Finding a Hidden Edge

TL;DR

This paper investigates the problem of finding an edge in a hidden graph using subset queries, revealing that randomized algorithms significantly reduce query complexity compared to deterministic ones, and explores trade-offs between query complexity and adaptivity.

Contribution

It introduces new bounds for edge-finding in hidden graphs under different models and analyzes the impact of adaptivity and graph families on query complexity.

Findings

Randomized algorithms require 0(n) queries, much fewer than deterministic 2(n^2) queries.

Specific graph families like stars, cliques, and matchings have tailored query complexities.

Trade-offs between query complexity and the number of adaptive rounds are established.

Abstract

We consider the problem of finding an edge in a hidden undirected graph $G = (V, E)$ with $n$ vertices, in a model where we only allowed queries that ask whether or not a subset of vertices contains an edge. We study the non-adaptive model and show that while in the deterministic model the optimal algorithm requires $\binom{n}{2}$ queries (i.e., querying for any possible edge separately), in the randomized model $\tilde{\Theta}(n)$ queries are sufficient (and needed) in order to find an edge. In addition, we study the query complexity for specific families of graphs, including Stars, Cliques, and Matchings, for both the randomized and deterministic models. Lastly, for general graphs, we show a trade-off between the query complexity and the number of rounds, $r$, made by an adaptive algorithm. We present two algorithms with $O(rn^{2/r})$ and $\tilde{O}(rn^{1/r})$ sample complexity for the deterministic and randomized models, respectively.