Computations with polynomial evaluation oracle: ruling out superlinear SETH-based 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-ε)n time. In recent years, it has been proved that known algorithms for many problems are optimal under SETH. Despite 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-ε)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+ε 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 NPhard problems, proving any explicit exponential lower bound under SETH also implies strong circuit lower bounds.

In this paper, we combine the two barriers above. 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-ε)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.

Finally, by considering computations that make oracle calls to evaluating constant degree polynomials depending on a small number of variables, we show connections between nondeterministic time lower bounds and arithmetic circuit lower bounds. Namely, we prove that if any of MAX-k-SAT, Binary Permanent, or a variant of Set Cover problems cannot be solved in co-nondeterministic time 2 (1-ε)n , for any ε > 0, then one gets arbitrary large polynomial arithmetic circuit lower bounds.