Tighter Connections Between Formula-SAT and Shaving Logs
Amir Abboud, Karl Bringmann

TL;DR
This paper establishes new complexity-theoretic barriers linking the difficulty of shaving logarithmic factors in fundamental problems to the hardness of solving Formula-SAT faster than brute force, thereby setting limits on algorithmic improvements.
Contribution
It proves that achieving certain faster algorithms for problems like LCS would imply breakthroughs in Formula-SAT algorithms, thus establishing tight complexity barriers.
Findings
Shows that faster LCS algorithms imply new Formula-SAT algorithms.
Provides a nearly optimal reduction from SAT to LCS.
Establishes complexity barriers for shaving log factors in fundamental problems.
Abstract
A noticeable fraction of Algorithms papers in the last few decades improve the running time of well-known algorithms for fundamental problems by logarithmic factors. For example, the dynamic programming solution to the Longest Common Subsequence problem (LCS) was improved to 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 ? 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 (Formula-SAT) faster than exhaustive search. They show that an $O(n^2/\log^{1000}…
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