Lower Bounds for Swapping Arthur and Merlin

We prove a lower bound for swapping the order of Arthur and Merlin in two-round MerlinArthur games using black-box techniques. Namely, we show that any AM-game requires time (t2 ) to black-box simulate MA-games running in time t. Thus, the known simulations of MA by AM with quadratic overhead, dating back to Babai's original paper on Arthur-Merlin games, are tight within this setting. The black-box lower bound also yields an oracle relative to which MA-TIME[n] AM-TIME[o(n2 )]. Complementing our lower bounds for swapping Merlin in MA-games, we prove a time-space lower bound for simulations that drop Merlin entirely. We show that for any c < 2, there exists a positive d such that there is a language recognized by linear-time MA-games with one-sided error but not by probabilistic random-access machines with two-sided error that run in time nc and space nd . This improves recent results that give such lower bounds for problems in the second level of the polynomial-time hierarchy....

• Type: Conference
• Year: 2007
• Where: APPROX
• Authors: Scott Diehl
• Keywords: Algorithms, APPROX 2007, Black-box, Black-box Lower Bound, Lower Bounds