Abstract
A noticeable fraction of Algorithms papers in the last few decades improve the running time of wellknown algorithms for fundamental problems by logarithmic factors. For example, the {O}(n^2) dynamic programming solution to the Longest Common Subsequence problem (LCS) was improved to O(n^2/log^{2}n) in several ways and using a variety of ingenious tricks. This line of research, also known as the art of shaving log factors, lacks a tool for proving negative results. Specifically, how can we show that it is unlikely that LCS can be solved in time O(n^2/log^3n)?
Perhaps the only approach for such results was suggested in a recent paper of Abboud, Hansen, Vassilevska W. and Williams (STOC'16). The authors blame the hardness of shaving logs on the hardness of solving satisfiability on boolean formulas (FormulaSAT) faster than exhaustive search. They show that an O(n^2/log^{1000} n) algorithm for LCS would imply a major advance in circuit lower bounds. Whether this approach can lead to tighter barriers was unclear.
In this paper, we push this approach to its limit and, in particular, prove that a wellknown barrier from complexity theory stands in the way for shaving five additional log factors for fundamental combinatorial problems. For LCS, regular expression pattern matching, as well as the Fréchet distance problem from Computational Geometry, we show that an O(n^2/log^{7+epsilon}{n}) runtime would imply new FormulaSAT algorithms.
Our main result is a reduction from SAT on formulas of size s over n variables to LCS on sequences of length N=2^{n/2} * s^{1+o(1)}. Our reduction is essentially as efficient as possible, and it greatly improves the previously known reduction for LCS with N=2^{n/2} * s^c, for some c >= 100.
BibTeX  Entry
@InProceedings{abboud_et_al:LIPIcs:2018:9012,
author = {Amir Abboud and Karl Bringmann},
title = {{Tighter Connections Between FormulaSAT and Shaving Logs}},
booktitle = {45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)},
pages = {8:18:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959770767},
ISSN = {18688969},
year = {2018},
volume = {107},
editor = {Ioannis Chatzigiannakis and Christos Kaklamanis and D{\'a}niel Marx and Donald Sannella},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2018/9012},
URN = {urn:nbn:de:0030drops90129},
doi = {10.4230/LIPIcs.ICALP.2018.8},
annote = {Keywords: FineGrained Complexity, Hardness in P, FormulaSAT, Longest Common Subsequence, Frechet Distance}
}
Keywords: 

FineGrained Complexity, Hardness in P, FormulaSAT, Longest Common Subsequence, Frechet Distance 
Collection: 

45th International Colloquium on Automata, Languages, and Programming (ICALP 2018) 
Issue Date: 

2018 
Date of publication: 

04.07.2018 