Graph Connectivity with Noisy Queries

TL;DR

This paper develops efficient algorithms for identifying network connectivity and spanning trees using noisy edge queries, addressing various error models and graph classes, with theoretical bounds and practical implications.

Contribution

It introduces algorithms for finding spanning trees with noisy, potentially adversarial edge queries across different error regimes and graph families, with matching lower bounds.

Findings

Efficient algorithms for 2-sided error and false positive regimes with matching lower bounds.

Algorithms for false negative regime on bounded treewidth and sparse graphs.

Tight algorithms for planar graphs in all error regimes.

Abstract

Graph connectivity is a fundamental combinatorial optimization problem that arises in many practical applications, where usually a spanning subgraph of a network is used for its operation. However, in the real world, links may fail unexpectedly deeming the networks non-operational, while checking whether a link is damaged is costly and possibly erroneous. After an event that has damaged an arbitrary subset of the edges, the network operator must find a spanning tree of the network using non-damaged edges by making as few checks as possible. Motivated by such questions, we study the problem of finding a spanning tree in a network, when we only have access to noisy queries of the form "Does edge e exist?". We design efficient algorithms, even when edges fail adversarially, for all possible error regimes; 2-sided error (where any answer might be erroneous), false positives (where "no" answers are always correct) and false negatives (where "yes" answers are always correct). In the first two regimes we provide efficient algorithms and give matching lower bounds for general graphs. In the False Negative case we design efficient algorithms for large interesting families of graphs (e.g. bounded treewidth, sparse). Using the previous results, we provide tight algorithms for the practically useful family of planar graphs in all error regimes.