Structure in Approximation Classes Extended abstract P Crescenzi V Kann R Silvestri and L Trevisan Dipartimento di Scienze dellInformazione Universitadegli Studi di Roma La Sapienza Via Salaria Rome Italy Email fpilucsilvertrevisangdsiuniromait 2 Department of Numerical Analysis and Computing Science Royal Institute of Technology S Sto ckholm Sweden Email viggonadakthse Summary The study of the approximability prop erties of NPhard optimization problems has recently made great advances mainly due to the results obtained in the eld of pro of checking The last imp ortant breakthrough has b een obtained in where the APXcompleteness of several imp ortant optimization problems is proved thus reconciling two distinct views of ap proximation classes syntactic and computational In this pap er we obtain new results on the structure of two imp ortant computationallydened classes the class NPO that is the class of optimization problems whose underlying decision problem is in NP and the class APX that is the class of constantfactor approximable NPO problems In particular we give the rst ex amples of natural APXintermediate problems and the rst examples of natural NPOcomplete problems Moreover we state new connections b etween the approximability prop erties and the query complexity of NPO problems Introduction In his pioneering pap er on the approximation of combinatorial optimization problems David Johnson formally introduced the notion of approximable problem prop osed approxi mation algorithms for several problems and suggested a p ossible classication of optimization problems on grounds of their approximability prop erties Since then it was clear that even though all NPhard optimization problems are manyone p olynomialtime reducible to each other they do not share the same approximability prop erties The main reason of this fact is that manyone reductions not always preserve the ob jective function and even if this happ ens they rarely preserve the quality of the solutions It is then clear that a stronger kind of re ducibility has to b e 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 b e able to come back from go o d solutions in B to go o d solutions in A Surprisingly the rst denition of this kind of reducibility was given as long as years after Johnsons pap er and after that at least seven dierent denitions of approximation preserving reducibility app eared in the liter ature see Fig These denitions are identical with resp ect to the overall scheme but dier essentially in the way they preserve approximability they range from the Strict reducibility in which the error cannot increase to the PTASreducibility in which there are basically no restrictions see also Chapter of PTASreducibility Preducibility Areducibility HY HY H H H H H H Continuous reducibility Lreducibility Ereducibility Strict reducibility Figure The taxonomy of approximation preserving reducibilities By means of these reducibiliti es several notions of completeness in approximation classes have b een introduced and basically two dierent approaches were followed On the one hand the attention was fo cused on computationally dened classes of problems whose approxima bility prop erties were well understo o d such as NPO and APX along this line of research however almost all completeness results dealt either with articial optimization problems or with problems for which lower b ounds on the quality of the approximation were easily obtain able On the other hand researchers fo cused on the logical denability of optimization problems and introduced several syntactically dened classes for which natural completeness results were obtained unfortunately the approximability prop erties of the prob lems in these latter classes were not related to standard complexitytheoretic conjectures A rst step towards the reconciling of these two approaches consisted of proving lower b ounds on the approximability of complete problems for syntactically dened classes unless P NP or some other unlikely condition More recently another step has b een p erformed since the closure of syntactically dened classes with resp ect to approximation preserving reducibility has b een proved to b e equal to the more familiar computationally dened classes In spite of this imp ortant achievement b eyond APX we are still forced to distinguish b e tween maximization and minimization problems as long as we are interested in completeness pro ofs Indeed a result of states that it is not p ossible to rewrite every NP maximization problem as an NP minimization problem unless NPcoNP A natural question is thus whether this duality extends to approximation preserving reductions Finally even though the existence of intermediate articial problems that is problems for which lower b ounds on their approximation are not obtainable by completeness results was proved in a natural question arises do natural intermediate problems exist Observe that this question is also op en in the eld of decision problems even though the existence of articial NPintermediate problems has b een already proved For example it is known that the graph isomorphism problem cannot b e NPcomplete unless the p olynomialtime hierarchy collapses but no similar result has ever b een obtained showing that the problem do es not b elong to P Summary of the Results The rst goal of this pap er is to dene an approximation preserving reducibility such that all reductions that have app eared in the literature still hold and such that it can b e used for as many approximation classes as p ossible In spite of the fact that the Lreducibili ty has b een the most widely used so far we will give strong evidence that it cannot b e used to obtain completeness results in computationally dened classes such as APX logAPX that is the class of problems approximable within a logarithmic factor and p olyAPX that is the class of problems approximable within a p olynomial factor Indeed on the one hand the Lreducibil ity is to o weak and is not approximation preserving unless P NP coNP on the other it is to o strict and do es not allow to reduce problems which are known to b e NP easy to approximate to problems which are known to b e hard to approximate unless P NPO log n P The weakness of the Lreducibil ity is essentially shared by all reducibilitie s of Fig but the Strict reducibility and the Ereducibili ty while the strictness of the Lreducibili ty is shared by all of them but the PTASreducibility The reducibility we prop ose is a combination of the Ereducibility and of the PTASreducibility and as far as we know it is the strictest reducibility that allows to obtain all approximation completeness results that have app eared in the literature such as for example the APXcompleteness of the maximum satisability problem and the p olyAPXcompleteness of the maximum clique problem The second group of results refers to the existence of natural complete problems in NPO In deed b oth and provide examples of natural complete problems for the class of minimiza tion and maximization NP problems resp ectively In Sect we will show the existence of both maximization and minimization NPOcomplete natural problems In particular we prove that Maximum Programming Minimum Programming and Minimum Weighted Independent Dominating Set are NPOcomplete This result shows how making use of a natural approximation preserving reducibility is enough p owerful to encompass the duality problem raised in Moreover the same result can also b e obtained when restricting our selves to the class NPO PB that is the class of p olynomially b ounded NPO problems In particular we prove that Maximum PB Programming Minimum PB Pro gramming and Minimum Independent Dominating Set are NPO PBcomplete Indeed this result can also b e obtained as a consequence of Theorem a of However our proof does not make use of the PCP model The third group of results refers to the existence of natural APXintermediate problems In particular in Sect we will prove that Minimum Bin Packing and other natural NPO problems cannot b e APXcomplete unless the p olynomialtime hierarchy collapses Since it is wellknown that this problem b elongs to APX and that it do es not b elong to PTAS that is the class of NPO problems with p olynomialtime approximation schemes unless PNP our result thus yields the rst example of a natural APXintermediate problem under a natural complexitytheoretic conjecture Roughly sp eaking the pro of of our result is structured into two main steps In the rst step we show that if Minimum Bin Packing is APXcomplete then the problem of answering any set of k nonadaptive queries to an NPcomplete problem can b e reduced to the problem of approximating an instance of Minimum Bin Packing within a ratio dep ending on k In the second step we show that the problem of approximating an instance of Minimum Bin Packing within a given p erformance ratio can b e solved in p olynomialtime by means of a constant number of nonadaptive queries to an NPcomplete problem These two steps will imply the collapse of the query hierarchy which in turn implies the collapse of the p olynomialtime hierarchy As a side eect of our pro of we will show that if a problem is APXcomplete then it does not admit an asymptotic approximation scheme as far as we know no general technique to obtain this kind of results was previously known In the last group of results we state new connections b etween