66,219 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
Metrical Service Systems with Multiple Servers
We study the problem of metrical service systems with multiple servers
(MSSMS), which generalizes two well-known problems -- the -server problem,
and metrical service systems. The MSSMS problem is to service requests, each of
which is an -point subset of a metric space, using servers, with the
objective of minimizing the total distance traveled by the servers.
Feuerstein initiated a study of this problem by proving upper and lower
bounds on the deterministic competitive ratio for uniform metric spaces. We
improve Feuerstein's analysis of the upper bound and prove that his algorithm
achieves a competitive ratio of . In the randomized
online setting, for uniform metric spaces, we give an algorithm which achieves
a competitive ratio , beating the deterministic lower
bound of . We prove that any randomized algorithm for
MSSMS on uniform metric spaces must be -competitive. We then
prove an improved lower bound of on
the competitive ratio of any deterministic algorithm for -MSSMS, on
general metric spaces. In the offline setting, we give a pseudo-approximation
algorithm for -MSSMS on general metric spaces, which achieves an
approximation ratio of using servers. We also prove a matching
hardness result, that a pseudo-approximation with less than servers is
unlikely, even for uniform metric spaces. For general metric spaces, we
highlight the limitations of a few popular techniques, that have been used in
algorithm design for the -server problem and metrical service systems.Comment: 18 pages; accepted for publication at COCOON 201
Edit Distance for Pushdown Automata
The edit distance between two words is the minimal number of word
operations (letter insertions, deletions, and substitutions) necessary to
transform to . The edit distance generalizes to languages
, where the edit distance from to
is the minimal number such that for every word from
there exists a word in with edit distance at
most . We study the edit distance computation problem between pushdown
automata and their subclasses. The problem of computing edit distance to a
pushdown automaton is undecidable, and in practice, the interesting question is
to compute the edit distance from a pushdown automaton (the implementation, a
standard model for programs with recursion) to a regular language (the
specification). In this work, we present a complete picture of decidability and
complexity for the following problems: (1)~deciding whether, for a given
threshold , the edit distance from a pushdown automaton to a finite
automaton is at most , and (2)~deciding whether the edit distance from a
pushdown automaton to a finite automaton is finite.Comment: An extended version of a paper accepted to ICALP 2015 with the same
title. The paper has been accepted to the LMCS journa
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