Single-sink Fractionally Subadditive Network Design

TL;DR

This paper investigates a generalized network design problem related to Steiner trees, providing complexity results, improved approximation algorithms, and structural insights, with conjectures on potential constant-factor solutions.

Contribution

It resolves the complexity for the case of two subsets, improves the approximation ratio from 2 to 1.5, and offers structural results limiting existing techniques.

Findings

The problem is NP-hard for k=2.

A 1.5-approximation algorithm is developed.

Structural barriers prevent better than O(log n) approximations.

Abstract

We study a generalization of the Steiner tree problem, where we are given a weighted network $G$ together with a collection of $k$ subsets of its vertices and a root $r$. We wish to construct a minimum cost network such that the network supports one unit of flow to the root from every node in a subset simultaneously. The network constructed does not need to support flows from all the subsets simultaneously. We settle an open question regarding the complexity of this problem for $k=2$, and give a $\frac{3}{2}$-approximation algorithm that improves over a (trivial) known 2-approximation. Furthermore, we prove some structural results that prevent many well-known techniques from doing better than the known $O(\log n)$-approximation. Despite these obstacles, we conjecture that this problem should have an $O(1)$-approximation. We also give an approximation result for a variant of the problem where the solution is required to be a path.