重建超图的最优查询复杂度
Optimal Query Complexity for Reconstructing Hypergraphs
摘要：
在本文中，我们考虑的问题，重建一个隐藏的加权超图的常秩使用加法查询。证明了：设$G$是一个n阶加权常秩隐超图，其超边为$m$.对于任何$m$，存在使用$$ O(\frac{m\log n}{\log m}) $$加法查询来找到图的边及其权重的非自适应算法。这解决了[S. Choi，J. H. Kim.查找图的最佳查询复杂度界限。{ \em STOC}，749- 758，2008年7月]。当超图的权重是小于$O(poly(n^d/m))$的整数时，其中$d$是超图的秩（并且因此对于未加权超图），存在使用$$ O(\frac{m\log \frac{n^d}{m}}{\log m}). $$加法查询来找到图的边及其权重的非自适应算法。使用信息论界，上述查询复杂度是紧的。
英文摘要：
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.
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