Comparing imperfection ratio and imperfection index for graph classes

Koster, Arie M. C. A.; Wagler, Annegret K.

RAIRO - Operations Research - Recherche Opérationnelle, Tome 42 (2008) no. 4, pp. 485-500.

Perfect graphs constitute a well-studied graph class with a rich structure, reflected by many characterizations with respect to different concepts. Perfect graphs are, for instance, precisely those graphs $G$ where the stable set polytope $\mathrm{STAB}\left(G\right)$ coincides with the fractional stable set polytope $\mathrm{QSTAB}\left(G\right)$. For all imperfect graphs $G$ it holds that $\mathrm{STAB}\left(G\right)\subset \mathrm{QSTAB}\left(G\right)$. It is, therefore, natural to use the difference between the two polytopes in order to decide how far an imperfect graph is away from being perfect. We discuss three different concepts, involving the facet set of $\mathrm{STAB}\left(G\right)$, the disjunctive index of $\mathrm{QSTAB}\left(G\right)$, and the dilation ratio of the two polytopes. Including only certain types of facets for $\mathrm{STAB}\left(G\right)$, we obtain graphs that are in some sense close to perfect graphs, for example minimally imperfect graphs, and certain other classes of so-called rank-perfect graphs. The imperfection ratio has been introduced by Gerke and McDiarmid [12] as the dilation ratio of $\mathrm{STAB}\left(G\right)$ and $\mathrm{QSTAB}\left(G\right)$, whereas Aguilera et al. [1] suggest to take the disjunctive index of $\mathrm{QSTAB}\left(G\right)$ as the imperfection index of $G$. For both invariants there exist no general upper bounds, but there are bounds known for the imperfection ratio of several graph classes [7,12]. Outgoing from a graph-theoretical interpretation of the imperfection index, we prove that there exists no upper bound on the imperfection index for those graph classes with a known bounded imperfection ratio. Comparing the two invariants on those classes, it seems that the imperfection index measures imperfection much more roughly than the imperfection ratio; we, therefore, discuss possible directions for refinements.

Classification : 05C17, 90C57
Mots clés : perfect graphs, imperfection ratio, imperfection index

[1] N.E. Aguilera, M.S. Escalante, and G.L. Nasini, A generalization of the Perfect Graph Theorem under the disjunctive index. Math. Oper. Res. 27 (2002) 460-469. | MR | Zbl

[2] E. Balas, G. Cornuéjols, and S. Ceria, A lift-and-project cutting plane algorithm for mixed 0-1 programs. Math. Program. 58 (1993) 295-324. | MR | Zbl

[3] R.G. Bland, H.-C. Huang, and L.E. Trotter, Jr., Graphical Properties Related to Minimal Imperfection. Discrete Math. 27 (1979) 11-22. | MR | Zbl

[4] S. Ceria, Lift-and-project cuts and perfect graphs. Math. Program. 98 (2003) 309-317. | MR | Zbl

[5] M. Chudnovsky, N. Robertson, P. Seymour, and R. Thomas, The Strong Perfect Graph Theorem. Ann. Math. 164 (2006) 51-229. | MR | Zbl

[6] V. Chvátal, On Certain Polytopes Associated with Graphs. J. Comb. Theory (B) 18 (1975) 138-154. | MR | Zbl

[7] S. Coulonges, A. Pêcher, and A. Wagler, Characterizing and bounding the imperfection ratio for some graph classes. Math. Program. (to appear). | MR

[8] W.H. Cunningham, Polyhedra for composed independence systems, in: Bonn Workshop on Combinatorial Optimization, edited by A. Bachem, M. Grötschel, and B. Korte, Ann. Discrete Math. 16 (1982) 57-67. | MR | Zbl

[9] J.R. Edmonds, Maximum Matching and a Polyhedron with (0, 1) Vertices. J. Res. Nat. Bur. Standards 69B (1965) 125-130. | MR | Zbl

[10] A.M.H. Gerards and A. Schrijver, Matrices with the Edmonds-Johnson Property. Combinatorica 6 (1986) 403-417. | MR | Zbl

[11] A.M.H. Gerards, G. Maróti, and A. Schrijver, Note on: N.E. Aguilera, M.S. Escalante, and G.L. Nasini, “A generalization of the Perfect Graph Theorem under the disjunctive index”. Math. Oper. Res. 28 (2003) 884-885. | MR | Zbl

[12] S. Gerke and C. Mcdiarmid, Graph imperfection I, II. J. Comb. Theor. (B) 83 (2001) 58-78, 79-101. | MR | Zbl

[13] M. Grötschel, L. Lovász, and A. Schrijver, Geometric algorithms and combinatorial optimization. Springer-Verlag (1988). | EuDML | MR | Zbl

[14] M. Larsen, J. Propp, and D.H. Ullman, The fractional chromatic number of Mycielski graphs. J. Graph Theory 20 (1995) 411-416. | Zbl

[15] L. Lipták and L. Tunçel, Lift-and-project ranks and antiblocker duality. Oper. Res. Lett. 33 (2005) 35-41. | MR | Zbl

[16] L. Lovász, Normal hypergraphs and the weak perfect graph conjecture. Discrete Math. 2 (1972) 253-267. | MR | Zbl

[17] J. Mycielski, Sur le coloriage des graphs. Colloq. Math. 3 (1955) 161-162. | EuDML | MR | Zbl

[18] G.L. Nasini, El índice de imperfección de un grafo y su complemento, in Anales de las XXX JAIIO-SIO'01 (2001) 101-106.

[19] B. Reed and J. Ramirez-Alfonsin, Perfect Graphs, Wiley (2001). | MR | Zbl

[20] F.B. Shepherd, Near-Perfect Matrices. Math. Program. 64 (1994) 295-323. | MR | Zbl

[21] A. Wagler, Relaxing perfectness: which graphs are “almost” perfect? in The sharpest cut, impact of Manfred Padberg and his Work, edited by M. Grötschel , MPS/SIAM series on Optimization 4 (2004). | MR | Zbl

[22] A. Wagler, Antiwebs are rank-perfect. A Quaterly Journal of Operations Research 2 (2004) 149-152. | MR | Zbl

[23] A. Wagler, On the stable set polytopes of classes of near-bipartite graphs. A Quaterly Journal of Operations Research 3 (2005) 329-336. | Zbl