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 article, 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).

The field of fine-grained complexity aims at proving conditional lower bounds on the time complexity of computational problems. One of the most popular and successfully used assumptions, Strong Exponential Time Hypothesis (SETH), implies that SAT cannot be solved in 2(1-ɛ)n time. In recent years, it has been proved that known algorithms for many problems are optimal under SETH. Despite the wide applicability of SETH, for many problems, there are no known SETH-based lower bounds, so the quest for new reductions continues.

The Strong Exponential Time Hypothesis (SETH) asserts that for every ε > 0 there exists k such that k-SAT requires time (2 — ε)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 λn-hardness of these problems from SETH for any λ > 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 λ > 1, we conditionally rule out fine-grained reductions implying SETH-based lower bounds of λk: 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). * The full version of the paper can be accessed at https://arxiv.org/abs/2205.07709