## Structure in Approximation Classes (1996)

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Citations: | 74 - 14 self |

### BibTeX

@MISC{Crescenzi96structurein,

author = {P. Crescenzi and V. Kann and R. Silvestri and L. Trevisan},

title = {Structure in Approximation Classes},

year = {1996}

}

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### Abstract

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 APX-intermediate problems. Moreover, we state new connections between the approximability properties and the query complexity of NPO problems.

### Citations

10922 |
Computers and Intractability: A Guide to the Theory of NP-Completeness,” Freeman and Co
- Garey, Johnson
- 1979
(Show Context)
Citation Context ...he basic concepts of computational complexity theory. For the definitions of most of the complexity classes used in the paper we refer the reader to one of the books on the subject (see, for example, =-=[2, 5, 18, 37]-=-). We now give some standard definitions in the field of optimization and approximation theory. Definition 1.1. An NP optimization problem A is a 4-tuple (I,sol,m,type) such that the following hold: 1... |

2343 | Computational Complexity
- Papadimitriou
- 1994
(Show Context)
Citation Context ...he basic concepts of computational complexity theory. For the definitions of most of the complexity classes used in the paper we refer the reader to one of the books on the subject (see, for example, =-=[2, 5, 18, 37]-=-). We now give some standard definitions in the field of optimization and approximation theory. Definition 1.1. An NP optimization problem A is a 4-tuple (I,sol,m,type) such that the following hold: 1... |

717 | Proof Verification and the Hardness of Approximation Problems
- Arora, Lund, et al.
- 1998
(Show Context)
Citation Context ...the reconciling of these two approaches consisted of proving lower bounds (modulo P �= NP or some other likely condition) on the approximability of complete problems for syntactically defined classes =-=[1, 33]-=-. More recently, another step has been performed since the closure of syntactically defined classes with respect to an approximation preserving reducibility which has been proved to be equal to the mo... |

682 |
Approximation algorithms for combinatorial problems
- Johnson
- 1974
(Show Context)
Citation Context ..., approximation algorithms AMS subject classifications. 03D30, 68Q15, 68Q20 PII. S0097539796304220 1. 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 t... |

572 |
Optimization, approximation, and complexity classes
- Papadimitriou, Yannakakis
- 1991
(Show Context)
Citation Context ... for Computer Science, MIT, Cambridge, MA 02139 (luca@theory.lcs.mit.edu). 1759s1760 P. CRESCENZI, V. KANN, R. SILVESTRI, AND L. TREVISAN PTAS-reducibility [16] ✻ P-reducibility [35] ✟ L-reducibility =-=[38]-=- E-reducibility [28] ✟✟✟✯ ✟ ✟✟✟✯ A-reducibility [35] ❍❨ ❍❨ ❍ ❍ ❍ ❍ ❍ ❍ Continuous reducibility [41] ✻ Strict reducibility [35] Fig. 1.1. The taxonomy of approximation preserving reducibilities. By mea... |

475 |
The complexity of theorem proving procedures
- Cook
- 1971
(Show Context)
Citation Context ... ∈ IA} xB := f(x); yB := r-approximate solution of xB; yA := g(x, yB); mx := max{mA(x, yA),mB(xB,yB)/α}; guess y ∈ solA(x); if mA(x, y) ≥ mx then accept else reject; end; By applying Cook’s reduction =-=[10]-=- to the above algorithm, it easily follows that, for any x ∈ IA, a satisfiable Boolean formula φx can be constructed in polynomial time in the length of x so that any satisfying assignment for φx enco... |

382 |
On the hardness of approximating minimization problems
- Lund, Yannakakis
- 1993
(Show Context)
Citation Context ...the reconciling of these two approaches consisted of proving lower bounds (modulo P �= NP or some other likely condition) on the approximability of complete problems for syntactically defined classes =-=[1, 33]-=-. More recently, another step has been performed since the closure of syntactically defined classes with respect to an approximation preserving reducibility which has been proved to be equal to the mo... |

188 |
The complexity of optimization problems
- Krentel
- 1988
(Show Context)
Citation Context ...which asks at most f(n) queries to an NP-complete oracle, where n is the input size. The class QH is defined to be the union � k>1 PNP[k] . Similarly, we can define the class of functions FP NP[f(n)] =-=[30]-=-. The following result has been proved in [23, 24]. Theorem 4.2. If there exists a constant k such that QH=P NP[k] , then the polynomial-time hierarchy collapses. The query-complexity of the “nonconst... |

