Logical Definability of NP Optimization Problems

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

doi:10.1006/inco.1994.1100

Cited by 71 publications

(40 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In this paper, we attempt to characterize the optimization versions of P via expressions in second order logic, many of them using universal Horn formulae with successor relations. These results nicely complement those of Kolaitis and Thakur [13] for polynomially bounded NP-optimization problems. The polynomially bounded versions of maximization and minimization problems are treated first, and then the maximization problems in the "not necessarily polynomially bounded" class.…”

supporting

confidence: 82%

“…A few attempts to characterize approximation classes in terms of logic are: Papadimitriou and Yannakakis in 1991 [17], Panconesi and Ranjan in 1993 [15], Kolaitis and Thakur in 1994 and 1995 [13,14], and Khanna et al in 1998 [11].…”

Section: Introductionmentioning

confidence: 99%

“…In [13,15,17], the authors characterize approximation hardness in terms of quantifier complexity -the number and types of quantifiers that appear at the beginning of a second-order formula in prenex normal form (PNF). For a formula in PNF, all quantifiers appear at the beginning, followed by a quantifier-free formula.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Untitled

Manyem¹

2008

Chicago J. of Theoretical Comp. Sci.

View full text Add to dashboard Cite

The characterization of important complexity classes by logical descriptions has been an important and prolific area of Descriptive complexity. However, the central focus of the research has been the study of classes like P, NP, L and NL, corresponding to decision problems (e.g. the characterization of NP by Fagin [5] and of P by Grädel [7]). In contrast, optimization problems have received much less attention. Optimization problems corresponding to the NP class have been characterized in terms of logic expressions by Papadimitriou and Yannakakis, Panconesi and Ranjan, Kolaitis and Thakur, Khanna et al, and by Zimand. In this paper, we attempt to characterize the optimization versions of P via expressions in second order logic, many of them using universal Horn formulae with successor relations. These results nicely complement those of Kolaitis and Thakur [13] for polynomially bounded NP-optimization problems. The polynomially bounded versions of maximization and minimization problems are treated first, and then the maximization problems in the "not necessarily polynomially bounded" class.

show abstract

supporting

confidence: 82%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Untitled

Manyem¹

2008

Chicago J. of Theoretical Comp. Sci.

View full text Add to dashboard Cite

show abstract

“…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%

Inapproximability of Combinatorial Optimization Problems

Trevisan

2014

Paradigms of Combinatorial Optimization

View full text Add to dashboard Cite

show abstract

“…Kolaitis and Thakur [29] proved that MAX-SNP ⊂ MAX-NP, as Maximum Satisfiability is in MAX-NP, but not in MAX-SNP.…”

Section: Existence Of Optimum-asymptotic Approximation Schemesmentioning

confidence: 99%

Structure of Polynomial-Time Approximation

Leeuwen

2011

Theory Comput Syst

View full text Add to dashboard Cite

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.

show abstract

Logical Definability of NP Optimization Problems

Cited by 71 publications

References 0 publications

Untitled

Untitled

Inapproximability of Combinatorial Optimization Problems

Structure of Polynomial-Time Approximation

Contact Info

Product

Resources

About