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Due to the lack of unconditional polynomial lower bounds, it is now in fashion to prove conditional lower bounds in order to advance our understanding of the class P. The vast majority of these lower bounds are based on one of three famous hypotheses: the 3-SUM conjecture, the all pairs shortest paths (APSP) conjecture, and the Strong Exponential Time Hypothesis. Only circumstantial evidence is known in support of these hypotheses, and no formal relationship between them is known. In hopes of obtaining “less conditional" and therefore more reliable lower bounds, we consider the conjecture that at least one of the above three hypotheses is true. We design novel reductions from 3-SUM, APSP, and CNF-SAT, and derive interesting consequences of this very plausible conjecture, including tight $n^{3-o(1)}$ lower bounds for purely combinatorial problems about the triangles in unweighted graphs; new $n^{1-o(1)}$ lower bounds for the amortized update and query times of dynamic algorithms for Single-Source Reachabil...

Matching Triangles and Basing Hardness on an Extremely Popular Conjecture