Fine-grained Complexity Meets IP = PSPACE

Lijie Chen, Shafi Goldwasser, Kaifeng Lyu, Guy N. Rothblum, Aviad Rubinstein

Abstract: 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 an exact to an 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 LCS(a, b) with (a, b) ∊ A × 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: Under the NC-SETH assumption (a significantly more relaxed assumption than SETH), solving any of the problems in this class requires essentially quadratic time. Modest improvements on the running time of known algorithms (shaving log factors) would imply that NEXP is not in non-uniform NC1. Finally, we leverage our techniques to show new barriers for deterministic approximation algorithms for LCS. A very important consequence of our results is that they continue to hold in the data structure setting. In particular, it shows that a data structure for approximate Nearest Neighbor Search for LCS (NNSLCS) implies a data structure for exact NNSLCS and a data structure for answering regular expression queries with essentially the same complexity. 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.