### Recherche et téléchargement d’archives de revues mathématiques numérisées

Table des matières de ce fascicule | Article suivant
Monnot, Jérôme
Differential approximation of NP-hard problems with equal size feasible solutions. RAIRO - Operations Research - Recherche Opérationnelle, 36 no. 4 (2002), p. 279-297
Texte intégral djvu | pdf | Analyses MR 1997926 | Zbl 1037.90059
Mots clés: approximate algorithms, differential ratio, performance ratio, analysis of algorithms

URL stable: http://www.numdam.org/item?id=RO_2002__36_4_279_0

Voir cet article sur le site de l'éditeur

Résumé

In this paper, we focus on some specific optimization problems from graph theory, those for which all feasible solutions have an equal size that depends on the instance size. Once having provided a formal definition of this class of problems, we try to extract some of its basic properties; most of these are deduced from the equivalence, under differential approximation, between two versions of a problem $\pi$ which only differ on a linear transformation of their objective functions. This is notably the case of maximization and minimization versions of $\pi$, as well as general minimization and minimization with triangular inequality versions of $\pi$. Then, we prove that some well known problems do belong to this class, such as special cases of both spanning tree and vehicles routing problems. In particular, we study the strict rural postman problem (called $SRPP$) and show that both the maximization and the minimization versions can be approximately solved, in polynomial time, within a differential ratio bounded above by $1/2$. From these results, we derive new bounds for standard ratio when restricting edge weights to the interval $[a,ta]$ (the $SRPP[t]$ problem): we respectively provide a $2/(t+1)$- and a $(t+1)/2t$-standard approximation for the minimization and the maximization versions.

