The 1-versus-2 queries problem revisited

• Authors: Rahul Tripathi
• Venue: ISAAC'07 Proceedings of the 18th international conference on Algorithms and computation
• Year: 2007

The 1-versus-2 queries problem, which has been extensively studied in computational complexity theory, asks in its generality whether every efficient algorithm that makes at most 2 queries to a Σkp-complete set Lk has an efficient simulation that makes at most 1 query to Lk. We obtain solutions to this problem under hypotheses weaker than previously considered. We prove that: 1. For each k ≥ 2, PttΣkp[2] ⊆ ZPPΣkp[1] ⇒ PH = Σkp, and 2. PttNP[2] ⊆ ZPPNP[1] ⇒ PH = S2p. Here, for a complexity class C and integer j ≥ 1, we model ZPPC[j] to be the class of problems solvable by zero-error randomized algorithms that always run in polynomial time, make at most j queries to C, and succeed with probability only 1/2 + 1/poly(ċ). This same model of ZPPC[j], also considered in [CC06], subsumes the class of problems solvable by randomized algorithms that always answer correctly in expected polynomial time and make at most j queries to C. Hemaspaandra, Hemaspaandra, and Hempel [HHH98], for k 2, and Buhrman and Fortnow [BF99], for k = 2, had obtained the same consequence as of ours in (1) using the stronger hypothesis PttΣkp[2] ⊆ PΣkp[1]. Fortnow, Pavan, and Sengupta [FPS] had obtained the same consequence as of ours in (2) using the stronger hypothesis PttNP[2] ⊆ PNP[1]. Our results may also be viewed as steps towards obtaining solutions to the most general form of the 1-versus-2 queries problem: For any k ≥ 1, whether PttΣkp[2] can be simulated in BPPΣkp[1].