Erasure-Resilient Sublinear-Time Graph Algorithms

TL;DR

This paper develops sublinear-time algorithms for partially erased graphs, focusing on connectivity testing and average degree estimation, with thresholds for erasure resilience and improvements over previous models.

Contribution

It introduces erasure-resilient algorithms for graph properties, achieving optimal dependence on parameters and bridging query complexities between existing models.

Findings

Connectivity can be tested efficiently if erasures are below a threshold.

Testing becomes linear in graph size when erasures reach the threshold.

New algorithms improve upon previous methods in standard property testing.

Abstract

We investigate sublinear-time algorithms that take partially erased graphs represented by adjacency lists as input. Our algorithms make degree and neighbor queries to the input graph and work with a specified fraction of adversarial erasures in adjacency entries. We focus on two computational tasks: testing if a graph is connected or $\varepsilon$-far from connected and estimating the average degree. For testing connectedness, we discover a threshold phenomenon: when the fraction of erasures is less than $\varepsilon$, this property can be tested efficiently (in time independent of the size of the graph); when the fraction of erasures is at least $\varepsilon,$ then a number of queries linear in the size of the graph representation is required. Our erasure-resilient algorithm (for the special case with no erasures) is an improvement over the previously known algorithm for connectedness in the standard property testing model and has optimal dependence on the proximity parameter $\varepsilon$. For estimating the average degree, our results provide an "interpolation" between the query complexity for this computational task in the model with no erasures in two different settings: with only degree queries, investigated by Feige (SIAM J. Comput. `06), and with degree queries and neighbor queries, investigated by Goldreich and Ron (Random Struct. Algorithms `08) and Eden et al. (ICALP `17). We conclude with a discussion of our model and open questions raised by our work.