Strong computational lower bounds via parameterized complexity

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Abstract

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(nk) 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)no(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 no(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)no(1/ϵ) for any function f unless an unlikely collapse occurs in parameterized complexity theory.

Keywords

Parameterized computation
Computational complexity
Lower bound
Clique
Polynomial time approximation scheme

Cited by (0)

A preliminary version of this paper “Linear FPT reductions and computational lower bounds” was presented at The 36th ACM Symposium on Theory of Computing, STOC 2004, Chicago, June 13–15, 2004 (see [J. Chen, X. Huang, I. Kanj, G. Xia, Linear FPT reductions and computational lower bounds, in: Proc. 36th ACM Symposium on Theory of Computing, STOC '04, 2004, pp. 212–221]).

1

Supported in part by USA National Science Foundation under Grants CCR-0311590 and CCF-0430683, and by China National Natural Science Foundation under Grants Nos. 60373083 and 60433020 while this author was at College of Information Science and Engineering, Central-South University, Changsha, Hunan 410083, PR China.

2

Supported in part by USA National Science Foundation under Grant CCR-0000206.

3

Supported in part by DePaul University Competitive Research Grant.

4

Supported in part by USA National Science Foundation under Grants CCR-0311590 and CCF-0430683.