M. Grotschel, L. Lov'asz, and A. Schrijver. Relaxations of vertex packing. JCT B, pages 330--343, 1986.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... representations of G (cf. Definition 6) Orthogonal representations were introduced by Lov asz [Lo79] in the study of the Shannon capacity of a graph; they arise in connection with various topics in combinatorial optimization and discrete geometry, like the vertex packing polytope of a graph (cf. [GLS86, GLS88]) connectivity properties of graphs (cf. LSS89] spectral invariants of graphs (cf. CdV] vdH96] k Blocks. Following [AHMR88] a graph G is called a k block if G has order k and every proper induced subgraph of G has order k Gamma 1. For instance, the circuit C n is an (n Gamma 2) block ....

M. Grotschel, L. Lov'asz and A. Schrijver. Relaxations of vertex packing. Journal of Combinatorial Theory B, 40:330--343, 1986.

....different form. If x denotes the incidence vector of a stable set then we have that X i (c T v i ) 2 x i 1: 6) In other words, the orthonormal representation constraints (6) are valid inequalities for STAB(G) the convex hull of incidence vectors of stable sets of G. Grotschel et al. [23] show that if we let TH(G) fx : x satisfies (6) and x 0g, then #(G) maxf P i x i : x 2 TH(G)g. Yet more formulations of # are known (it seems all paths lead to # ) we strongly urge the reader to read Lov asz s original article or [23,24] for additional results. Schrijver [59] proposed a ....

....incidence vectors of stable sets of G. Grotschel et al. 23] show that if we let TH(G) fx : x satisfies (6) and x 0g, then #(G) maxf P i x i : x 2 TH(G)g. Yet more formulations of # are known (it seems all paths lead to # ) we strongly urge the reader to read Lov asz s original article or [23,24] for additional results. Schrijver [59] proposed a strengthening of #(G) by adding simple inequalities. We describe this improved upper bound on ff(G) in terms of the various formulations discussed above (other formulations of #(G) can also be similarly improved) The validity of these ....

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M. Grotschel, L. Lov'asz, and A. Schrijver. Relaxations of vertex packing. Journal of Combinatorial Theory B, 40:330--343, 1986.

....is an orthogonal labeling of dimension 1. Therefore every constraint of QSTAB(G) is one of the constraints of TH(G) Note: QSTAB first defined by Shannon [18] and the first systematic study of STAB was undertaken by Padberg [17] TH was first defined by Grotschel, Lovasz, and Schrijver in [6]. 3. Monotonicity. Suppose G and G # are graphs on the same vertex set V ,withG # G # (i.e. u v in G implies u v in G # ) Then every stable set in G # is stable in G, hence STAB(G) # STAB(G # ) every clique in G is a clique in G # , hence QSTAB(G) # QSTAB(G # ) every ....

M. Grotschel, L. Lovasz. and A. Schrijver, "Relaxations of vertex packing," Journal of Combinatorial Theory B40 (1986), 330--343.

....a stable set of G, then S is a clique of G. So, MSS is trivially equivalent to finding a maximum cardinality clique of G. MSS is known to be NP hard [13] for arbitrary graphs, while it is polynomially solvable for special classes of graphs, for instance, perfect graphs and t perfect graphs [16], circle graphs and their complements [14] circular arc graphs and their complements [15,17] claw free graphs [22] graphs without long odd cycles [19] Even though there is little hope to find a polynomial time algorithm for arbitrary graphs, many efforts have been made to implement efficient ....

M. Grotschel, L.Lovasz, and A. Schrijver. Relaxation of vertex packing. J. of Comb. Theory B, 40:330--343, 1986.

....in [21] Observe that finding a MSS in G is equivalent to finding a maximum cardinality clique in its complement. The MSS Problem is known to be NP hard [10] for arbitrary graphs, while it is polynomially solvable for special classes of graphs, for instance, perfect graphs and t perfect graphs [13], circle graphs and their complements [11] circular arc graphs and their complements [12, 14] claw free graphs [19] graphs without long odd cycles [15] Even though there is little hope to find a polynomial time algorithm for arbitrary graphs, many efforts have been made to implement efficient ....

M. Groetschel, L. Lov'asz, A. Schrijver, Relaxation of vertex packing, J. of Comb. Theory B, 40 (1986), 330--343.

....f0; 1g; i = 1; n: The advantage of formulation (2) over (1) is a smaller gap between the optimal values of (2) and its linear relaxation. However, since the number of constraints in (2) is exponential, solving the linear relaxation of (2) is not an easy problem. In fact, Grotschel et al. [151, 152] have shown that the linear relaxation problem of (2) is NP hard on general graphs. They have also shown that the same problem is polynomially solvable on perfect graphs. Furthermore, they have shown that a graph is perfect if and only if the optimal solution to the linear relaxation of (2) ....

M. Grotschel, L. Lov'asz and A. Schrijver, Relaxations of vertex packing, J. Combin. Theory B, Vol. 40: 330--343, 1986.

....is an orthogonal labeling of dimension 1. Therefore every constraint of QSTAB(G) is one of the constraints of TH(G) Note: QSTAB first defined by Shannon [18] and the first systematic study of STAB was undertaken by Padberg [17] TH was first defined by Grotschel, Lov asz, and Schrijver in [6]. 3. Monotonicity. Suppose G and G 0 are graphs on the same vertex set V , with G G 0 (i.e. u Gamma Gamma v in G implies u Gamma Gamma v in G 0 ) Then every stable set in G 0 is stable in G, hence STAB(G) STAB(G 0 ) every clique in G is a clique in G 0 , hence QSTAB(G) ....

M. Grotschel, L. Lov'asz. and A. Schrijver, "Relaxations of vertex packing," Journal of Combinatorial Theory B40 (1986), 330--343.

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M. Grotschel, L. Lov'asz, and A. Schrijver. Relaxations of vertex packing. JCT B, pages 330--343, 1986.

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M, Grotschel, Lov'asz and A. Schrijver, "Relaxations of vertex packing," Journal of combinatorial Theory, Series B 40 (1986) 330-343.

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M, Grotschel, Lov'asz and A. