Network construction with subgraph connectivity constraints

Abstract

We consider the problem introduced by Korach and Stern (Mathematical Programming, 98:345–414, 2003) of building a network given connectivity constraints. A network designer is given a set of vertices $V$ and constraints $S_i \subseteq V$, and seeks to build the lowest cost set of edges $E$ such that each $S_i$ induces a connected subgraph of $(V,E)$. First, we answer a question posed by Korach and Stern (Discrete Applied Mathematics, 156:444–450, 2008): for the offline version of the problem, we prove an $\varOmega (\log n)$ hardness of approximation result for uniform cost networks (where edge costs are all $1$) and give an algorithm that almost matches this bound, even in the arbitrary cost case. Then we consider the online problem, where the constraints must be satisfied as they arrive. We give an $O(n\log n)$-competitive algorithm for the arbitrary cost online problem, which has an $\varOmega (n)$-competitive lower bound. We look at the uniform cost case as well and give an $O(n^{2/3}\log ^{2/3} n)$-competitive algorithm against an oblivious adversary, as well as an $\varOmega (\sqrt{n})$-competitive lower bound against an adaptive adversary. We also examine cases when the underlying network graph is known to be a star or a path and prove matching upper and lower bounds of $\Theta (\log n)$ on the competitive ratio for them.

Appendix: updating a pq-tree

We briefly describe the patterns in Booth and Lueker (1976) for updating pq-trees, as broken down into $10$ cases.

This can be used as a guide for tracking the changes in Eq. 2.

• L This pattern simply relabels some leaf nodes.

• P1 This pattern simply relabels a p-node.

• P2 This pattern moves some children of a p-node into their own p-node.

• P3 This pattern moves some children of a p-node into their own p-node and creates a parent q-node.

• P4 This pattern moves some children of a p-node to be children of a newly created p-node, whose parent is a q-node that is a child of the original p-node.

• P5 This pattern moves some children of a p-node into their own p-node that is the child of the original p-node, which becomes transformed to a q-node.

• P6 This pattern moves some children of a p-node to their own p-node that is moved to be the child of a newly created q-node formed by merging two q-nodes.

• Q1 This pattern simply relabels a q-node.

• Q2 This pattern deletes a q-node and moves its children to become children of its parent q-node.

• Q3 This pattern deletes two q-nodes and merges their children to become children of their parent q-node.