Adaptive Algorithm for Finding Connected Dominating Sets in Uncertain Graphs

TL;DR

This paper introduces an adaptive algorithm for identifying minimum-weight connected dominating sets in uncertain graphs where node states are probabilistic, providing theoretical guarantees on its performance relative to optimal solutions.

Contribution

It formulates a stochastic variant of the CDS problem and proposes an adaptive algorithm with provable approximation guarantees for uncertain graphs.

Findings

Algorithm achieves an $O(eta \, \log(1/\delta))$ approximation ratio.

Provides theoretical performance bounds for the adaptive approach.

Addresses uncertainty in node activation in wireless network models.

Abstract

The problem of finding a minimum-weight connected dominating set (CDS) of a given undirected graph has been studied actively, motivated by operations of wireless ad hoc networks. In this paper, we formulate a new stochastic variant of the problem. In this problem, each node in the graph has a hidden random state, which represents whether the node is active or inactive, and we seek a CDS of the graph that consists of the active nodes. We consider an adaptive algorithm for this problem, which repeat choosing nodes and observing the states of the nodes around the chosen nodes until a CDS is found. Our algorithms have a theoretical performance guarantee that the sum of the weights of the nodes chosen by the algorithm is at most $O(\alpha \log (1/\delta))$ times that of any adaptive algorithm in expectation, where $\alpha$ is an approximation factor for the node-weighted polymatroid Steiner tree problem and $\delta$ is the minimum probability of possible scenarios on the node states.