Approximation algorithms for Node-weighted Steiner Problems: Digraphs with Additive Prizes and Graphs with Submodular Prizes

In the \emph{budgeted rooted node-weighted Steiner tree} problem, we are given a graph $G$ with $n$ nodes, a predefined node $r$, two weights associated to each node modelling costs and prizes. The aim is to find a tree in $G$ rooted at $r$ such that the total cost of its nodes is at most a given budget $B$ and the total prize is maximized. In the \emph{quota rooted node-weighted Steiner tree} problem, we are given a real-valued quota $Q$, instead of the budget, and we aim at minimizing the cost of a tree rooted at $r$ whose overall prize is at least $Q$. For the case of directed graphs with additive prize function, we develop a technique resorting on a standard flow-based linear programming relaxation to compute a tree with good trade-off between prize and cost, which allows us to provide very simple polynomial time approximation algorithms for both the budgeted and the quota problems. For the \emph{budgeted} problem, our algorithm achieves a bicriteria $(1+\epsilon, O(\frac{1}{\epsilon^{2}n{2/3}\ln{n}))$-approximation,} for any $\epsilon \in (0, 1]$. For the \emph{quota} problem, our algorithm guarantees a bicriteria approximation factor of $(2, O(n^{{2/3}\ln{n}))$.} Next, by using the flow-based LP, we provide a surprisingly simple polynomial time $O((1+\epsilon)\sqrt{n} \ln {n})$-approximation algorithm for the node-weighted version of the directed Steiner tree problem, for any $\epsilon>0$. For the case of undirected graphs with monotone submodular prize functions over subsets of nodes, we provide a polynomial time $O(\frac{1}{\epsilon^{3}\sqrt{n}\log{n})$-approximation} algorithm for the budgeted problem that violates the budget constraint by a factor of at most $1+\epsilon$, for any $\epsilon \in (0, 1]$. Our technique allows us to provide a good approximation also for the quota problem.