171 |
Structural Complexity I
- Balcázar, Díaz, et al.
- 1988
(Show Context)
Citation Context ...he basic concepts of computational complexity theory. For the definitions of most of the complexity classes used in the paper we refer the reader to one of the books on the subject (see, for example, =-=[2, 5, 18, 37]-=-). We now give some standard definitions in the field of optimization and approximation theory. Definition 1.1. An NP optimization problem A is a 4-tuple (I,sol,m,type) such that the following hold: 1... |

120 |
An efficient approximation scheme for the one-dimensional binpacking problem
- Karmarkar, Karp
- 1982
(Show Context)
Citation Context ...t collapse, then Minimum Bin Packing, Minimum Degree Spanning Tree, and Minimum Edge Coloring are APX-intermediate. Proof. From Proposition 4.6 and from the fact that Minimum Bin Packing is in PTAS ∞ =-=[27]-=-, it follows that AQH(Minimum Bin Packing) ⊆ P NP[h] for a given h. If Minimum Bin Packing is APX-complete, then from Proposition 4.8 it follows that QH ⊆ P NP[h] . From Theorem 4.2 we thus have the c... |

116 | On Syntactic Versus Computational Views of Approximability
- Khanna, Motwani, et al.
- 1994
(Show Context)
Citation Context ...e, MIT, Cambridge, MA 02139 (luca@theory.lcs.mit.edu). 1759s1760 P. CRESCENZI, V. KANN, R. SILVESTRI, AND L. TREVISAN PTAS-reducibility [16] ✻ P-reducibility [35] ✟ L-reducibility [38] E-reducibility =-=[28]-=- ✟✟✟✯ ✟ ✟✟✟✯ A-reducibility [35] ❍❨ ❍❨ ❍ ❍ ❍ ❍ ❍ ❍ Continuous reducibility [41] ✻ Strict reducibility [35] Fig. 1.1. The taxonomy of approximation preserving reducibilities. By means of these reducibi... |

114 |
The NP-completeness of edge-coloring
- Holyer
- 1981
(Show Context)
Citation Context ... QH ⊆ P NP[h] . From Theorem 4.2 we thus have the collapse of the polynomial-time hierarchy. The proofs for Minimum Degree Spanning Tree and Minimum Edge Coloring are identical and use the results of =-=[20, 17]-=-. Observe that the previous result does not seem to be obtainable by using the hypothesis P �= NP, as shown by the following theorem. Theorem 4.11. If NP = co-NP, then Minimum Bin Packing is APX-compl... |

108 |
On the approximation of maximum satisfiability
- Yannakakis
- 1994
(Show Context)
Citation Context ...nstruction,sSTRUCTURE IN APPROXIMATION CLASSES 1777 an r-approximate solution for xφ is indeed an optimum solution provided that r<3/2. Let T be a 4/3-approximate algorithm for Maximum Satisfiability =-=[44, 19]-=-. The reduction from Maximum Satisfiability to Minimum Bin Packing is defined as follows: f(φ, r)=t1(φ); � T(φ) if r ≥ 4/3, g(φ, y, r)= t2(t1(φ),y) otherwise. It is easy to verify that the above is an... |

94 | Introduction to the Theory of Complexity
- Bovet, Crescenzi
- 1993
(Show Context)
Citation Context |

73 |
On the Complexity of Approximating the Independent Set Problem
- Berman, Schnitger
- 1992
(Show Context)
Citation Context ...PO and NPO PB (from now on, unless otherwise specified, we will always refer to the AP-reducibility). Previously, completeness results have been obtained just for Max NPO, Min NPO, Max PB, and Min PB =-=[14, 35, 4, 26]-=-. One example of such a result is the following theorem. Theorem 3.1. Minimum Weighted Ones is Min NPO-complete and Maximum Weighted Ones is Max NPO-complete (with respect to the E-reducibility), even... |

69 | New 3/4-approximation algorithms for the maximum satisfiabihty problem
- Goemans, Wiffiamson
- 1994
(Show Context)
Citation Context ...nstruction,sSTRUCTURE IN APPROXIMATION CLASSES 1777 an r-approximate solution for xφ is indeed an optimum solution provided that r<3/2. Let T be a 4/3-approximate algorithm for Maximum Satisfiability =-=[44, 19]-=-. The reduction from Maximum Satisfiability to Minimum Bin Packing is defined as follows: f(φ, r)=t1(φ); � T(φ) if r ≥ 4/3, g(φ, y, r)= t2(t1(φ),y) otherwise. It is easy to verify that the above is an... |

