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A Survey on the Structure of Approximation Classes

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A Survey on the Structure of Approximation Classes COMPUTERSCIENCEREVIEW 4 (2010) 19±40 available at www.sciencedirect.com journal homepage: www.elsevier.com/locate/cosrev Survey A survey on the structure of approximation classes Bruno Escoffier∗, Vangelis Th. Paschos LAMSADE, CNRS and Université Paris­Dauphine, Place du Maréchal De Lattre de Tassigny, 75775 Paris Cedex 16, France ARTICLEINFO ABSTRACT Article history: The structure of approximability classes by the introduction of approximation preserving Received 15 December 2008 reductions has been one of the main research programmes in theoretical computer science Received in revised form during the last thirty years. This paper surveys the main results achieved in this domain. 13 November 2009 c 2009 Elsevier Inc. All rights reserved. Accepted 13 November 2009 1. Introduction In a second time, the objective is to structure every such sub­class by exhibiting “hard” problems for each of them. The issue of polynomial approximation theory is the con­ This requires a processing of the same type as for NP­ struction of algorithms for NP­hard problems that compute completeness but adapted to the combinatorial characteris­ “good” (under some predefined quality criterion) feasible so­ tics of each of the approximability classes handled. This can lutions for them in polynomial time. This issue appears very be done by the introduction of notions of approximability pre­ early (in the seminal paper by Johnson [1]), only three years serving reductions. Sections 4–8 are dedicated to such reduc­ after the celebrated Cook’s theorem [2]. For over thirty years, tions. polynomial approximation has been a major research pro­ The technique of transforming a problem into another in gram in theoretical computer science that has motivated nu­ such a way that the solution of the latter entails, somehow, merous studies by researchers from all over the world. the solution of the former, is a classical mathematical tech­ One of the main goals of this program is, in a first time, nique that has found wide application in computer science to provide a structure for the principal class of optimization since the seminal works of Cook [2] and Karp [3] who intro­ problems, called NPO in what follows, and formally defined in duced particular kinds of transformations (called reductions) Section 2. Providing such a structure for NPO, consists first of with the aim of studying the computational complexity of dividing it into sub­classes, called approximability classes, the combinatorial decision problems. The interesting aspect of a problems belonging to each of them sharing common approx­ imability properties (e.g., they are approximable within con­ reduction between two problems consists of its twofold appli­ stant approximation ratios). Indeed, even if NP­hard problems cation: on one side it allows to transfer positive results (res­ can be considered equivalent regarding their optimal reso­ olution techniques) from one problem to the other one and, lution by polynomial time algorithms, they behave very dif­ on the other side, it may also be used for deriving negative ferently regarding their approximability. For instance, some (hardness) results. In fact, as a consequence of such semi­ NP­hard problems are “well” approximable (e.g., within con­ nal work, by making use of a specific kind of reduction, the stant approximation ratios, or within ratios arbitrarily close polynomial­time Karp­reducibility, it has been possible to es­ to 1) while there exist other problems for which such approx­ tablish a complexity partial order among decision problems, imability qualities are impossible unless a highly improbable which, for example, allows us to state that, modulo poly­ complexity hypothesis is true (for example P D NP). nomial time transformations, the SATISFIABILITY problem is ∗ Corresponding author. E­mail addresses: escoffi[email protected] (B. Escoffier), [email protected] (V.T. Paschos). 