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

TL;DR

This paper develops a unified framework to derive nonuniform circuit lower bounds from uniform or nondeterministic conditional assumptions. By connecting SAT-time lower bounds, OV reductions, and tensor/rank phenomena, it shows win-win outcomes: under plausible hypotheses, either E^NP requires large series-parallel circuits or there exist explicit monotone functions of high monotone circuit size; similarly, high matrix rigidity or high tensor rank can be obtained via small generators. The core technique translates uniform nondeterministic bounds into generators of combinatorial objects (monotone functions, rigid matrices, high-rank tensors), yielding conditional lower bounds and new constructions. Overall, the work deepens the bridge between fine-grained complexity and nonuniform circuit lower bounds, offering concrete conditional results and open directions for improving monotone and threshold-size lower bounds with potential impact on arithmetic circuits and complexity theory at large.

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 $ω(n)$. 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 $2^{Ω(n / \log n)}$. This implies win-win circuit lower bounds: either $\text{E}^{\text{NP}}$ requires series-parallel circuits of size $ω(n)$ or coNP requires monotone circuits of size $2^{Ω(n / \log n)}$. 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.

Theorems & Definitions (51)

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