68 |
Approximating the minimum-degree spanning tree to within one from the optimal degree
- Fürer, Raghavachari
- 1992
(Show Context)
Citation Context ... QH ⊆ P NP[h] . From Theorem 4.2 we thus have the collapse of the polynomial-time hierarchy. The proofs for Minimum Degree Spanning Tree and Minimum Edge Coloring are identical and use the results of =-=[20, 17]-=-. Observe that the previous result does not seem to be obtainable by using the hypothesis P �= NP, as shown by the following theorem. Theorem 4.11. If NP = co-NP, then Minimum Bin Packing is APX-compl... |

68 |
On the Approximability of NP-complete Optimization Problems
- Kann
- 1992
(Show Context)
Citation Context ... 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]). ∗Received by the editors May 24, 1996; accepted for publication (in revised form) January 7, 1998; published electronically May 13, 1999. An extended abstract of this paper was presented a... |

67 |
The polynomial time hierarchy collapses if the boolean hierarchy collapses
- Kadin
- 1988
(Show Context)
Citation Context ...ete oracle, where n is the input size. The class QH is defined to be the union � k>1 PNP[k] . Similarly, we can define the class of functions FP NP[f(n)] [30]. The following result has been proved in =-=[23, 24]-=-. Theorem 4.2. If there exists a constant k such that QH=P NP[k] , then the polynomial-time hierarchy collapses. The query-complexity of the “nonconstructive” approximation of several NP-hard optimiza... |

63 | Bounded queries to SAT and the boolean hierarchy - Beigel - 1991 |

60 |
Graph isomorphism is in the low hierarchy
- Schöning
- 1988
(Show Context)
Citation Context ...rve that this question is also open in the field of decision problems: for example, it is known that the graph isomorphism problem cannot be NP-complete unless the polynomial-time hierarchy collapses =-=[40]-=-, but no result has ever been obtained giving evidence that the problem does not belong to P. The first goal of this paper is to define an approximation preserving reducibility that can be used for as... |

49 | Approximation properties of NP minimization classes - Kolaitis, Thakur - 1995 |

49 |
On the structure of polynomial-time reducibility
- Ladner
- 1975
(Show Context)
Citation Context ... already been shown in [14] (assuming P �= NP) where an artificial such problem is obtained by diagonalization techniques similar to those developed to prove the existence of NP-intermediate problems =-=[31]-=-. In this section, we prove that “natural” APXintermediate problems exist: for instance, we will show that Minimum Bin Packing is APX-intermediate. In order to prove this result, we will establish new... |

45 | Quantifiers and approximation - Panconesi, Ranjan - 1993 |

35 | On approximation preserving reductions: Complete problems and robust measures
- P, Mannila
- 1987
(Show Context)
Citation Context ...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 ident... |

33 |
Decision trees downward closure
- Impagliazzo, Naor
- 1988
(Show Context)
Citation Context ...λy.y, α = 1, and β =1. Observe that in [32] it is shown that the third statement of the above theorem holds if and only if the γ-reducibility is different from the many-one reducibility. Moreover, in =-=[21]-=- it is shown that the latter hypothesis is somewhat intermediate between P �=NP∩co-NP and P �= NP. In other words, there is strong evidence that, even though the L-reducibility is suitable for proving... |

26 |
Bounded query computations
- Wagner
- 1988
(Show Context)
Citation Context ...ntained in P MC1 , where MC stands for Maximum Clique [30], and from the fact that if there exists a constant k such that P NP[log log n+k] =P NP[log n] , then the polynomial-time hierarchy collapses =-=[42]-=-, it follows that the next result solves an open question posed in [7]. Informally, this result states that it is not possible to reduce the problem of finding a maximum clique to the problem of findi... |

25 | On approximation scheme preserving reducibility and its applications
- Crescenzi, Trevisan
(Show Context)
Citation Context ...Italy (silver@dsi.uniroma1.it). ¶ Laboratory for Computer Science, MIT, Cambridge, MA 02139 (luca@theory.lcs.mit.edu). 1759s1760 P. CRESCENZI, V. KANN, R. SILVESTRI, AND L. TREVISAN PTAS-reducibility =-=[16]-=- ✻ P-reducibility [35] ✟ L-reducibility [38] E-reducibility [28] ✟✟✟✯ ✟ ✟✟✟✯ A-reducibility [35] ❍❨ ❍❨ ❍ ❍ ❍ ❍ ❍ ❍ Continuous reducibility [41] ✻ Strict reducibility [35] Fig. 1.1. The taxonomy of app... |

