  In this paper we study the fine-grained complexity of finding exact and
approximate solutions to problems in P. Our main contribution is showing
reductions from exact to approximate solution for a host of such problems.
  As one (notable) example, we show that the Closest-LCS-Pair problem (Given
two sets of strings $A$ and $B$, compute exactly the maximum $\textsf{LCS}(a,
b)$ with $(a, b) \in A \times B$) is equivalent to its approximation version
(under near-linear time reductions, and with a constant approximation factor).
More generally, we identify a class of problems, which we call BP-Pair-Class,
comprising both exact and approximate solutions, and show that they are all
equivalent under near-linear time reductions.
  Exploring this class and its properties, we also show:
  $\bullet$ Under the NC-SETH assumption (a significantly more relaxed
assumption than SETH), solving any of the problems in this class requires
essentially quadratic time.
  $\bullet$ Modest improvements on the running time of known algorithms
(shaving log factors) would imply that NEXP is not in non-uniform
$\textsf{NC}^1$.
  $\bullet$ Finally, we leverage our techniques to show new barriers for
deterministic approximation algorithms for LCS.
  At the heart of these new results is a deep connection between interactive
proof systems for bounded-space computations and the fine-grained complexity of
exact and approximate solutions to problems in P. In particular, our results
build on the proof techniques from the classical IP = PSPACE result.
