Combinatorial search in two and more rounds.

In a combinatorial search problem we wish to identify an unknown element by binary tests, where the edges of a hypergraph specify the available tests. We show that, for rather general cases of this problem, the worst-case minimum number of tests, even if adaptive testing is permitted, can already be achieved in a small number of rounds of parallel tests. In particular, the maximum number of necessary rounds grows only as the square root of the number of elements, and two rounds are enough if, e.g., the test number is close to the number of elements, or the hypergraph is a graph. We also provide polynomial-time, hardness, and parameterized results on the computational complexity of finding optimal strategies for some cases, including graphs and tree hypergraphs. • We study multi-round combinatorial search on test hypergraphs in full generality. • We show a tight bound on the number of rounds needed for the optimal test number. • In several natural cases, strategies with optimal test number need only two rounds. • Related computational problems are NP-hard but fixed-parameter tractable. • The case of graphs can be completely solved in polynomial time. [ABSTRACT FROM AUTHOR]