Autour de nouvelles notions pour l'analyse des algorithmes d'approximation : formalisme unifié et classes d'approximation
RAIRO - Operations Research - Recherche Opérationnelle, Tome 36 (2002) no. 3, pp. 237-277.

The main objective of the polynomial approximation is the development of polynomial time algorithms for NP-hard problems, these algorithms guaranteeing feasible solutions lying “as near as possible” to the optimal ones. This work is the fist part of a couple of papers where we introduce the key-concepts of the polynomial approximation and present the main lines of a new formalism. Our purposes are, on the one hand, to present this theory and its objectives and, on the other hand, to discuss the appropriateness and the pertinence of its constitutive elements, as people knew them until now, and to propose their enrichment. Henceforth, these papers are addressed to both domain researchers and non-specialist readers. We particularly quote the great theoretical and operational interest in constructing an internal structure for the class NPO (of the optimization problems in NP). In this fist part, we focus on some basic tools allowing the individual evaluation of the approximability properties of any NP-hard problem. We present and discuss notions as algorithmic chain, approximation level, hardness threshold and two notions of limits (with respect to algorithmic chains and with respect to problems instances). The notions dealt in the paper are presented together with several illustrative examples.

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Demange, Marc; Paschos, Vangelis. Autour de nouvelles notions pour l'analyse des algorithmes d'approximation : formalisme unifié et classes d'approximation. RAIRO - Operations Research - Recherche Opérationnelle, Tome 36 (2002) no. 3, pp. 237-277. doi : 10.1051/ro:2003005. http://www.numdam.org/articles/10.1051/ro:2003005/

[1] 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. | Zbl

[2] 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). | Zbl

[3] C. Berge, Graphs and hypergraphs. North Holland, Amsterdam (1973). | MR | Zbl

[4] P. Berman et M. Fürer, Approximating maximum independent set in bounded degree graphs, in Proc. Symposium on Discrete Algorithms (1994) 365-371. | MR | Zbl

[5] B.B. Boppana et M.M. Halldórsson, Approximating maximum independent sets by excluding subgraphs. BIT 32 (1992) 180-196. | MR | Zbl

[6] V. Chvátal, A greedy-heuristic for the set covering problem. Math. Oper. Res. 4 (1979) 233-235. | MR | Zbl

[7] S.A. Cook, The complexity of theorem-proving procedures, in Proc. STOC'71 (1971) 151-158. | Zbl

[8] M. Demange, P. Grisoni et V.T. Paschos, Differential approximation algorithms for some combinatorial optimization problems. Theoret. Comput. Sci. 209 (1998) 107-122. | MR | Zbl

[9] M. Demange, J. Monnot et V.T. Paschos, Bridging gap between standard and differential polynomial approximation: The case of bin-packing. Appl. Math. Lett. 12 (1999) 127-133. | MR | Zbl

[10] , Maximizing the number of unused bins. Found. Comput. Decision Sci. 26 (2001) 169-186. | MR

[11] M. Demange et V.T. Paschos, On an approximation measure founded on the links between optimization and polynomial approximation theory. Theoret. Comput. Sci. 158 (1996) 117-141. | MR | Zbl

[12] , Valeurs extrémales d'un problème d'optimisation combinatoire et approximation polynomiale. Math. Inf. Sci. Humaines 135 (1996) 51-66. | Numdam | Zbl

[13] , 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). | Numdam | Zbl

[14] , Improved approximations for maximum independent set via approximation chains. Appl. Math. Lett. 10 (1997) 105-110. | MR | Zbl

[15] , Towards a general formal framework for polynomial approximation. LAMSADE, Université Paris-Dauphine, Cahier du LAMSADE 177 (2001).

[16] R. Duh et M. Fürer, Approximation of k-set cover by semi-local optimization, in Proc. STOC'97 (1997) 256-265. | Zbl

[17] U. Feige et J. Kilian, Zero knowledge and the chromatic number, in Proc. Conference on Computational Complexity (1996) 278-287.

[18] W. Fernandez De La Vega, Sur la cardinalité maximum des couplages d'hypergraphes aléatoires uniformes. Discrete Math. 40 (1982) 315-318. | Zbl

[19] M.R. Garey et D.S. Johnson, Computers and intractability. A guide sto the theory of NP-completeness. W.H. Freeman, San Francisco (1979). | MR | Zbl

[20] M.M. Halldórsson, A still better performance guarantee for approximate graph coloring. Inform. Process. Lett. 45 (1993) 19-23. | MR | Zbl

[21] , Approximations via partitioning. JAIST Research Report IS-RR-95-0003F, Japan Advanced Institute of Science and Technology, Japan (1995).

[22] , 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. | MR

[23] 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.

[24] , Improved approximations of independent sets in bounded-degree graphs via subgraph removal. Nordic J. Comput. 1 (1994) 475-492. | MR | Zbl

[25] R. Hassin et S. Lahav, Maximizing the number of unused colors in the vertex coloring problem. Inform. Process. Lett. 52 (1994) 87-90. | MR | Zbl

[26] J. Håstad, Clique is hard to approximate within n 1-ϵ . Acta Math. 182 (1999) 105-142. | MR | Zbl

[27] D.S. Hochbaum, Efficient bounds for the stable set, vertex cover and set packing problems. Discrete Appl. Math. 6 (1983) 243-254. | MR | Zbl

[28] , Approximation algorithms for NP-hard problems. PWS, Boston (1997).

[29] O.H. Ibarra et C.E. Kim, Fast approximation algorithms for the knapsack and sum of subset problems. J. Assoc. Comput. Mach. 22 (1975) 463-468. | MR | Zbl

[30] D.S. Johnson, Approximation algorithms for combinatorial problems. J. Comput. System Sci. 9 (1974) 256-278. | MR | Zbl

[31] 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. | MR

[32] S. Khanna, R. Motwani, M. Sudan et U. Vazirani, On syntactic versus computational views of approximability. SIAM J. Comput. 28 (1998) 164-191. | MR | Zbl

[33] H.R. Lewis et C.H. Papadimitriou, Elements of the theory of computation. Prentice-Hall (1981). | Zbl

[34] C. Lund et M. Yannakakis, On the hardness of approximating minimization problems. J. Assoc. Comput. Mach. 41 (1994) 960-981. | MR | Zbl

[35] R. Motwani, Lecture notes on approximation algorithms, Vol. I. Stanford University (1993).

[36] G.L. Nemhauser, L.A. Wolsey et M.L. Fischer, An analysis of approximations for maximizing submodular set functions. Math. Programming 14 (1978) 265-294. | MR | Zbl

[37] C.H. Papadimitriou et K. Steiglitz, Combinatorial optimization: Algorithms and complexity. Prentice Hall, New Jersey (1981). | MR | Zbl

[38] 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. | Zbl

[39] D. Simchi-Levi, New worst-case results for the bin-packing problem. Naval Res. Logistics 41 (1994) 579-585. | Zbl

[40] P. Turán, On an extremal problem in graph theory (in Hungarian). Mat. Fiz. Lapok 48 (1941) 436-452. | Zbl

[41] V. Vazirani, Approximation algorithms. Springer, Heildelberg (2001). | MR | Zbl

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