4 research outputs found
Hamming Approximation of NP Witnesses
Given a satisfiable 3-SAT formula, how hard is it to find an assignment to
the variables that has Hamming distance at most n/2 to a satisfying assignment?
More generally, consider any polynomial-time verifier for any NP-complete
language. A d(n)-Hamming-approximation algorithm for the verifier is one that,
given any member x of the language, outputs in polynomial time a string a with
Hamming distance at most d(n) to some witness w, where (x,w) is accepted by the
verifier. Previous results have shown that, if P != NP, then every NP-complete
language has a verifier for which there is no
(n/2-n^(2/3+d))-Hamming-approximation algorithm, for various constants d > 0.
Our main result is that, if P != NP, then every paddable NP-complete language
has a verifier that admits no (n/2+O(sqrt(n log n)))-Hamming-approximation
algorithm. That is, one cannot get even half the bits right. We also consider
natural verifiers for various well-known NP-complete problems. They do have
n/2-Hamming-approximation algorithms, but, if P != NP, have no
(n/2-n^epsilon)-Hamming-approximation algorithms for any constant epsilon > 0.
We show similar results for randomized algorithms
Approximating solution structure of the Weighted Sentence Alignment problem
We study the complexity of approximating solution structure of the bijective
weighted sentence alignment problem of DeNero and Klein (2008). In particular,
we consider the complexity of finding an alignment that has a significant
overlap with an optimal alignment. We discuss ways of representing the solution
for the general weighted sentence alignment as well as phrases-to-words
alignment problem, and show that computing a string which agrees with the
optimal sentence partition on more than half (plus an arbitrarily small
polynomial fraction) positions for the phrases-to-words alignment is NP-hard.
For the general weighted sentence alignment we obtain such bound from the
agreement on a little over 2/3 of the bits. Additionally, we generalize the
Hamming distance approximation of a solution structure to approximating it with
respect to the edit distance metric, obtaining similar lower bounds
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Hamming Approximation of NP Witnesses
Given a satisfiable 3-SAT formula, how hard is it to find an assignment to
the variables that has Hamming distance at most n/2 to a satisfying assignment?
More generally, consider any polynomial-time verifier for any NP-complete
language. A d(n)-Hamming-approximation algorithm for the verifier is one that,
given any member x of the language, outputs in polynomial time a string a with
Hamming distance at most d(n) to some witness w, where (x,w) is accepted by the
verifier. Previous results have shown that, if P != NP, then every NP-complete
language has a verifier for which there is no
(n/2-n^(2/3+d))-Hamming-approximation algorithm, for various constants d > 0.
Our main result is that, if P != NP, then every paddable NP-complete language
has a verifier that admits no (n/2+O(sqrt(n log n)))-Hamming-approximation
algorithm. That is, one cannot get even half the bits right. We also consider
natural verifiers for various well-known NP-complete problems. They do have
n/2-Hamming-approximation algorithms, but, if P != NP, have no
(n/2-n^epsilon)-Hamming-approximation algorithms for any constant epsilon > 0.
We show similar results for randomized algorithms
The intrinsic hardness of structure approximation
A natural question when dealing with problems that are computationally hard to solve
is whether it is possible to compute a solution which is close enough to the optimal
solution for practical purposes. The usual notion of "closeness" is that the value of the
solution is not far from the optimal value. In this M.Sc thesis, we will focus on the
generalization of this notion where closeness is defined with respect to a given distance
function. This framework, named "Structure approximation", was introduced in 2007
paper by Hamilton, Müller, van Rooij and Wareham [HMvRW07], who posed the
question of complexity of approximation of NP-hard optimization problems in this
setting.
In this thesis, we will survey what is known about the complexity of structure approximation,
in particular recasting results on Hamming approximation of NP witnesses in
the Hamilton/Müller/van Rooij/Wareham framework. Specifically, the lower bounds
in the NP witness setting imply that at least for some natural choices of the distance
function any non-trivial structure approximation is as hard to compute as an exact
solution to the original problem. In particular, results of Sheldon and Young (2013)
state, in the language of structure approximation, that it is not possible to compute
more than n/2 + nᵋ bits of an optimal solution correctly in polynomial time without
being able to solve the original problem in polynomial time. Moreover, for some
problems and solution representations, even a polynomial-time algorithm for finding η/2−O([root of]η ln η) bits of an optimal solution can be used to solve the problem exactly
in polynomial time.
In addition to the lower bounds results, we will discuss algorithms and design techniques
that can be used to achieve some degree of structure approximation for several
well-known problems, and look at some traditional approximation algorithms to see
whether the approximate solution they provide is also a structure approximation.
We make a number of observations throughout the thesis, in particular extending
Hamming approximation lower bounds to edit distance approximation with the same
n/2 + nᵋ parameters for several problems.
Finally, we apply these techniques to analyse the structure approximation complexity
of the Phrase Alignment problem in linguistics, in particular Weighted Sentence
Alignment (WSA) problem of DeNero and Klein [DK08]. We discuss several ways
of defining a natural witness for this problem and their implications for its structure
approximability. In a search of the most compact natural witness representation, we
define a still NP-hard restriction of the WSA, a Partition Weighted Sentence Alignment,
and show that the n/2 + nᵋ lower bounds for the Hamming distance and edit
distance apply in this case; this implies structure inapproximability of the WSA itself
with somewhat weaker parameters