Abstract
One of the major challenges of the research in circuit complexity is proving super-polynomial lower bounds for de Morgan formulas. Karchmer et al. (Comput Complex 5(3/4):191–204, 1995b) suggested to approach this problem by proving that formula complexity behaves “as expected” with respect to the composition of functions \({f\diamond g}\) . They showed that this conjecture, if proved, would imply super-polynomial formula lower bounds.
The first step toward proving the KRW conjecture was made by Edmonds et al. (Comput Complex 10(3):210–246, 2001), who proved an analogue of the conjecture for the composition of “universal relations.” In this work, we extend the argument of Edmonds et al. (2001) further to \({f\diamond g}\) where f is an arbitrary function and g is the parity function. While this special case of the KRW conjecture was already proved implicitly in Håstad’s work on random restrictions (Håstad in SIAM J Comput 27(1):48–64, 1998), our proof seems more likely to be generalizable to other cases of the conjecture. In particular, our proof uses an entirely different approach, based on communication complexity technique of Karchmer & Wigderson in (SIAM J Discrete Math 3(2):255–265, 1990). In addition, our proof gives a new structural result, which roughly says that the naive way for computing \({f\diamond g}\) is the only optimal way. Along the way, we obtain a new proof of the state-of-the-art formula lower bound of n 3-o(1) due to Håstad (1998).
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Dinur, I., Meir, O. Toward the KRW Composition Conjecture: Cubic Formula Lower Bounds via Communication Complexity. comput. complex. 27, 375–462 (2018). https://doi.org/10.1007/s00037-017-0159-x
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DOI: https://doi.org/10.1007/s00037-017-0159-x
Keywords
- Communication Complexity
- de-Morgan formulas
- lower bounds
- Karchmer–Wigderson relations
- Karchmer–Wigderson games
- KRW conjecture