Fine-grained reduction

In computational complexity theory, a fine-grained reduction is a transformation from one computational problem to another, used to relate the difficulty of improving the time bounds for the two problems. Intuitively, it provides a method for solving one problem efficiently by using the solution to the other problem as a subroutine. If problem $$A$$ can be solved in time $$a(n)$$ and problem $$B$$ can be solved in time $$b(n)$$, then the existence of an $$(a,b)$$-reduction from problem $$A$$ to problem $$B$$ implies that any significant speedup for problem $$B$$ would also lead to a speedup for problem $$A$$.

Definition
Let $$A$$ and $$B$$ be computational problems, specified as the desired output for each possible input. Let $$a$$ and $$b$$ both be time-constructible functions that take an integer argument $$n$$ and produce an integer result. Usually, $$a$$ and $$b$$ are the time bounds for known or naive algorithms for the two problems, and often they are monomials such as $$n^2$$.

Then $$A$$ is said to be $$(a,b)$$-reducible to $$B$$ if, for every real number $$\epsilon>0$$, there exists a real number $$\delta>0$$ and an algorithm that solves instances of problem $$A$$ by transforming it into a sequence of instances of problem $$B$$, taking time $$O\bigl(a(n)^{1-\delta}\bigr)$$ for the transformation on instances of size $$n$$, and producing a sequence of instances whose sizes $$n_i$$ are bounded by $$\sum_i b(n_i)^{1-\epsilon}0$$ such that $$B$$ can be solved in time $$O\bigl(b(n)^{1-\epsilon}\bigr)$$. Then, with these assumptions, there also exists $$\delta>0$$ such that $$A$$ can be solved in time $$O\bigl(a(n)^{1-\delta}\bigr)$$. Namely, let $$\delta$$ be the value given by the $$(a,b)$$-reduction, and solve $$A$$ by applying the transformation of the reduction and using the fast algorithm for $$B$$ for each resulting subproblem.

Equivalently, if $$A$$ cannot be solved in time significantly faster than $$a(n)$$, then $$B$$ cannot be solved in time significantly faster than $$b(n)$$.

History
Fine-grained reductions were defined, in the special case that $$a$$ and $$b$$ are equal monomials, by Virginia Vassilevska Williams and Ryan Williams in 2010. They also showed the existence of $$(n^3,n^3)$$-reductions between several problems including all-pairs shortest paths, finding the second-shortest path between two given vertices in a weighted graph, finding negative-weight triangles in weighted graphs, and testing whether a given distance matrix describes a metric space. According to their results, either all of these problems have time bounds with exponents less than three, or none of them do.

The term "fine-grained reduction" comes from later work by Virginia Vassilevska Williams in an invited presentation at the 10th International Symposium on Parameterized and Exact Computation.

Although the original definition of fine-grained reductions involved deterministic algorithms, the corresponding concepts for randomized algorithms and nondeterministic algorithms have also been considered.