Bibliographie

[1] A. Aggarwal, H. Imai, N. Katoh and S. Suri, Finding $k$ points with minimum diameter and related problems. J. Algorithms 12 (1991) 38-56.  MR 1088115 |  Zbl 0715.68082
[2] A. Aiello, E. Burattini, M. Massarotti and F. Ventriglia, A new evaluation function for approximation. Sem. IRIA (1977).
[3] L. Alfandari and V.Th. Paschos, Approximating the minimum rooted spanning tree with depth two. Int. Trans. Oper. Res. 6 (1999) 607-622.  MR 1724458
[4] G. Ausiello, P. Crescenzi, G. Gambosi, V. Kann, A. Marchetti Spaccamela and M. Protasi, Complexity and approximation: Combinatorial optimization problems and their approximability properties. Springer Verlag (1999).  MR 1734026 |  Zbl 0937.68002
[5] G. Ausiello, P. Crescenzi and M. Protasi, Approximate solutions of NP-optimization problems. Theoret. Comput. Sci. 150 (1995) 1-55.  MR 1357119 |  Zbl 0874.68145
[6] G. Ausiello, A. D’Atri and M. Protasi, Structure preserving reductions among convex optimization problems. J. Comput. System Sci. 21 (1980) 136-153.  Zbl 0441.68049
[7] N. Christofides, Worst-case analysis of a new heuristic for the traveling salesman problem. Technical Report 338 Grad School of Indistrial Administration, CMU (1976).
[8] G. Cornuejols, M.L. Fisher and G.L. Memhauser, Location of bank accounts to optimize float: An analytic study of exact and approximate algorithms. Management Sci. 23 (1977) 789-810.  MR 444049 |  Zbl 0361.90034
[9] P. Crescenzi and V. Kann, A compendium of NP-optimization problems. Available on www address: http://www.nada.kth.se/ viggo/problemlist/compendium.html (1998).
[10] P. Crescenzi and A. Panconesi, Completness in approximation classes. Inform. and Comput. 93 (1991) 241-262.  MR 1115527 |  Zbl 0734.68039
[11] M. Demange, D. de Werra and J. Monnot, Weighted node coloring: When stable sets are expensive (extended abstract), in Proc. WG 02. Springer Verlag, Lecture Notes in Comput. Sci. 2573 (2002) 114-125.  MR 2063804 |  Zbl 1022.68092
[12] M. Demange, P. Grisoni and V.Th. Paschos, Differential approximation algorithms for some combinatorial optimization problems. Theoret. Comput. Sci. 209 (1998) 107-122.  MR 1647498 |  Zbl 0912.68061
[13] M. Demange, J. Monnot and V.Th. Paschos, Bridging gap between standard and differential polynomial approximation: The case of bin-packing. Appl. Math. Lett. 12 (1999) 127-133.  MR 1750071 |  Zbl 0942.68144
[14] M. Demange, J. Monnot and V.Th. Paschos, Differential approximation results for the Steiner tree problem. Appl. Math. Lett. (to appear).  MR 1986043 |  Zbl 1046.90096
[15] M. Demange and V.Th. Paschos, On an approximation measure founded on the links between optimization and polynomial approximation theory. Theoret. Comput. Sci. 156 (1996)117-141.  MR 1388966 |  Zbl 0871.90069
[16] H.A. Eiselt, M. Gendreau and G. Laporte, Arc routing problems, part II: The rural postman problem. Oper. Res. (Survey, Expository and Tutorial) 43 (1995) 399-414.  MR 1327413 |  Zbl 0853.90042
[17] L. Engebretsen and M. Karpinski, Approximation hardness of TSP with bounded metrics. Available on www address: http://www.nada.kth.se/~enge/enge.bib (2002). ECCC Report TR00-089 (2000).  MR 2065863
[18] M.L. Fisher, G.L. Nemhauser and L.A. Wolsey, An analysis of approximations for finding a maximum weight hamiltonian circuit. Oper. Res. 27 (1979) 799-809.  MR 542983 |  Zbl 0412.90070
[19] G.N. Frederickson, Approximation algorithm for some postman problems. J. ACM 26 (1979) 538-554.  MR 535270 |  Zbl 0405.90076
[20] M.R. Garey and D.S. Johnson, Computers and intractability. A guide to the theory of NP-completeness. CA. Freeman (1979).  MR 519066 |  Zbl 0411.68039
[21] N. Garg, A $3$-approximation for the minimum tree spanning $k$ vertices. In Proc. FOCS (1996) 302-309.  MR 1450628
[22] L. Gouveia, Multicommodity flow models for spanning tree with hop constraints. Eur. J. Oper. Res. 95 (1996) 178-190.  Zbl 0947.90513
[23] L. Gouveia, Using variable redefinition for computing lower bounds for minimum spanning and Steiner trees with hop constraints. J. Comput. 10 (1998) 180-188.  MR 1637563 |  Zbl 1054.90622
