Non deterministic polynomial optimization problems and their approximations

A. Paz, S. Moran
1981 Theoretical Computer Science  
Abstcaet. A unified and general framework for the study of nondeterministic polynomial optimization problems (NPDP) is presented and some prop,erties of NPOP's are investigated. A characterization of NPOP'E with regard to their approximability properties is given by proving necessary and sufficient conditions for two approximability schemes. Known approximability results are shown to fit within the general frame developed in the paper. Finally NPOP's are classified and studied wit!. regard to
more » ... e possibility or impossibility of 'reducing* certain types of NPOP's to other types in .a sense specified in the text. problem at hand) number oat of the set of numbers associated with it. Clearly, if every element in the: set of ar NPOP has only two numbers associated with it, 0 or 1, then such an NPOP is equi\/alenlt to the recognition problem for an 1NP set. The conjecture that P f NP is widely believed to be true and NP completf: problems are generally believed to be intractable. This prompted many researchers CO develop and study polynomial approximation schemes for NP-problems. Such schemes are more natural and easier to define and study in the context of optim zation problems where both the output and its approximation are numerical. In this paper we suggest a formal definition of NPOP and propose a framework for their study. The definition is similar to other definitions which appeared in the literature both in explicit and implicit form (see references at the end of the paper) and has the property that almost all of the known results concerning NPOP in their various forms, can be stated within its framework. In addition Some new and important results are stated and proved, results provid' :lg some new Llsight concerning the nature of the various approximation schemes for NPOP investigated by several authors (see e.g. [l, J, 5, 6, 9, 10, 13, 15, 16, 21-231) and providing a characterization of two different approximation schemes. studied in the literature. We believe that these results are important in several ways: A unified framework for the study of NPOP, their approximation, and their reductions of different types is set. (As in the NP case reductions may be useful for getting new approximation algoritlms out of known such algorithms.) A characterization of various approximation schemes may be useful for the findiug of proper approximation schemes to fit new or existing N?VP. Finaiiy such a characterization may also lead in the future to a universal algorithm which will fit automatically an appbmoximation algorithm to an NPUP provided such an algorithm exists and provided that the NPOP in case is known to have certain properties. The paper is divided into five sections including this introductory first section. Section 2 provides the basic definitions. In the third section some general properties of NPOP and their relation to NP sets are studied. Section 4., which is the core of tht paper, deals with the approximability properties of NPOP and provides a full characterization of two known approximation s'chemes. The last section deals with reducibility properties of NPOP. The paper has two appendices; the first one provides formal definitions for Ihe different NPOP quoted in the paper and the second one contains an NPOP version of Cook's reducibility result for NP. optimization problems 2.
doi:10.1016/0304-3975(81)90081-5 fatcat:cqzybi4pxzdcto2yiogaqgj6ba