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Free keywords:
Computer Science, Computational Complexity, cs.CC,Computer Science, Data Structures and Algorithms, cs.DS
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 $O(n^2)$ 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 $O(n^2/\log^3 n)$?
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
$O(n^2/\log^{1000} n)$ 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 well-known 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 Fr\'echet distance problem
from Computational Geometry, we show that an $O(n^2/\log^{7+\varepsilon} n)$
runtime would imply new Formula-SAT algorithms.
Our main result is a reduction from SAT on formulas of size $s$ over $n$
variables to LCS on sequences of length $N=2^{n/2} \cdot s^{1+o(1)}$. Our
reduction is essentially as efficient as possible, and it greatly improves the
previously known reduction for LCS with $N=2^{n/2} \cdot s^c$, for some $c \geq
100$.