[24] L. Gouveia and E. Janssen, Designing reliable tree with two cable technologies. Eur. J. Oper. Res. 105 (1998) 552-568.  Zbl 0955.90009
[25] N. Guttmann–Beck, R. Hassin, S. Khuller and B. Raghavachari, Approximation algorithms with bounded performance guarantees for the clustered traveling salesman problem. Algorithmica 4 (2000) 422-437.  Zbl 0963.68226
[26] M.M. Halldorsson, K. Iwano, N. Katoh and T. Tokuyama, Finding subsets maximizing minimum structures, in Proc. SODA (1995) 150-159.  MR 1321846 |  Zbl 0848.68071
[27] R. Hassin and S. Khuller, $z$-approximations. J. Algorithms 41 (2002) 429-442.  MR 1869261 |  Zbl 1014.68222
[28] R. Hassin and S. Rubinstein, An approximation algorithm for maximum packing of 3-edge paths. Inform. Process. Lett. 6 (1997) 63-67.  MR 1474580
[29] D. Hochbaum, Approximation algorithms for NP-hard problems. P.W.S (1997).
[30] J.A. Hoogeveen, Analysis of christofides’ heuristic: Some paths are more difficult than cycles. Oper. Res. Lett. 10 (1991) 291-295.  Zbl 0748.90071
[31] D.S. Johnson and C.H. Papadimitriou, Performance guarantees for heuristics, edited by E.L. Lawler, J.K. Lenstra, A.H.G. Rinnooy Kan and D.B. Shmoys, The Traveling Salesman Problem: A guided tour of Combinatorial Optimization. Wiley, Chichester (1985) 145-180.  MR 811472 |  Zbl 0574.90086
[32] R.M. Karp, Reducibility among combinatorial problems, edited by R.E Miller and J.W. Thatcher, Complexity of Computer Computations. Plenum Press, NY (1972) 85-103.  MR 378476
[33] D.G. Kirkpatrick and P. Hell, The complexity of a generalized matching problem, in Proc. STOC (1978) 240-245.  MR 521059
[34] G. Kortsarz and D. Peleg, Approximating shallow-light trees, in Proc. SODA (1997) 103-110.  MR 1447655
[35] S.R. Kosaraju, J.K. Park and C. Stein, Long tours and short superstrings, in Proc. FOCS (1994) 166-177.
[36] T. Magnanti and L. Wolsey, Optimal trees, Network models. North-Holland, Handbooks Oper. Res. Management Sci. 7 (1995) 503-615.  MR 1420874 |  Zbl 0839.90135
[37] P. Manyem and M.F.M. Stallmann, Some approximation results in multicasting, Working Paper. North Carolina State University (1996).
[38] J.S.B. Mitchell, Guillotine subdivisions approximate polygonal subdivisions: Part II – A simple polynomial-time approximation scheme for geometric TSP, $k$-MST, and related problems. SIAM J. Comput. 28 (1999) 1298-1309.  Zbl 0940.68062
[39] J. Monnot, Families of critical instances and polynomial approximation, Ph.D. Thesis. LAMSADE, Université Paris IX, Dauphine (1998) (in French).
[40] J. Monnot, The maximum $f$-depth spanning tree problem. Inform. Process. Lett. 80 (2001) 179-187.  MR 1859339 |  Zbl 1032.68087
[41] J. Monnot, V.Th. Paschos and S. Toulouse, Differential approximation results for traveling salesman problem with distance $1$ and $2$ (extended abstract). Proc. FCT 2138 (2001) 275-286.  MR 1914277 |  Zbl 1003.90035
[42] J. Monnot, V.Th. Paschos and S. Toulouse, Approximation polynomiale des problèmes NP-difficiles: optima locaux et rapport différentiel. Édition HERMÈS Science Lavoisier (2003).  Zbl 1140.90011
[43] J.S. Naor and B. Schieber, Improved approximations for shallow-light spanning trees, in Proc. FOCS (1997) 536-541.
[44] C.S. Orloff, A fundamental problem in vehicle routing. Networks 4 (1974) 35-64.  MR 332133 |  Zbl 0368.90130
[45] P. Orponen and H. Mannila, On approximation preserving reductions: Complete problems and robust measures, Technical Report C-1987-28. Department of Computer Science, University of Helsinki (1987).
[46] R. Ravi, R. Sundaram, M.V. Marathe, D.J. Rosenkrants and S.S. Ravi, Spanning tree short or small. SIAM J. Discrete Math. 9 (1996) 178-200.  MR 1386876 |  Zbl 0855.05058
[47] S. Sahni and T. Gonzalez, P-complete approximation problems. J. ACM 23 (1976) 555-565.  MR 408313 |  Zbl 0348.90152
[48] S.A. Vavasis, Approximation algorithms for indefinite quadratic programming. Math. Programming 57 (1972) 279-311.  MR 1195028 |  Zbl 0845.90095
[49] E. Zemel, Measuring the quality of approximate solutions to zero-one programming problems. Math. Oper. Res. 6 (1981) 319-332.  MR 629633 |  Zbl 0538.90065
Copyright Cellule MathDoc 2014 | Crédit | Plan du site