Polynomial Time Optimal Query Algorithms for Finding Graphs with Arbitrary Real Weights

We consider the problem of finding the edges of a hidden weighted graph and their weights by using a certain type of queries as few times as possible, with focusing on two types of queries with additive property. For a set of vertices, the additive query asks the sum of weights of the edges with both ends in the set. For a pair of disjoint sets of vertices, the cross-additive query asks the sum of weights of the edges crossing between the two sets. These queries are related to DNA sequencing and finding Fourier coefficients of pseudo-Boolean functions, and have been paid attention to in computational learning. In this paper, we achieve an ultimate goal of recent years for graph finding, by constructing the first polynomial time algorithms with optimal query complexity for the general class of graphs with n vertices and at most m edges in which the weights of edges are arbitrary real numbers. The algorithms are randomized and their query complexities are O ( m logn logm ) which improve the best known bounds by a factor of logm. To build a key component for graph finding, we consider coin weighing with a spring scale which itself has been paid attention to in a long history of combinatorial search. We construct the first polynomial time algorithm with optimal query complexity for the general case in which the weight differences between counterfeit and authentic coins are arbitrary real numbers. We also construct the first polynomial time optimal query algorithm for finding Fourier coefficients of a certain class of pseudo-Boolean functions.