25 |
Lecture notes on approximation algorithms
- Motwani
(Show Context)
Citation Context ...ermediate problems. In section 4, we will prove that Minimum Bin Packing (and other natural NPO problems) cannot be APX-complete unless the polynomial-time hierarchy collapses. Since it is well known =-=[34]-=- that this problem belongs to APX and that it does not belong to PTAS (that is, the class of NPO problems with polynomialtime approximation schemes) unless P = NP, our result yields the first example ... |

23 | Polynomially bounded minimization problems that are hard to approximate
- Kann
- 1994
(Show Context)
Citation Context ...PO and NPO PB (from now on, unless otherwise specified, we will always refer to the AP-reducibility). Previously, completeness results have been obtained just for Max NPO, Min NPO, Max PB, and Min PB =-=[14, 35, 4, 26]-=-. One example of such a result is the following theorem. Theorem 3.1. Minimum Weighted Ones is Min NPO-complete and Maximum Weighted Ones is Max NPO-complete (with respect to the E-reducibility), even... |

16 | On bounded queries and approximation
- Chang, Gasarch, et al.
- 1997
(Show Context)
Citation Context ...uery complexity (that is, the number of queries to an NP oracle needed to solve a given problem) has been shown to be a very useful tool for understanding the complexity of approximation problems. In =-=[7, 9]-=- upper and lower bounds have been proved on the number of queries needed to approximate certain optimization problems (such as Maximum Satisfiability and Maximum Clique): these results deal with the c... |

13 |
A compendium of NP optimization problems. Technical Report, Dipartimento di Scienze dell'Informazione, Universita di Roma \La Sapienza
- Crescenzi, Kann
- 1995
(Show Context)
Citation Context ...clusions are strict if and only if P �= NP. 1.2. A list of NPO problems. We here define the NP optimization problems that will be used in the paper (for a much larger list of NPO problems we refer to =-=[12, 13]-=-). Observe that in the following definitions we will not mention the type of the problem since it will be specified by the name of the problem itself.s1764 P. CRESCENZI, V. KANN, R. SILVESTRI, AND L. ... |

12 | On the query complexity of clique size and maximum satisfiability
- Chang
- 1996
(Show Context)
Citation Context ...uery complexity (that is, the number of queries to an NP oracle needed to solve a given problem) has been shown to be a very useful tool for understanding the complexity of approximation problems. In =-=[7, 9]-=- upper and lower bounds have been proved on the number of queries needed to approximate certain optimization problems (such as Maximum Satisfiability and Maximum Clique): these results deal with the c... |

11 |
Bounded queries to SAT and the Boolean hierarchy, Theoret
- Beigel
- 1991
(Show Context)
Citation Context ....,xk. We are now able to prove the following result. Proposition 4.8. For any APX-complete problem A, QH ⊆ AQH(A). Proof. Let L ∈ QH, then L ∈ P NP[h] for some h. It is well known (see, for instance, =-=[3]-=-) that L can be reduced to the problem of answering k =2 h−1 nonadaptive queries to NP. More formally, there exist two polynomial-time computable functions t1 and t2 such that: 1. For any x, t1(x) =(x... |

8 |
A short guide to approximation preserving reductions
- Crescenzi
- 1997
(Show Context)
Citation Context ...serve 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]-=-). ∗Received by the editors May 24, 1996; accepted for publication (in revised form) January 7, 1998; published electronically May 13, 1999. An extended abstract of this paper was presented at the 1st... |

8 |
Assembly line balancing as generalized bin packing
- Wee, Magazine
- 1982
(Show Context)
Citation Context ...ed if and only if xi is a yes-instance. The optimum measure thus is at most 4k − 1 so that any (1+1/(4k))-approximate solution is an optimum solution. Since Minimum Ordered Bin Packing belongs to APX =-=[43]-=- and A is APXcomplete, there then exists an AP-reduction (f1,g1,α) from Minimum Ordered Bin Packing to A. We can then define x = f(x1,...,xk)=f1(w,1+1/(4αk)) and r =1+1/(4αk). For any r-approximate so... |

6 |
On the Efficiency and Optimization
- Wu, Wang
- 1996
(Show Context)
Citation Context ...ity. This result is not obtainable by means of the L-reducibility: indeed, it is easy to prove that Minimum Bin Packing is not L-reducible to Maximum 3-Satisfiability unless P = NP (see, for example, =-=[6]-=-). The E-reducibility is still somewhat too strict. Indeed, in [16] it has been shown that there exist natural PTAS problems, such as Maximum Knapsack, which are not E-reducible to polynomially bounde... |

