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Autour de nouvelles notions pour l'analyse desalgorithmes d'approximation : formalisme unifié et classes d'approximation

Published online by Cambridge University Press:  15 April 2003

Marc Demange  and
Vangelis Paschos
Marc Demange
Affiliation:
ESSEC, Cergy-Pontoise, France; [email protected].
Vangelis Paschos
Affiliation:
LAMSADE, Université Paris-Dauphine, France; [email protected].
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Abstract

Cet article est le premier d'une série de deux articles oùnous présentons les principales caractéristiques d'un nouveauformalisme pour l'approximation polynomiale (algorithmiquepolynomiale à garanties de performances pour les problèmesNP-difficiles). Ce travail est l'occasion d'unregard critique sur ce domaine et de discussions sur la pertinencedes notions usuelles. Il est aussi l'occasion de se familiariseravec l'approximation polynomiale, de comprendre ses enjeux etses méthodes. Ces deux articles s'adressent donc autant auxspécialistes qu'aux non spécialistes de ce domaine. Nous insistons toutparticulièrementsur l'intérêt, tant théorique qu'opérationnel, de mettre enévidence une structure au sein de la classe NPO desproblèmes d'optimisation de NP. Dans ce premier article, nousnous intéressons aux outils qui permettent d'évaluer,dans l'absolu, les propriétés d'approximation de problèmesdifficiles. Nous discutons notamment les notions de chaînesd'approximation, de niveau d'approximation, d'ordre de difficultéainsi que deux notions de limites (par rapport à une suited'algorithmes et par rapport aux instances). Chaque notion estlargement discutée et illustrée par de nombreux exempleschoisis essentiellement pour leur valeur pédagogique.
Mots Clés. Complexité, difficulté intrinsèque, analyse des algorithmes et des problèmes,algorithmes d'approximation. Classification Mathématique. 68Q15, 68Q17, 68Q25, 68W25.

Type
Research Article
Information
RAIRO - Operations Research , Volume 36 , Issue 3 , July 2002 , pp. 237 - 277
Copyright
© EDP Sciences, 2002

