-GST problem admits a ρ ratio approximation, then the DB-Steiner k-tree problem, admits an O(log2k⋅ρ) expected approximation. We also show that if there are k groups, there exists an algorithm that is able to coverk/4 of the groups with minimum maximal degree, then there is a deterministic O(logn⋅ρ) approximation for DB-Steiner k-tree problem. Using the work of Guo et al. (2020) we derive an O(log3n) approximation for DB-Steiner k-tree problem on general graphs, that runs in quasi-polynomial time.