NDLI: Formulas vs. circuits for small distance connectivity

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Formulas vs. circuits for small distance connectivity

Content Provider	ACM Digital Library
Author	Rossman, Benjamin
Abstract	We give the first super-polynomial separation in the power of bounded-depth boolean formulas vs. circuits. Specifically, we consider the problem Distance k(n) Connectivity, which asks whether two specified nodes in a graph of size n are connected by a path of length at most k(n). This problem is solvable (by the recursive doubling technique) on circuits of depth O(log k) and size $O(kn^{3}).$ In contrast, we show that solving this problem on formulas of depth log n/(log log $n)^{O(1)}$ requires size nΩ(log k) for all k(n) ≤ log log n. As corollaries: (i) It follows that polynomial-size circuits for Distance k(n) Connectivity require depth Ω(log k) for all k(n) ≤ log log n. This matches the upper bound from recursive doubling and improves a previous Ω(log log k) lower bound of Beame, Impagliazzo and Pitassi [BIP98]. (ii) We get a tight lower bound of $s^{Ω(d)}$ on the size required to simulate size-s depth-d circuits by depth-d formulas for all s(n) = $n^{O(1)}$ and d(n) ≤ log log log n. No lower bound better than $s^{Ω(1)}$ was previously known for any d(n) ≮ O(1). Our proof technique is centered on a new notion of pathset complexity, which roughly speaking measures the minimum cost of constructing a set of (partial) paths in a universe of size n via the operations of union and relational join, subject to certain density constraints. Half of our proof shows that bounded-depth formulas solving Distance k(n) Connectivity imply upper bounds on pathset complexity. The other half is a combinatorial lower bound on pathset complexity.
Starting Page	203
Ending Page	212
Page Count	10
File Format	PDF MP4
ISBN	9781450327107
DOI	10.1145/2591796.2591828
Language	English
Publisher	Association for Computing Machinery (ACM)
Publisher Date	2014-05-31
Publisher Place	New York
Access Restriction	Subscribed
Content Type	Audio Text
Resource Type	Article

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