The field of fine-grained complexity aims at proving conditional lower bounds on the time complexity of computational problems. One of the most popular assumptions, Strong Exponential Time Hypothesis (SETH), implies that SAT cannot be solved in $2^{(1-\epsilon)n}$ 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. Two barriers for proving SETH-based lower bounds are known. Carmosino et al. (ITCS 2016) introduced the Nondeterministic Strong Exponential Time Hypothesis (NSETH) stating that TAUT cannot be solved in time $2^{(1-\epsilon)n}$ even if one allows nondeterminism. They used this hypothesis to show that some natural fine-grained reductions would be difficult to obtain: proving that, say, 3-SUM requires time $n^{1.5+\epsilon}$ under SETH, breaks NSETH and this, in turn, implies strong circuit lower bounds. Recently, Belova et al. (SODA 2023) introduced the so-called polynomial formulations to show that for many NP-hard problems, proving any explicit exponential lower bound under SETH also implies strong circuit lower bounds. We prove that for a range of problems from P, including $k$-SUM and triangle detection, proving superlinear lower bounds under SETH is challenging as it implies new circuit lower bounds. To this end, we show that these problems can be solved in nearly linear time with oracle calls to evaluating a polynomial of constant degree. Then, we introduce a strengthening of SETH stating that solving SAT in time $2^{(1-\varepsilon)n}$ is difficult even if one has constant degree polynomial evaluation oracle calls. This hypothesis is stronger and less believable than SETH, but refuting it is still challenging: we show that this implies circuit lower bounds.

翻译：细粒度复杂度理论旨在证明计算问题时间复杂度的条件性下界。最广泛使用的假设之一——强指数时间假设（SETH）表明，SAT问题无法在$2^{(1-\epsilon)n}$时间内求解。近年来，已证明许多问题的已知算法在SETH假设下是最优的。尽管SETH具有广泛适用性，但许多问题尚无基于SETH的下界，因此寻找新的归约方法仍需持续推进。已知证明基于SETH的下界存在两大障碍。Carmosino等人（ITCS 2016）引入非确定性强指数时间假设（NSETH），指出即使允许非确定性，TAUT问题也无法在$2^{(1-\epsilon)n}$时间内求解。他们利用该假设表明，某些自然的细粒度归约难以实现：例如，证明3-SUM在SETH下需要$n^{1.5+\epsilon}$时间将导致NSETH失效，进而蕴含强电路下界。近期，Belova等人（SODA 2023）提出所谓的多项式表述，证明对于许多NP难问题，在SETH下证明任何显式指数下界均会导出强电路下界。我们证明，对于$k$-SUM和三角形检测等P类问题，在SETH下证明超线性下界具有挑战性，因为这将蕴含新的电路下界。为此，我们表明这些问题可通过调用常数次多项式评估预言机，在近线性时间内求解。接着，我们引入SETH的强化版本：即使允许调用常数次多项式评估预言机，在$2^{(1-\varepsilon)n}$时间内求解SAT仍然困难。该假设比SETH更强且可信度更低，但证伪它仍具挑战性：我们证明这将导出的电路下界。