Improved Approximation Algorithms for (Budgeted) Node-weighted Steiner Problems

Abstract

Moss and Rabani study constrained node-weighted Steiner tree problems with two independent weight values associated with each node, namely, cost and prize (or penalty). They give an $O(\log n)$-approximation algorithm for the node-weighted prize-collecting Steiner tree problem (PCST)---where the goal is to minimize the cost of a tree plus the penalty of vertices not covered by the tree. They use the algorithm for PCST to obtain a bicriteria $(2, O(\log n))$-approximation algorithm for the budgeted node-weighted Steiner tree problem---where the goal is to maximize the prize of a tree with a given budget for its cost. Their solution may cost up to twice the budget, but collects a factor $\Omega(\frac{1}{\log n})$ of the optimal prize. We improve these results from at least two aspects. Our first main result is a primal-dual $O(\log h)$-approximation algorithm for a more general problem, node-weighted prize-collecting Steiner forest (PCSF), where we have $h$ demands each requesting the connectivity of a pair of vertices. Our algorithm can be seen as a greedy algorithm which reduces the number of demands by choosing a structure with minimum cost-to-reduction ratio. This natural style of argument leads to a much simpler algorithm than that of Moss and Rabani for PCST. Our second main contribution is for the budgeted node-weighted Steiner tree problem, which is also an improvement to the work of Moss and Rabani. In the unrooted case, we improve upon an existing $O(\log^2 n)$-approximation by Guha et al., and present an $O(\log n)$-approximation algorithm without any budget violation. For the rooted case, where a specified vertex has to appear in the solution tree, we improve the bicriteria result of Moss and Rabani to the bicriteria approximation ratio of $(1+\epsilon, O(\log n)/\epsilon^2)$ for any positive (possibly subconstant) $\epsilon$. That is, for any permissible budget violation $1+\epsilon$, we present an algorithm achieving a trade off in the guarantee for the prize. Indeed, we show that this is almost tight for the natural linear-programming relaxation used by us as well as in the previous works.