Optimal Query Complexity for Reconstructing Hypergraphs

TL;DR

This paper presents optimal non-adaptive algorithms for reconstructing weighted hypergraphs of constant rank using additive queries, achieving query complexities that match information-theoretic bounds and solving an open problem from 2008.

Contribution

It introduces the first non-adaptive algorithms with optimal query complexity for hypergraph reconstruction, extending previous work to weighted and unweighted cases.

Findings

Query complexity of O(m log n / log m) for hypergraph reconstruction.

Query complexity of O(m log(n^d/m) / log m) for weighted hypergraphs with bounded weights.

Results are tight according to information-theoretic bounds.

Abstract

In this paper we consider the problem of reconstructing a hidden weighted hypergraph of constant rank using additive queries. We prove the following: Let $G$ be a weighted hidden hypergraph of constant rank with n vertices and $m$ hyperedges. For any $m$ there exists a non-adaptive algorithm that finds the edges of the graph and their weights using $$ O(\frac{m\log n}{\log m}) $$ additive queries. This solves the open problem in [S. Choi, J. H. Kim. Optimal Query Complexity Bounds for Finding Graphs. {\em STOC}, 749--758,~2008]. When the weights of the hypergraph are integers that are less than $O(poly(n^d/m))$ where $d$ is the rank of the hypergraph (and therefore for unweighted hypergraphs) there exists a non-adaptive algorithm that finds the edges of the graph and their weights using $$ O(\frac{m\log \frac{n^d}{m}}{\log m}). $$ additive queries. Using the information theoretic bound the above query complexities are tight.