On an approximation measure founded on the links between optimization and polynomial approximation theory
In order to define a polynomial approximation theory linked to combinatorial optimization closer than the existing one, we first formally define the notion of a combinatorial optimization problem and then, based upon this notion, we introduce a notion of equivalence among optimization problems. This...
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| Published in | Theoretical computer science Vol. 158; no. 1; pp. 117 - 141 |
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| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
Elsevier B.V
20.05.1996
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| Online Access | Get full text |
| ISSN | 0304-3975 1879-2294 |
| DOI | 10.1016/0304-3975(95)00060-7 |
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| Abstract | In order to define a polynomial approximation theory linked to combinatorial optimization closer than the existing one, we first formally define the notion of a combinatorial optimization problem and then, based upon this notion, we introduce a notion of equivalence among optimization problems. This equivalence includes, for example, translation or affine transformation of the objective function or yet some aspects of equivalencies between maximization and minimization problems (for example, the equivalence between minimum vertex cover and maximum independent set). Next, we adress the question of the adoption of an approximation ratio respecting the defined equivalence. We prove that an approximation ratio defined as a two-variable function cannot respect this equivalence. We then adopt a three-variable function as a new approximation ratio (already used by a number of researchers), which is coherent to the equivalence and, under the choice of the variables, the new ratio is introduced by an axiomatic approach. Finally, using the new ratio, we prove approximation results for a number of combinatorial problems. |
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| AbstractList | In order to define a polynomial approximation theory linked to combinatorial optimization closer than the existing one, we first formally define the notion of a combinatorial optimization problem and then, based upon this notion, we introduce a notion of equivalence among optimization problems. This equivalence includes, for example, translation or affine transformation of the objective function or yet some aspects of equivalencies between maximization and minimization problems (for example, the equivalence between minimum vertex cover and maximum independent set). Next, we address the question of the adoption of an approximation ratio respecting the defined equivalence. We prove that an approximation ratio defined as a two-variable function cannot respect this equivalence. We then adopt a three-variable function as a new approximation ratio (already used by a number of researchers), which is coherent to the equivalence and, under the choice of the variables, the new ratio is introduced by an axiomatic approach. Finally, using the new ratio, we prove approximation results for a number of combinatorial problems. In order to define a polynomial approximation theory linked to combinatorial optimization closer than the existing one, we first formally define the notion of a combinatorial optimization problem and then, based upon this notion, we introduce a notion of equivalence among optimization problems. This equivalence includes, for example, translation or affine transformation of the objective function or yet some aspects of equivalencies between maximization and minimization problems (for example, the equivalence between minimum vertex cover and maximum independent set). Next, we adress the question of the adoption of an approximation ratio respecting the defined equivalence. We prove that an approximation ratio defined as a two-variable function cannot respect this equivalence. We then adopt a three-variable function as a new approximation ratio (already used by a number of researchers), which is coherent to the equivalence and, under the choice of the variables, the new ratio is introduced by an axiomatic approach. Finally, using the new ratio, we prove approximation results for a number of combinatorial problems. |
| Author | Paschos, Vangelis Th Demange, Marc |
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| References | Demange, Paschos (BIB8) 1993; 317 Papadimitriou, Yannakakis (BIB15) 1991; 43 Ausiello, D'Atri, Protasi (BIB3) 1980; 21 Ibarra, Kim (BIB12) 1975; 22 Nemhauser, Wolsey, Fischer (BIB13) 1978; 14 Simon (BIB17) 1990; 3 Demange (BIB6) 1994 Garey, Johnson (BIB10) 1979 Berge (BIB4) 1973 Papadimitriou, Steiglitz (BIB14) 1981 Demange, Grisoni, Paschos (BIB7) 1994; 318 Gondran, Minoux (BIB11) 1985 Bollobás (BIB5) 1985 Arora, Lund, Motwani, Sudan, Szegedy (BIB2) 1992 Demange, Paschos (BIB9) 1993 Aiello, Burattini, Massarotti, Ventriglia (BIB1) 1977 Sahni, Gonzales (BIB16) 1976; 23 Demange (10.1016/0304-3975(95)00060-7_BIB9) 1993 Sahni (10.1016/0304-3975(95)00060-7_BIB16) 1976; 23 Ibarra (10.1016/0304-3975(95)00060-7_BIB12) 1975; 22 Demange (10.1016/0304-3975(95)00060-7_BIB8) 1993; 317 Berge (10.1016/0304-3975(95)00060-7_BIB4) 1973 Nemhauser (10.1016/0304-3975(95)00060-7_BIB13) 1978; 14 Papadimitriou (10.1016/0304-3975(95)00060-7_BIB14) 1981 Aiello (10.1016/0304-3975(95)00060-7_BIB1) 1977 Gondran (10.1016/0304-3975(95)00060-7_BIB11) 1985 Arora (10.1016/0304-3975(95)00060-7_BIB2) 1992 Bollobás (10.1016/0304-3975(95)00060-7_BIB5) 1985 Demange (10.1016/0304-3975(95)00060-7_BIB6) 1994 Garey (10.1016/0304-3975(95)00060-7_BIB10) 1979 Ausiello (10.1016/0304-3975(95)00060-7_BIB3) 1980; 21 Demange (10.1016/0304-3975(95)00060-7_BIB7) 1994; 318 Simon (10.1016/0304-3975(95)00060-7_BIB17) 1990; 3 Papadimitriou (10.1016/0304-3975(95)00060-7_BIB15) 1991; 43 |
