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

2024-11-05

AI Summary

This work addresses the longstanding barrier in complexity theory concerning non-uniform lower bound proofs, focusing on three fundamental measures: monotone circuit size, matrix rigidity, and tensor rank. Leveraging the Nondeterministic Strong Exponential Time Hypothesis (NSETH) and its variants, we establish the first unified framework that systematically translates co-nondeterministic time lower bounds—such as the impossibility of solving k-SAT or MAX-3-SAT in O(2^{(1/2+ε)n}) co-nondeterministic time—into constructive combinatorial lower bounds. Our approach integrates pseudorandomness, circuit complexity, algebraic analysis, and explicit construction techniques. Key results include: assuming no O(2^{(1/2+ε)n}) co-nondeterministic algorithm for k-SAT, there exists an explicit family of Boolean functions requiring monotone circuits of size 2^{Ω(n/log n)}; moreover, we obtain explicit families of highly rigid matrices and high-rank 3D tensors. This reveals a “win-win” lower-bound structure, simultaneously advancing all three central complexity measures.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Machine Learning: Matrix & Tensor Methods

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

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 $ ext{E}^{ ext{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, $ ext{ETHR} circ ext{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 $ ext{E}^{ ext{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.

Problem

• Proving nonuniform complexity lower bounds from uniform assumptions
• Exploring connections between circuit size, matrix rigidity, and tensor rank
• Deriving win-win lower bounds under nondeterministic time hypotheses

Innovation

• Using uniform nondeterministic lower bounds
• Constructing generators for combinatorial objects
• Proving win-win circuit lower bounds