Approximation properties of NP minimization classes

Kolaitis, Phokion G.; Thakur, M.N.

doi:10.1109/sct.1991.160280

Cited by 36 publications

(11 citation statements)

References 8 publications

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“…It should be noted here that similar results can be proved for a syntactically defined class of minimization problems, called MIN F + 1 [30], which includes Minimum Vertex Cover and many vertex-deletion and edge-deletion problems in graphs such as Minimum Feedback Arc Set. Cai and Chen [4] proved that the standard parameterizations of all problems in this class are in FPT w .…”

Section: Theorem 613 If a Problem P Is Max-snp-complete (Under The Lsupporting

confidence: 65%

Structure of Polynomial-Time Approximation

Leeuwen

2011

Theory Comput Syst

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Approximation schemes are commonly classified as being either a polynomial-time approximation scheme (ptas) or a fully polynomial-time approximation scheme (fptas). To properly differentiate between approximation schemes for concrete problems, several subclasses have been identified: (optimum-)asymptotic schemes (ptas ∞ , fptas ∞ ), efficient schemes (eptas), and size-asymptotic schemes. We explore the structure of these subclasses, their mutual relationships, and their connection to the classic approximation classes. We prove that several of the classes are in fact equivalent. Furthermore, we prove the equivalence of eptas to so-called convergent polynomial-time approximation schemes. The results are used to refine the hierarchy of polynomial-time approximation schemes considerably and demonstrate the central position of eptas among approximation schemes.We also present two ways to bridge the hardness gap between asymptotic approximation schemes and classic approximation schemes. First, using notions from fixedparameter complexity theory, we provide new characterizations of when problems have a ptas or fptas. Simultaneously, we prove that a large class of problems (including all MAX-SNP-complete problems) cannot have an optimum-asymptotic approximation scheme unless P = NP, thus strengthening results of Arora et al. (J. ACM 45(3):501-555, 1998). Secondly, we distinguish a new property exhibited by many optimization problems: pumpability. With this notion, we considerably generalize several problem-specific approaches to improve the effectiveness of approximation schemes with asymptotic behavior.

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Section: Theorem 613 If a Problem P Is Max-snp-complete (Under The Lsupporting

confidence: 65%

Structure of Polynomial-Time Approximation

Leeuwen

2011

Theory Comput Syst

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“…In particular, we prove that Maximum 0 − 1 Programming and Minimum 0−1 Programming are NPO-complete. This result shows that making use of a natural approximation preserving reducibility is powerful enough to encompass the "duality" problem raised in [29]. (Indeed, in [28] it was shown that this duality does not arise in APX, log-APX, poly-APX, and other subclasses of NPO.)…”

Section: Introductionmentioning

confidence: 89%

Structure in Approximation Classes

Crescenzi¹,

Kann²,

Silvestri³

et al. 1999

SIAM J. Comput.

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Abstract. The study of the approximability properties of NP-hard optimization problems has recently made great advances mainly due to the results obtained in the field of proof checking. The last important breakthrough proves the APX-completeness of several important optimization problems and thus reconciles "two distinct views of approximation classes: syntactic and computational" [S. Khanna et al., in Proc. 35th IEEE Symp. on Foundations of Computer Science, IEEE Computer Society Press, Los Alamitos, CA, 1994, pp. 819-830]. In this paper we obtain new results on the structure of several computationally-defined approximation classes. In particular, after defining a new approximation preserving reducibility to be used for as many approximation classes as possible, we give the first examples of natural NPO-complete problems and the first examples of natural APXintermediate problems. Moreover, we state new connections between the approximability properties and the query complexity of NPO problems. Key words. complexity classes, reducibilities, approximation algorithms AMS subject classifications. 03D30, 68Q15, 68Q20PII. S00975397963042201. Introduction. In his pioneering paper on the approximation of combinatorial optimization problems [22], David Johnson formally introduced the notion of an approximable problem, proposed approximation algorithms for several problems, and suggested a possible classification of optimization problems on the grounds of their approximability properties. Since then it has been clear that, even though the decision versions of most NP-hard optimization problems are many-one polynomial-time reducible to each other, they do not share the same approximability properties. The main reason is that many-one reductions usually do not preserve the objective function and, even when they do, they rarely preserve the quality of the solutions. It is then clear that a stronger kind of reducibility has to be used. Indeed, an approximation preserving reduction not only has to map instances of a problem A to instances of a problem B, but it also has to be able to come back from "good" solutions for B to "good" solutions for A. Surprisingly, the first definition of this kind of reducibility [35] was given a full 13 years after Johnson's paper; after that, at least seven different approximation preserving reducibilities appeared in the literature (see Fig. 1.1). These reducibilities are identical with respect to the overall scheme but differ essentially in the way they preserve approximability: they range from the Strict reducibility in which the error cannot increase to the PTAS-reducibility in which there are basically no restrictions (see also Chapter 3 of [25] and [11]).

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“…Other papers followed with logic-based definitions of optimization classes having natural complete problems, such as the work of Kolaitis and Thakur [KT94,KT95] and of Panconesi and Ranjan [PR90].…”

Section: Complexity Classes Of Optimization Problemsmentioning

confidence: 99%