Fine-grained Complexity Meets IP = PSPACE

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 NC 1 .• 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 (NNS LCS ) implies a data structure for exact NNS LCS 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 boundedspace 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. * MIT, lijieche@mit.edu.The study of the fine-grained hardness of problems in P is one of the most interesting developments of the last few years in complexity theory. The study was initially aimed at the complexity of exact versions of important problem in P, such as Longest Common Subsequence (LCS), Edit Distance, All Pair Shortest Path (APSP), and 3-SUM. This was the natural starting point. There are several main thrusts of the study: establishing equivalence classes of problems that are "equivalent" to each other in the sense that a substantial improvement in one would imply a similar improvement in the other; showing fine-grained hardness under complexity assumptions, most notably the SETH1; and showing implications of even slight algorithmic improvements, such as "shaving-logs" off algorithms for P time problems, to circuit lower bounds.However, for many of these problems, approximate solutions are of interest as well, as they originate in natural problems which arise in pattern matching and bioinformatics [AVW14, BI15, BI16, BGL17, BK18], dynamic data structures [Pat10, AV14, AV14, HKNS15, KPP16, AD16, HLNV17, GKLP17], graph algorithms [RW13, GIKW17, AVY15, KT17], computational geometry [Bri14, Wil18a, DKL18, Che18] and ...

Asymmetric Streaming Algorithms for Edit Distance and LCS

“…More recently constant factor approximation algorithms with truly subquadratic runtimes are obtained for edit distance (a question which was open for a few decades): first a quantum algorithm [18], then a classic solution [23], and finally for far strings, near linear time solutions are also given [51,44]. LCS has also received tremendous attention in recent years [38,52,53,1,4,26]. Only trivial solutions were known for LCS until very recently: a 2 approximate solution when the alphabet is 0/1 and an O( √ n) approximate solution for general alphabets in linear time.…”

An Equivalence Class for Orthogonal Vectors

“…Lemma 4.4 (Theorem 4.1 of[Rub18]). If Bichrom.-ℓ p -Closest-Pair or ℓ p -Furthest-Pair can be approximated in truly subquadratic time for any p ∈ [1, 2] or Max-IP can be additively approximated in truly subquadratic time, then OV is in truly subquadratic time 18. …”

Fine-Grained Completeness for Optimization in P

“…Studying the fine-grained approximability of polynomial-time optimization problems (hardness of approximation in P), is a recent and influential trend: After a breakthrough result by Abboud, Rubinstein, and Williams [3] establishing the Distributed PCP in P framework, a number of works gave strong conditional lower bounds, including results for nearest neighbor search [28] or a tight characterization of the approximability of maximum inner product [13,15]. Further results include work on approximating graph problems [25,7,10,22], the Fréchet distance [8], LCS [1, 2], monochromatic inner product [23], earth mover distance [26], as well as equivalences for fine-grained approximation in P [15,14,10]. Related work studies the inapproximability of parameterized problems, ruling out certain approximation guarantees within running time f (k)n g (k) under parameter k (such as FPT time f (k) poly(n), or n o(k) ), see [17] for a recent survey.…”