The minimum degree Group Steiner problem

The DB-GST problem is given an undirected graph G(V,E), and a collection of groups S={S_i}_i=1^q,S_i⊆V, find a tree that contains at least one vertex from every group S_i, 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 d_v, 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, deg_T(v)≤d_v, 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.