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References

S. Arora, C. Lund, R. Motwani, M. Sudan et M. Szegedy, Proof verification and intractability of approximation problems, in Proc. FOCS'92 (1992) 14-23.
G. Ausiello, P. Crescenzi, G. Gambosi, V. Kann, A. Marchetti-Spaccamela et M. Protasi, Complexity and approximation. Combinatorial optimization problems and their approximability properties. Springer, Heildelberg (1999).
C. Berge, Graphs and hypergraphs. North Holland, Amsterdam (1973).
P. Berman et M. Fürer, Approximating maximum independent set in bounded degree graphs, in Proc. Symposium on Discrete Algorithms (1994) 365-371.
Boppana, B.B. et Halldórsson, M.M., Approximating maximum independent sets by excluding subgraphs. BIT 32 (1992) 180-196. CrossRef
Chvátal, V., A greedy-heuristic for the set covering problem. Math. Oper. Res. 4 (1979) 233-235. CrossRef
S.A. Cook, The complexity of theorem-proving procedures, in Proc. STOC'71 (1971) 151-158.
Demange, M., Grisoni, P. et Paschos, V.T., Differential approximation algorithms for some combinatorial optimization problems. Theoret. Comput. Sci. 209 (1998) 107-122. CrossRef
Demange, M., Monnot, J. et Paschos, V.T., Bridging gap between standard and differential polynomial approximation: The case of bin-packing. Appl. Math. Lett. 12 (1999) 127-133. CrossRef
height 2pt depth -1.6pt width 23pt, Maximizing, the number of unused bins. Found. Comput. Decision Sci. 26 (2001) 169-186.
Demange, M. et Paschos, V.T., On an approximation measure founded on the links between optimization and polynomial approximation theory. Theoret. Comput. Sci. 158 (1996) 117-141. CrossRef
height 2pt depth -1.6pt width 23pt, Valeurs, extrémales d'un problème d'optimisation combinatoire et approximation polynomiale. Math. Inf. Sci. Humaines 135 (1996) 51-66.
height 2pt depth -1.6pt width 23pt, Autour de nouvelles notions pour l'analyse des algorithmes d'approximation : de la structure de NPO à la structure des instances. RAIRO: Oper. Res. (à paraître).
height 2pt depth -1.6pt width 23pt, Improved, approximations for maximum independent set via approximation chains. Appl. Math. Lett. 10 (1997) 105-110.
height 2pt depth -1.6pt width 23pt, Towards a general formal framework for polynomial approximation. LAMSADE, Université Paris-Dauphine, Cahier du LAMSADE 177 (2001).
R. Duh et M. Fürer, Approximation of k-set cover by semi-local optimization, in Proc. STOC'97 (1997) 256-265.
U. Feige et J. Kilian, Zero knowledge and the chromatic number, in Proc. Conference on Computational Complexity (1996) 278-287.
Fernandez de, W. la Vega, Sur la cardinalité maximum des couplages d'hypergraphes aléatoires uniformes. Discrete Math. 40 (1982) 315-318. CrossRef
M.R. Garey et D.S. Johnson, Computers and intractability. A guide sto the theory of NP-completeness. W.H. Freeman, San Francisco (1979).
Halldórsson, M.M., A still better performance guarantee for approximate graph coloring. Inform. Process. Lett. 45 (1993) 19-23. CrossRef
height 2pt depth -1.6pt width 23pt, Approximations via partitioning. JAIST Research Report IS-RR-95-0003F, Japan Advanced Institute of Science and Technology, Japan (1995).
height 2pt depth -1.6pt width 23pt, Approximating k-set cover and complementary graph coloring, in Proc. International Integer Programming and Combinatorial Optimization Conference. Springer Verlag, Lecture Notes in Comput. Sci. 1084 (1996) 118-131.
M.M. Halldórsson et J. Radhakrishnan, Greed is good: Approximating independent sets in sparse and bounded-degree graphs, in Proc. STOC'94 (1994) 439-448.
height 2pt depth -1.6pt width 23pt, Improved, approximations of independent sets in bounded-degree graphs via subgraph removal. Nordic J. Comput. 1 (1994) 475-492.
Hassin, R. et Lahav, S., Maximizing the number of unused colors in the vertex coloring problem. Inform. Process. Lett. 52 (1994) 87-90. CrossRef
Håstad, J., Clique is hard to approximate within n1-ε. Acta Math. 182 (1999) 105-142. CrossRef
Hochbaum, D.S., Efficient bounds for the stable set, vertex cover and set packing problems. Discrete Appl. Math. 6 (1983) 243-254. CrossRef
height 2pt depth -1.6pt width 23pt, Approximation algorithms for NP-hard problems. PWS, Boston (1997).
Ibarra, O.H. et Kim, C.E., Fast approximation algorithms for the knapsack and sum of subset problems. J. Assoc. Comput. Mach. 22 (1975) 463-468. CrossRef
Johnson, D.S., Approximation algorithms for combinatorial problems. J. Comput. System Sci. 9 (1974) 256-278. CrossRef
R.M. Karp, Reducibility among combinatorial problems, dans Complexity of computer computations, édité par R.E. Miller et J.W. Thatcher. Plenum Press, New York (1972) 85-103.
Khanna, S., Motwani, R., Sudan, M. et Vazirani, U., On syntactic versus computational views of approximability. SIAM J. Comput. 28 (1998) 164-191. CrossRef
H.R. Lewis et C.H. Papadimitriou, Elements of the theory of computation. Prentice-Hall (1981).
Lund, C. et Yannakakis, M., On the hardness of approximating minimization problems. J. Assoc. Comput. Mach. 41 (1994) 960-981. CrossRef
R. Motwani, Lecture notes on approximation algorithms, Vol. I. Stanford University (1993).
Nemhauser, G.L., Wolsey, L.A. et Fischer, M.L., An analysis of approximations for maximizing submodular set functions. Math. Programming 14 (1978) 265-294. CrossRef
C.H. Papadimitriou et K. Steiglitz, Combinatorial optimization: Algorithms and complexity. Prentice Hall, New Jersey (1981).
R. Raz et S. Safra, A sub-constant error probability low-degree test and a sub-constant error probability PCP characterization of NP, in Proc. STOC'97 (1997) 475-484.
Simchi-Levi, D., New worst-case results for the bin-packing problem. Naval Res. Logistics 41 (1994) 579-585. 3.0.CO;2-G>CrossRef
Turán, P., On an extremal problem in graph theory (in Hungarian). Mat. Fiz. Lapok 48 (1941) 436-452.
V. Vazirani, Approximation algorithms. Springer, Heildelberg (2001).