Separation of the Monotone NC Hierarchy
- Cite this article as:
- Raz, R. & McKenzie, P. Combinatorica (1999) 19: 403. doi:10.1007/s004930050062
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, for the monotone depth of functions in monotone-P. As a result we achieve the separation of the following classes.
1. monotone-NC ≠ monotone-P.
2. For every i≥1, monotone-\(\)≠ monotone-\(\).
3. More generally: For any integer function D(n), up to \(\) (for some ε>0), we give an explicit example of a monotone Boolean function, that can be computed by polynomial size monotone Boolean circuits of depth D(n), but that cannot be computed by any (fan-in 2) monotone Boolean circuits of depth less than Const·D(n) (for some constant Const).
Only a separation of monotone-\(\) from monotone-\(\) was previously known.
Our argument is more general: we define a new class of communication complexity search problems, referred to below as DART games, and we prove a tight lower bound for the communication complexity of every member of this class. As a result we get lower bounds for the monotone depth of many functions. In particular, we get the following bounds:
1. For st-connectivity, we get a tight lower bound of \(\). That is, we get a new proof for Karchmer–Wigderson's theorem, as an immediate corollary of our general result.
2. For the k-clique function, with \(\), we get a tight lower bound of Ω(k log n). This lower bound was previously known for k≤ log n . For larger k, however, only a bound of Ω(k) was previously known.