A survey of lower bounds for satisfiability and related problems
Foundations and Trends® in Theoretical Computer Science
Hardness amplification proofs require majority
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
Bounds on the Size of Small Depth Circuits for Approximating Majority
ICALP '09 Proceedings of the 36th International Colloquium on Automata, Languages and Programming: Part I
Towards a Study of Low-Complexity Graphs
ICALP '09 Proceedings of the 36th International Colloquium on Automata, Languages and Programming: Part I
Alternation-Trading Proofs, Linear Programming, and Lower Bounds
ACM Transactions on Computation Theory (TOCT)
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We prove new results on the circuit complexity of Approximate Majority, which is the problem of computing Majority of a given bit string whose fraction of 1's is bounded away from 1/2 (by a constant). We then apply these results to obtain new relationships between probabilistic time, BPTime (t), and alternating time, \Sigma_{O(1)}Time (t). Our main results are the following: 1. We prove that (2^n^0.1)-size depth-3 circuits for Approximate Majority on n bits have bottom fan-in \Omega(log n). As a corollary we obtain that BPTime (t) \not\subseteq\Sigma_{2}Time (o(t^2)) with respect to some oracle. This complements the result that BPTime (t) \subseteq\Sigma_{2}Time (t^2 \cdot poly log t) with respect to every oracle (Sipser and Gacs, STOC '83; Lautemann, IPL '83). 2. We prove that Approximate Majority is computable by uniform polynomial-size circuits of depth 3. Prior to our work, the only known polynomial-size depth-3 circuits for Approximate Majority were non-uniform (Ajtai, Ann. Pure Appl. Logic '83). We also prove that BPTime (t) \subseteq \Sigma_{3}Time (t \cdot poly log t). This complements our results in (1). 3. We prove new lower bounds for solving QSAT_3 \in \Sigma_{3}Time (n \cdot poly log n) on probabilistic computational models. In particular, we prove that solving QSAT_3 requires time n^{1+\Omega(1)} on Turing machines with a random-access input tape and a sequentialaccess work tape that is initialized with random bits. No lower bound was previously known on this model (for a function computable in linear space).