The Strong Exponential Time Hypothesis (SETH) asserts that for every $\varepsilon>0$ there exists $k$ such that $k$-SAT requires time $(2-\varepsilon)^n$. The field of fine-grained complexity has leveraged SETH to prove quite tight conditional lower bounds for dozens of problems in various domains and complexity classes, including Edit Distance, Graph Diameter, Hitting Set, Independent Set, and Orthogonal Vectors. Yet, it has been repeatedly asked in the literature whether SETH-hardness results can be proven for other fundamental problems such as Hamiltonian Path, Independent Set, Chromatic Number, MAX-$k$-SAT, and Set Cover. In this paper, we show that fine-grained reductions implying even $\lambda^n$-hardness of these problems from SETH for any $\lambda>1$, would imply new circuit lower bounds: super-linear lower bounds for Boolean series-parallel circuits or polynomial lower bounds for arithmetic circuits (each of which is a four-decade open question). We also extend this barrier result to the class of parameterized problems. Namely, for every $\lambda>1$ we conditionally rule out fine-grained reductions implying SETH-based lower bounds of $\lambda^k$ for a number of problems parameterized by the solution size $k$. Our main technical tool is a new concept called polynomial formulations. In particular, we show that many problems can be represented by relatively succinct low-degree polynomials, and that any problem with such a representation cannot be proven SETH-hard (without proving new circuit lower bounds).
翻译:强烈的市值时间假说(SETH) 断言, 每1美元 瓦列普西隆0美元, 就会有1美元 。 然而, 文献中反复询问, 每1美元 硬度 是否能够证明其他基本问题, 如汉密尔顿路徑、 独立赛、 铬数字、 美元 数字 、 MAX- 美元 SAT 和 Set Cover。 微量复杂程度 的域域让 SETH 能够证明相当严格的条件下限, 包括编辑距离、 图形仪表、 Hitting Set、 独立赛特 和 Orthogoal 矢量。 然而, 文献中反复询问, 每1美元 硬度 硬度 硬度 标准, 我们的超级硬度 硬度 硬度 硬度 标准 无法证明 其它基本问题, 我们的硬度 硬度 硬度 硬度 标准, 也代表我们 硬度 硬度 标准 的硬度 的硬度 。