# Near-optimal small-depth lower bounds for small distance connectivity

@article{Chen2016NearoptimalSL, title={Near-optimal small-depth lower bounds for small distance connectivity}, author={Xi Chen and Igor Carboni Oliveira and Rocco A. Servedio and Li-Yang Tan}, journal={Proceedings of the forty-eighth annual ACM symposium on Theory of Computing}, year={2016} }

We show that any depth-d circuit for determining whether an n-node graph has an s-to-t path of length at most k must have size nΩ(k1/d/d) when k(n) ≤ n1/5, and nΩ(k1/5d/d) when k(n)≤ n. The previous best circuit size lower bounds were nkexp(−O(d)) (by Beame, Impagliazzo, and Pitassi (Computational Complexity 1998)) and nΩ((logk)/d) (following from a recent formula size lower bound of Rossman (STOC 2014)). Our lower bound is quite close to optimal, as a simple construction gives depth-d circuits…

## 17 Citations

### A Near-Optimal Depth-Hierarchy Theorem for Small-Depth Multilinear Circuits

- Computer Science, Mathematics2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS)
- 2018

Two separating examples may be viewed as algebraic analogues of variants of the Graph Reachability problem studied by Chen, Oliveira, Servedio and Tan (STOC 2016), who used them to prove lower bounds for constant-depth Boolean circuits.

### On the Probabilistic Degree of an n-variate Boolean Function

- Computer Science, MathematicsElectron. Colloquium Comput. Complex.
- 2021

This paper shows that if the probabilistic degree of OR is ( log n ) c, then the minimum possible probabilism degree of such an f is at least ( logn ) c / ( c + 1 ) − o ( 1 ) , and it is shown that this is tight up to (log n ) o (1 ) factors.

### Prediction from Partial Information and Hindsight, with Application to Circuit Lower Bounds

- Computer Science, Mathematicscomputational complexity
- 2019

This work proves a stronger result that says, roughly, that the average coordinate looks random to an adversary that is allowed to query other coordinates of the sequence, even if the adversary is non-deterministic.

### Dynamic complexity of Reachability: How many changes can we handle?

- Computer Science, MathematicsICALP
- 2020

This paper shows that, for changes of polylogarithmic size, first-order update formulas suffice for maintaining undirected reachability, and (2) directed reachability under insertions, and for classes of directed graphs for which efficient parallel algorithms can compute non-zero circulation weights, reachability can be maintained with update formulas that may use "modulo 2" quantifiers under changes ofpolylogarital size.

### Shortest path length with bounded-alternation (min, +) formulas

- MathematicsElectron. Colloquium Comput. Complex.
- 2018

Bounded-depth (min,+) formulas computing the shortest path polynomial are studied, and lower bounds parameterized by certain fan-in restrictions on + gates except those at the bottom level are obtained.

### Smaller ACC0 Circuits for Symmetric Functions

- Computer ScienceITCS
- 2022

This paper shows how to construct MODm circuits computing symmetric functions with non-prime power m, with size-depth tradeoffs that beat the longstanding lower bounds for AC0[m] circuits when m is a prime power, and shows that depth-3 CC0 circuits can compute any symmetric function in subexponential size.

### Shortest path length with bounded-alternation $$(\min ,+)$$(min,+) formulas

- GeologyInternational Journal of Advances in Engineering Sciences and Applied Mathematics
- 2018

AbstractWe study bounded-depth $$(\min ,+)$$(min,+) formulas computing the shortest path polynomial. For depth 2d with $$d \ge 2$$d≥2,
we obtain lower bounds parameterized by certain fan-in…

### Improved pseudorandom generators from pseudorandom multi-switching lemmas

- Computer Science, MathematicsAPPROX-RANDOM
- 2019

The pseudorandom multi-switching lemma is derandomized, a randomness-efficient algorithm for sampling restrictions that simultaneously simplify all circuits in a family that achieves the parameters obtained by the (full randomness) multi- Switching lemmas of Impagliazzo, Matthews, and Paturi and H{\aa}stad.

### Dynamic Complexity of Expansion

- Mathematics, Computer ScienceCSR
- 2021

The spectral graph theoretic material of this work is to maintain up to logarithmic powers of the transition matrix of a bounded degree undirected graph in $\DynACz$ under batch changes (insertions and deletions) of $O(\frac{n}}{\log{\log{n}})$ many edges.

### The Cayley Semigroup Membership Problem

- Mathematics
- 2022

The Cayley semigroup membership problem asks, given a multiplication table representing a semigroup 𝑆 , a subset 𝑋 of 𝑆 and an element 𝑡 of 𝑆 , whether 𝑡 can be expressed as a product of…

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