Tight lower bounds for certain parameterized NP-hard problems

Chen, Jianer; Chor, Benny; Fellows, Michael R.; Huang, Xiuzhen; Juedes, David; Kanj, Iyad A.; Xia, Ge

doi:10.1016/j.ic.2005.05.001

Cited by 171 publications

(188 citation statements)

References 25 publications

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“…Moreover, the reduction is a linear parameterized reduction, that is, the new parameter is bounded in a linear function of the old parameter. Hence, an n o( ) algorithm for Highly Connected Subgraph implies an n o(k) algorithm for Hitting Set which implies FPT = W[1] [5].…”

Section: It Remains To Show Thatmentioning

confidence: 99%

Finding Highly Connected Subgraphs

Hüffner

Komusiewicz

Sorge

2015

Lecture Notes in Computer Science

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Abstract. A popular way of formalizing clusters in networks are highly connected subgraphs, that is, subgraphs of k vertices that have edge connectivity larger than k/2 (equivalently, minimum degree larger than k/2). We examine the computational complexity of finding highly connected subgraphs. We show that the problem is NP-hard. Thus, we explore possible parameterizations, such as the solution size, number of vertices in the input, the size of a vertex cover in the input, and the number of edges outgoing from the solution (edge isolation). For some parameters, we find strong intractability results; among the parameters yielding tractability, the edge isolation seems to provide the best trade-off between running time bounds and a small value of the parameter in relevant instances.

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Section: It Remains To Show Thatmentioning

confidence: 99%

Finding Highly Connected Subgraphs

Hüffner

Komusiewicz

Sorge

2015

Lecture Notes in Computer Science

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“…We have seen that CSP problems constitute a rich source of NP-intermediate problems via different kinds of parameterizations, Hence, it appears feasible that methods for studying the complexity of parameterized problems will become highly relevant. In particular, linear fptreductions [10,11] have been used for proving particularly strong lower bounds which may be used for linking together NP-intermediate problems, parameterized problems, and lower bound assumptions.…”

Section: Future Workmentioning

confidence: 99%

Constructing NP-intermediate problems by blowing holes with parameters of various properties

Jönsson

Lagerkvist

Nordh

2015

Theoretical Computer Science

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The search for natural NP-intermediate problems is one of the holy grails within computational complexity. Ladner's original diagonalization technique for generating NP-intermediate problems, blowing holes, has a serious shortcoming: it creates problems with a highly artificial structure by arbitrarily removing certain problem instances. In this article we limit this problem by generalizing Ladner's method to use parameters with various characteristics. This allows one to define more fine-grained parameters, resulting in NP-intermediate problems where we only blow holes in a controlled subset of the problem. We begin by fully characterizing the problems that admit NP-intermediate subproblems for a broad and natural class of parameterizations, and extend the result further such that structural CSP restrictions based on parameters that are hard to compute (such as tree-width) are covered, thereby generalizing a result by Grohe. For studying certain classes of problems, including CSPs parameterized by constraint languages, we consider more powerful parameterizations. First, we identify a new method for obtaining constraint languages Γ such that CSP(Γ) are NP-intermediate. The sets Γ can have very different properties compared to previous constructions (by, for instance, Bodirsky & Grohe) and provides insights into the algebraic approach for studying the complexity of infinite-domain CSPs. Second, we prove that the propositional abduction problem parameterized by constraint languages admits NP-intermediate problems.This settles an open question posed by Nordh & Zanuttini.

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“…For ε := 1/k determine the algorithm A ε according to (2) in Proposition 5 in time f (k) and apply A ε to x. Altogether, we need time…”

Section: The Miniaturization Of An Arbitrary Problemmentioning

confidence: 99%

“…For part (2), it suffices to consider circuits C, whose underlying graph is connected. Since such a graph with n nodes has at least n − 1 edges, we see that C 0 ≥ i + 2 · (i − 1) · log i, where i is the number of input nodes of C. Thus, i ∈ o eff ( C 0 ).…”

Section: Proposition 12 (1) Formentioning

confidence: 99%

On Miniaturized Problems in Parameterized Complexity Theory

Chen

Flum

2004

Parameterized and Exact Computation

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We introduce a general notion of miniaturization of a problem that comprises the different miniaturizations of concrete problems considered so far. We develop parts of the basic theory of miniaturizations. Using the appropriate logical formalism, we show that the miniaturization of a definable problem in W[t] lies in W[t], too. In particular, the miniaturization of the dominating set problem is in W[2]. Furthermore we investigate the relation between f (k) · n o(k) time and subexponential time algorithms for the dominating set problem and for the clique problem. . IntroductionParameterized complexity theory provides a framework for a refined complexity analysis of algorithmic problems that are intractable in general. Central to the theory is the notion of fixed-parameter tractability, which relaxes the classical notion of tractability, polynomial time computability, by admitting algorithms whose runtime is exponential, but only in terms of some parameter that is usually expected to be small. Let FPT denote the class of all fixed-parameter tractable problems. A well-known example of a problem in FPT is the vertex cover problem, the parameter being the size of the vertex cover we ask for.As a complexity theoretic counterpart, a theory of parameterized intractability has been developed. In classical complexity, the notion of NP-completeness is central to a nice and simple theory for intractable problems. Unfortunately, the world of parameterized intractability is more complex: there is a big variety of seemingly different classes of parameterized intractability. For a long while, the smallest complexity class of parameterized intractable problems considered in the literature was W Most of them are "miniaturizations" of well-studied problems in parameterized complexity theory; for example, mini-CIRCSAT is the problem that takes a circuit C of size ≤ k · log m, where k is the parameter and m in unary is part of the input, and asks whether C is satisfiable. This problem is called a miniaturization of CIRCSAT, as the size (≤ k · log m) of C is small compared with m (under the basic assumption of parameterized complexity that the parameter k is small too). In [6], Downey et al. introduce the class MINI[1] as the class of parameterized problems fpt-reducible to mini-CIRCSAT. MINI[1] now provides very nice connections between classical complexity and parameterized complexity as it is known that FPT = MINI [1] if and only if n variable 3SAT can be solved in time 2 o(n) . This equivalence stated in [6] is based on a result of Cai and Juedes [1].Besides this "miniaturization route", a second route to MINI[1] has been considered by Fellows in [9]; he calls it the "renormalization route" to MINI[1]. He "renormalizes" the parameterized vertex cover problem and considers the so-called 1

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Tight lower bounds for certain parameterized NP-hard problems

Cited by 171 publications

References 25 publications

Finding Highly Connected Subgraphs

Finding Highly Connected Subgraphs

Constructing NP-intermediate problems by blowing holes with parameters of various properties

On Miniaturized Problems in Parameterized Complexity Theory

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