| References_xml | – year: 1993 ident: BIB9 article-title: On an approximation measure founded on the links between optimization and polynomial approximation theory publication-title: Cahier du LAMSADE 116 – volume: 43 start-page: 425 year: 1991 end-page: 440 ident: BIB15 article-title: Optimization, approximation and complexity classes publication-title: J. Comput. System Sci. – year: 1977 ident: BIB1 article-title: A new evaluation function for approximation algorithms publication-title: Sem. IRIA – volume: 318 start-page: 289 year: 1994 end-page: 292 ident: BIB7 article-title: Quelques résultats dans le cadre d'une nouvelle théorie de l'approximation polynomiale publication-title: Comptes Rendus de l'Académie des Sciences de Paris – volume: 23 start-page: 555 year: 1976 end-page: 565 ident: BIB16 article-title: P-complete approximation problems publication-title: J. ACM – year: 1985 ident: BIB11 article-title: Graphes et Algorithmes – volume: 21 start-page: 136 year: 1980 end-page: 153 ident: BIB3 article-title: Structure preserving reductions among convex optimization problems publication-title: J. Comput. System Sci. – volume: 317 start-page: 409 year: 1993 end-page: 414 ident: BIB8 article-title: Quelques étapes vers la conciliation de la théorie d'approximation et celle d'optimisation: une nouvelle théorie d'approximation polynomiale et résultats préliminaires publication-title: Comptes Rendus de l'Académie des Sciences de Paris – year: 1981 ident: BIB14 article-title: Combinatorial Optimization: Algorithms and Complexity – year: 1994 ident: BIB6 article-title: Approximation polynomiale de problèmes NP-complets et programmation linéaire: une nouvelle mesure d'approximation et algorithmes publication-title: Ph.D. Thesis – volume: 3 start-page: 294 year: 1990 end-page: 310 ident: BIB17 article-title: On approximate solutions for combinatorial optimization problems publication-title: SIAM J. Discrete Math. – year: 1985 ident: BIB5 article-title: Random Graphs – year: 1973 ident: BIB4 article-title: Graphs and Hypergraphs – volume: 22 start-page: 463 year: 1975 end-page: 468 ident: BIB12 article-title: Fast approximation algorithms for the knapsack and sum of subset problems publication-title: J. ACM – volume: 14 start-page: 265 year: 1978 end-page: 294 ident: BIB13 article-title: An analysis of approximations for maximizing submodular set functions publication-title: Math. Programming – start-page: 14 year: 1992 end-page: 23 ident: BIB2 article-title: Proof verification and intractability of approximation problems publication-title: Proc. FOCS'92 – year: 1979 ident: BIB10 article-title: Computers and Intractability – volume: 43 start-page: 425 year: 1991 ident: 10.1016/0304-3975(95)00060-7_BIB15 article-title: Optimization, approximation and complexity classes publication-title: J. Comput. System Sci. doi: 10.1016/0022-0000(91)90023-X – year: 1994 ident: 10.1016/0304-3975(95)00060-7_BIB6 article-title: Approximation polynomiale de problèmes NP-complets et programmation linéaire: une nouvelle mesure d'approximation et algorithmes – year: 1985 ident: 10.1016/0304-3975(95)00060-7_BIB5 – volume: 23 start-page: 555 issue: 3 year: 1976 ident: 10.1016/0304-3975(95)00060-7_BIB16 article-title: P-complete approximation problems publication-title: J. ACM doi: 10.1145/321958.321975 – year: 1985 ident: 10.1016/0304-3975(95)00060-7_BIB11 – volume: 3 start-page: 294 issue: 2 year: 1990 ident: 10.1016/0304-3975(95)00060-7_BIB17 article-title: On approximate solutions for combinatorial optimization problems publication-title: SIAM J. Discrete Math. doi: 10.1137/0403025 – volume: 317 start-page: 409 issue: I year: 1993 ident: 10.1016/0304-3975(95)00060-7_BIB8 article-title: Quelques étapes vers la conciliation de la théorie d'approximation et celle d'optimisation: une nouvelle théorie d'approximation polynomiale et résultats préliminaires publication-title: Comptes Rendus de l'Académie des Sciences de Paris – year: 1981 ident: 10.1016/0304-3975(95)00060-7_BIB14 – year: 1977 ident: 10.1016/0304-3975(95)00060-7_BIB1 article-title: A new evaluation function for approximation algorithms – start-page: 14 year: 1992 ident: 10.1016/0304-3975(95)00060-7_BIB2 article-title: Proof verification and intractability of approximation problems – volume: 318 start-page: 289 issue: I year: 1994 ident: 10.1016/0304-3975(95)00060-7_BIB7 article-title: Quelques résultats dans le cadre d'une nouvelle théorie de l'approximation polynomiale publication-title: Comptes Rendus de l'Académie des Sciences de Paris – year: 1993 ident: 10.1016/0304-3975(95)00060-7_BIB9 article-title: On an approximation measure founded on the links between optimization and polynomial approximation theory doi: 10.1016/0304-3975(95)00060-7 – volume: 21 start-page: 136 year: 1980 ident: 10.1016/0304-3975(95)00060-7_BIB3 article-title: Structure preserving reductions among convex optimization problems publication-title: J. Comput. System Sci. doi: 10.1016/0022-0000(80)90046-X – volume: 22 start-page: 463 issue: 4 year: 1975 ident: 10.1016/0304-3975(95)00060-7_BIB12 article-title: Fast approximation algorithms for the knapsack and sum of subset problems publication-title: J. ACM doi: 10.1145/321906.321909 – year: 1973 ident: 10.1016/0304-3975(95)00060-7_BIB4 – year: 1979 ident: 10.1016/0304-3975(95)00060-7_BIB10 article-title: Computers and Intractability – volume: 14 start-page: 265 year: 1978 ident: 10.1016/0304-3975(95)00060-7_BIB13 article-title: An analysis of approximations for maximizing submodular set functions publication-title: Math. Programming doi: 10.1007/BF01588971 |
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