4 |
Relative complexity of evaluating the optimum cost and constructing the optimum for maximization problems
- Crescenzi, Silvestri
- 1990
(Show Context)
Citation Context ...putation tree. 3 In order to talk about hardness with respect to these classes we will use the following reducibility, which is an extension of both metric reducibility [30] and onequery reducibility =-=[15]-=- and has been introduced in [8]. Definition 5.2. Let F and G be two partial multivalued functions. We say that F many-one reduces to G (in symbols, F ≤mvG) if there exist two polynomial-time algorithm... |

4 | On fl-reducibility versus polynomial time many-one reducibility - Long - 1981 |

3 |
On γ-reducibility versus polynomial time many-one reducibility
- Long
- 1981
(Show Context)
Citation Context ...that computes a satisfying assignment for each formula in S, contradicting the assumption. Moreover, it is easy to see that A L-reduces to B via f ≡ λx.x, g ≡ λxλy.y, α = 1, and β =1. Observe that in =-=[32]-=- it is shown that the third statement of the above theorem holds if and only if the γ-reducibility is different from the many-one reducibility. Moreover, in [21] it is shown that the latter hypothesis... |

3 |
Bounds for assembly line balancing heuristics
- Queyranne
- 1985
(Show Context)
Citation Context ...ume that the Uis are pairwise disjoint and that, for any i, � u∈Ui si(u) =2. Letw =(U, s, �) be an instance of Minimum Ordered Bin Packing defined as follows. (A similar construction has been used in =-=[39]-=-.) 1. U = � k i=1 Ui ∪{v1,...,vk−1}, where the vis are new items. 2. For any u ∈ Ui, s(u)=si(u) and s(vi)=1fori =1,...,k− 1. 3. For any i<j≤ k, for any u ∈ Ui, and for any u ′ ∈ Uj, u � vi � u ′ . Any... |

3 |
Continous reductions among combinatorial optimization problems
- Simon
- 1989
(Show Context)
Citation Context ...V. KANN, R. SILVESTRI, AND L. TREVISAN PTAS-reducibility [16] ✻ P-reducibility [35] ✟ L-reducibility [38] E-reducibility [28] ✟✟✟✯ ✟ ✟✟✟✯ A-reducibility [35] ❍❨ ❍❨ ❍ ❍ ❍ ❍ ❍ ❍ Continuous reducibility =-=[41]-=- ✻ Strict reducibility [35] Fig. 1.1. The taxonomy of approximation preserving reducibilities. By means of these reducibilities, several notions of completeness in approximation classes have been intr... |

2 |
Approximation on the web: A compendium of NP optimization problems
- Crescenzi, Kann
- 1997
(Show Context)
Citation Context ...clusions are strict if and only if P �= NP. 1.2. A list of NPO problems. We here define the NP optimization problems that will be used in the paper (for a much larger list of NPO problems we refer to =-=[12, 13]-=-). Observe that in the following definitions we will not mention the type of the problem since it will be specified by the name of the problem itself.s1764 P. CRESCENZI, V. KANN, R. SILVESTRI, AND L. ... |

2 |
Quantifiers and approximation, Theoret
- Panconesi, Ranjan
- 1993
(Show Context)
Citation Context ...) is in the domain of G and, for any y ∈ G(t1(x)), t2(x, y) ∈ F(x). The combinatorial property used to characterize poly-APX-complete problems is the well-known self-improvability (see, for instance, =-=[36]-=-). 3 We say that a multivalued partial function F is computable by a nondeterministic Turing machine N if, for every x in the domain of F, there exists a halting computation path of N(x) and any halti... |

1 |
A machine model for NP-approximation problems and the revenge of the Boolean hierarchy
- Chang
- 1994
(Show Context)
Citation Context ...ximation problems with respect to classes of partial multivalued functions and in terms of suitably defined combinatorial properties. The classes of functions we will refer to have been introduced in =-=[8]-=- as follows. Definition 5.1. FNP NP[q(n)] is the class of partial multivalued functions computable by nondeterministic polynomial-time Turing machines which ask at most q(n) queries to an NP oracle in... |

1 | A compendium of NP optimization problems", Technical Report SI/RR-95/02, Dipartimento di Scienze dell'Informazione, Universit`a di Roma "La Sapienza". The list is updated continuously. The latest version is available by anonymous ftp from n - Crescenzi, Kann - 1995 |