2,546 research outputs found
Variants of Constrained Longest Common Subsequence
In this work, we consider a variant of the classical Longest Common
Subsequence problem called Doubly-Constrained Longest Common Subsequence
(DC-LCS). Given two strings s1 and s2 over an alphabet A, a set C_s of strings,
and a function Co from A to N, the DC-LCS problem consists in finding the
longest subsequence s of s1 and s2 such that s is a supersequence of all the
strings in Cs and such that the number of occurrences in s of each symbol a in
A is upper bounded by Co(a). The DC-LCS problem provides a clear mathematical
formulation of a sequence comparison problem in Computational Biology and
generalizes two other constrained variants of the LCS problem: the Constrained
LCS and the Repetition-Free LCS. We present two results for the DC-LCS problem.
First, we illustrate a fixed-parameter algorithm where the parameter is the
length of the solution. Secondly, we prove a parameterized hardness result for
the Constrained LCS problem when the parameter is the number of the constraint
strings and the size of the alphabet A. This hardness result also implies the
parameterized hardness of the DC-LCS problem (with the same parameters) and its
NP-hardness when the size of the alphabet is constant
The zero exemplar distance problem
Given two genomes with duplicate genes, \textsc{Zero Exemplar Distance} is
the problem of deciding whether the two genomes can be reduced to the same
genome without duplicate genes by deleting all but one copy of each gene in
each genome. Blin, Fertin, Sikora, and Vialette recently proved that
\textsc{Zero Exemplar Distance} for monochromosomal genomes is NP-hard even if
each gene appears at most two times in each genome, thereby settling an
important open question on genome rearrangement in the exemplar model. In this
paper, we give a very simple alternative proof of this result. We also study
the problem \textsc{Zero Exemplar Distance} for multichromosomal genomes
without gene order, and prove the analogous result that it is also NP-hard even
if each gene appears at most two times in each genome. For the positive
direction, we show that both variants of \textsc{Zero Exemplar Distance} admit
polynomial-time algorithms if each gene appears exactly once in one genome and
at least once in the other genome. In addition, we present a polynomial-time
algorithm for the related problem \textsc{Exemplar Longest Common Subsequence}
in the special case that each mandatory symbol appears exactly once in one
input sequence and at least once in the other input sequence. This answers an
open question of Bonizzoni et al. We also show that \textsc{Zero Exemplar
Distance} for multichromosomal genomes without gene order is fixed-parameter
tractable if the parameter is the maximum number of chromosomes in each genome.Comment: Strengthened and reorganize
Distributed PCP Theorems for Hardness of Approximation in P
We present a new distributed model of probabilistically checkable proofs
(PCP). A satisfying assignment to a CNF formula is
shared between two parties, where Alice knows , Bob knows
, and both parties know . The goal is to have
Alice and Bob jointly write a PCP that satisfies , while
exchanging little or no information. Unfortunately, this model as-is does not
allow for nontrivial query complexity. Instead, we focus on a non-deterministic
variant, where the players are helped by Merlin, a third party who knows all of
.
Using our framework, we obtain, for the first time, PCP-like reductions from
the Strong Exponential Time Hypothesis (SETH) to approximation problems in P.
In particular, under SETH we show that there are no truly-subquadratic
approximation algorithms for Bichromatic Maximum Inner Product over
{0,1}-vectors, Bichromatic LCS Closest Pair over permutations, Approximate
Regular Expression Matching, and Diameter in Product Metric. All our
inapproximability factors are nearly-tight. In particular, for the first two
problems we obtain nearly-polynomial factors of ; only
-factor lower bounds (under SETH) were known before
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