Conditional Complexity Hardness: Monotone Circuit Size, Matrix Rigidity, and Tensor Rank

TL;DR

This paper develops methods to derive circuit lower bounds and construct combinatorial objects like high-rigidity matrices and high-rank tensors from uniform nondeterministic complexity assumptions, advancing complexity theory.

Contribution

It introduces new connections between uniform nondeterministic lower bounds and the construction of objects with high complexity measures, such as monotone functions, matrices, and tensors.

Findings

Conditional lower bounds imply existence of high-complexity functions.

High rigidity matrices and high-rank tensors can be constructed under certain complexity assumptions.

Results establish a win-win scenario for circuit lower bounds and object complexity.

Abstract

Proving complexity lower bounds remains a challenging task: we only know how to prove conditional uniform lower bounds and nonuniform lower bounds in restricted circuit models. Williams (STOC 2010) showed how to derive nonuniform lower bounds from uniform upper bounds: by designing a fast algorithm for checking satisfiability of circuits, one gets a lower bound for this circuit class. Since then, a number of results of this kind have been proved. For example, Jahanjou et al. (ICALP 2015) and Carmosino et al. (ITCS 2016) proved that if NSETH fails, then $\text{E}^{\text{NP}}$ has series-parallel circuit size $\omega(n)$. One can also derive nonuniform lower bounds from nondeterministic uniform lower bounds. Recent examples include lower bounds on tensor rank, arithmetic circuit size, $\text{ETHR} \circ \text{ETHR}$ circuit size under assumptions that various problems (like TSP, MAX-3-SAT, SAT, Set Cover) cannot be solved faster than in $2^n$ time. In this paper, we continue developing this line of research and show how uniform nondeterministic lower bounds can be used to construct generators of various types of combinatorial objects: Boolean functions of high circuit size, matrices of high rigidity, and tensors of high rank. Specifically, we prove the following. If $k$-SAT cannot be solved in input-oblivious co-nondeterministic time $O(2^{(1/2+\varepsilon)n})$, then there exists a monotone Boolean function family in coNP of monotone circuit size $2^{\Omega(n / \log n)}$. This implies win-win circuit lower bounds: either $\text{E}^{\text{NP}}$ requires series-parallel circuits of size $\omega(n)$ or coNP requires monotone circuits of size $2^{\Omega(n / \log n)}$. If MAX-3-SAT cannot be solved in co-nondeterministic time $O(2^{(1 - \varepsilon)n})$, then there exist small families of matrices with high rigidity as well as small families of three-dimensional tensors of high rank.