1574­0137/$ ­ see front matter c 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.cosrev.2009.11.001 20 COMPUTERSCIENCEREVIEW 4 (2010) 19±40 as hard as thousands of other combinatorial decision prob­ A decision problem is defined by a set D of instances and a lems, even though the precise complexity level of all these question Q. Given an instance I 2 D, one wishes to determine problems is still unknown. In the same spirit, approxima­ if the answer to Q is yes, or no, on I. More formally, a decision tion preserving reducibilities allow us to establish a preorder problem is a set D of instances partitioned into two sets DC among combinatorial optimization problems with respect to and D−, where DC is the set of positive instances (the ones their common approximability properties and independently where the answer to Q is yes) and D− is the set of negative of the particular approximability properties of each of them. instances, where the answer to Q is no. The solution of such a Strictly associated with the notion of reducibility is the problem consists of determining, given I 2 D, if I 2 DC or not notion of completeness. Problems that are complete in a com­ (I 2 D−). plexity class via a given reducibility are, in a sense, the hard­ A decision problem belongs to NP if there exists a est problems of such a class. Besides, given two complexity polynomial binary relation R and a polynomial p such that 0 classes C and C ⊆ C, if a problem Π is complete in C via re­ for any instance I 2 D, I 2 DC if and only if there exists x, 0 ductions that preserve membership to C , in order to establish jxj p.jIj/, such that R.I; x/ (in other words, given I and x, one 0 6 whether C ⊂ C it is “enough” to assess the actual complexity can answer in time polynomial with jIj C jxj if R.I; x/ or not). of Π (informally we say that Π is a candidate to separate C This amounts to saying that the class NP is exactly the set 0 and C ). of problems solvable in polynomial time by a non­deterministic The aim of this survey is to present and discuss the last Turing machine [13,10]. C − C − developments of the structural aspects of the polynomial D . ; / D . ; / Let D1 D1 D1 and D2 D2 D2 be two decision time approximation theory (as for example completeness problems. Then D1 reduces to D2 under Karp­reducibility in classes like Poly­APX, Log­APX, DPTAS). Several previous (denoted by D1 6K D2) if there exists a function f such that: surveys and books (see, for instance, [4–8]) partially deal with this issue and contain some of the concepts and results • 8I 2 D1, f.I/ 2 D2; C C • 2 , . / 2 presented here. This survey includes several recent results I D1 f I D2 ; about completeness in approximability classes and carries • f is computable in polynomial time. out a large discussion about completeness in differential The fundamental property of this reducibility is that, if approximability classes. Besides, it is organized in such a D2 2 P and if D1 6K D2, then D1 2 P. way to emphasize the several techniques (Cook’s like proofs, A problem D is said to be NP­complete if any problem of NP PCP theorem) that have been used to derive completeness reduces to D under Karp­reducibility. An immediate corollary of results. Finally, as such results have been obtained for each this definition is that an NP­complete problem D is polynomial if of the notorious approximation classes, we hope that the and only if P D NP. This stipulates that if P 6D NP, classes P survey provides a global structured picture of this research and NP­complete are disjoint. Moreover, as it is proved in [14], programme. these two classes do not partition NP. In other words, there In what follows, we assume that the reader is familiar with exist problems that are neither in P, nor NP­complete. These basic notions from complexity theory and polynomial time are the so­called NP­intermediate problems (with respect to P approximation theory although, for sake of completeness, we and under Karp­reducibility). Indeed, Ladner proves in [14] provide basic definitions about complexity and approxima­ that there exists an infinity of complexity­levels in the bility classes as well as about Karp and Turing reducibili­ complexity­hierarchy built by the Karp­reducibility. ties. Also, definitions of the problems discussed are omitted; Let D and D0 be two decision problems. Then, D reduces they can be found in [9] or in [4]. The paper is organized as to D0 under Turing­reducibility (denoted by D D0) if, given a follows. The first two sections have an introductory charac­ 6T polynomial oracle that solves D0, there exists a polynomial ter. In Section 2, we summarize the definitions of the basic time algorithm A solving D by running (calling) . A polyno­ complexity classes (NP, NPO) and we recall the notions of mial oracle for D0 is a kind of fictive algorithm that solves D0, two important reducibilities: Karp and Turing reducibilities. In C i.e., it correctly determines if an instance I 2 D0 . So, A would Section 3, basic notions about polynomial time approxima­ tion are presented together with the definition of the main remain polynomial even if it called on a polynomial num­ ber of instances of D0 derived from a single instance of D. As approximation classes of optimization problems. Section 4 0 deals with the notion of an approximation preserving reduc­ for Karp­reducibility, given two decision problems D and D , 0 2 0 2 tion and the notion of completeness in an approximation if D P and D 6T D , then D P.
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