Optimal query complexity bounds for finding graphs Original Research Article

We consider the problem of finding an unknown graph by using queries with an additive property. This problem was partially motivated by DNA shotgun sequencing and linkage discovery problems of artificial intelligence. Given a graph, an additive query asks the number of edges in a set of vertices while a cross-additive query asks the number of edges crossing between two disjoint sets of vertices. The queries ask the sum of weights for weighted graphs. For a graph G with n vertices and at most m edges, we prove that there exists an algorithm to find the edges of G using image queries of both types for all m. The bound is best possible up to a constant factor. For a weighted graph with a mild condition on weights, it is shown that image queries are enough provided image for a sufficiently large constant α, which is best possible up to a constant factor if image for any constant image. This settles, in particular, a conjecture of Grebinski [V. Grebinski, On the power of additive combinatorial search model, in: Proceedings of the 4th Annual International Conference on Computing and Combinatorics (COCOON 1998), Taipei, Taiwan, 1998, pp. 194–203] for finding an unweighted graph using additive queries. We also consider the problem of finding the Fourier coefficients of a certain class of pseudo-Boolean functions as well as a similar coin weighing problem.