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

TL;DR

This paper introduces simple approximation algorithms for various node-weighted Steiner problems in directed and undirected graphs, addressing budgeted and quota variants with additive and submodular prizes, achieving near-optimal guarantees.

Contribution

It develops flow-based LP techniques to obtain new approximation algorithms for budgeted and quota Steiner problems with additive and submodular prizes, improving existing results.

Findings

Bicriteria approximation for directed graphs with additive prizes

Polynomial-time approximation for directed Steiner tree with LP approach

Approximation algorithms for undirected submodular prize Steiner problems

Abstract

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.