Strong computational lower bounds via parameterized complexity
We develop new techniques for deriving strong computational lower bounds for a class of well-known NP-hard problems. This class includes weighted satisfiability, dominating set, hitting set, set cover, clique, and independent set. For example, although a trivial enumeration can easily test in time O...
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| Published in: | Journal of computer and system sciences Vol. 72; no. 8; pp. 1346 - 1367 |
|---|---|
| Main Authors: | , , , |
| Format: | Journal Article |
| Language: | English |
| Published: |
Elsevier Inc
01.12.2006
|
| Subjects: | |
| ISSN: | 0022-0000, 1090-2724 |
| Online Access: | Get full text |
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| Summary: | We develop new techniques for deriving strong computational lower bounds for a class of well-known NP-hard problems. This class includes
weighted satisfiability,
dominating set,
hitting set,
set cover,
clique, and
independent set. For example, although a trivial enumeration can easily test in time
O
(
n
k
)
if a given graph of
n vertices has a clique of size
k, we prove that unless an unlikely collapse occurs in parameterized complexity theory, the problem is not solvable in time
f
(
k
)
n
o
(
k
)
for
any function
f, even if we restrict the parameter values to be bounded by an arbitrarily small function of
n. Under the same assumption, we prove that even if we restrict the parameter values
k to be of the order
Θ
(
μ
(
n
)
)
for
any reasonable function
μ, no algorithm of running time
n
o
(
k
)
can test if a graph of
n vertices has a clique of size
k. Similar strong lower bounds on the computational complexity are also derived for other NP-hard problems in the above class. Our techniques can be further extended to derive computational lower bounds on polynomial time approximation schemes for NP-hard optimization problems. For example, we prove that the NP-hard
distinguishing substring selection problem, for which a polynomial time approximation scheme has been recently developed, has no polynomial time approximation schemes of running time
f
(
1
/
ϵ
)
n
o
(
1
/
ϵ
)
for any function
f unless an unlikely collapse occurs in parameterized complexity theory. |
|---|---|
| ISSN: | 0022-0000 1090-2724 |
| DOI: | 10.1016/j.jcss.2006.04.007 |