Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Improved Approximation for Longest Common Subsequence over Small
  Alphabets   This paper investigates the approximability of the Longest Common Subsequence
(LCS) problem. The fastest algorithm for solving the LCS problem exactly runs
in essentially quadratic time in the length of the input, and it is known that
under the Strong Exponential Time Hypothesis the quadratic running time cannot
be beaten. There are no such limitations for the approximate computation of the
LCS however, except in some limited scenarios. There is also a scarcity of
approximation algorithms. When the two given strings are over an alphabet of
size $k$, returning the subsequence formed by the most frequent symbol
occurring in both strings achieves a $1/k$ approximation for the LCS. It is an
open problem whether a better than $1/k$ approximation can be achieved in truly
subquadratic time ($O(n^{2-\delta})$ time for constant $\delta>0$).
  A recent result [Rubinstein and Song SODA'2020] showed that a $1/2+\epsilon$
approximation for the LCS over a binary alphabet is possible in truly
subquadratic time, provided the input strings have the same length. In this
paper we show that if a $1/2+\epsilon$ approximation (for $\epsilon>0$) is
achievable for binary LCS in truly subquadratic time when the input strings can
be unequal, then for every constant $k$, there is a truly subquadratic time
algorithm that achieves a $1/k+\delta$ approximation for $k$-ary alphabet LCS
for some $\delta>0$. Thus the binary case is the hardest. We also show that for
every constant $k$, if one is given two strings of \emph{equal} length over a
$k$-ary alphabet, one can obtain a $1/k+\epsilon$ approximation for some
constant $\epsilon>0$ in truly subquadratic time, thus extending the Rubinstein
and Song result to all alphabets of constant size.

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The categories of this paper are: mtrl-th
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The categories of this paper are: supr-con
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The categories of this paper are: solv-int
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The categories of this paper are: bayes-an
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The categories of this paper are: eess
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The categories of this paper are: nucl-ex
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The categories of this paper are: adap-org
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The categories of this paper are: plasm-ph
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The categories of this paper are: atom-ph
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The categories of this paper are: hep-lat
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The categories of this paper are: cond-mat
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The categories of this paper are: quant-ph
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The categories of this paper are: math-ph
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The categories of this paper are: hep-th
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The categories of this paper are: q-fin
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The categories of this paper are: q-alg