Some Hardness Results for Question/answer Games

Question/Answer games (Q/A games) are a generalization of Ulam’s game and they model information extraction in parallel. A Q/A game, G = (D, s, (q1, . . . , qr)), is played on a directed acyclic graph, D = (V, E), with a distinguished start vertex s, between two players, Paul and Carole. In the i-th round, Paul selects a set, Qi ⊆ V , of at most qi non-terminal vertices. Carole responds by choosing an outgoing edge from each vertex in Qi. At the end of r rounds, Paul wins if and only if Carole’s answers define a unique path from the root to one of the terminal vertices in D. We explore the complexity of determining if Carole wins a Q/A game G. We show that the problem is NP-hard if the game is restricted to one question per round, except for the last round. The problem remains NP-hard if we restrict the game to two rounds. The general version of the game is PSPACE-complete.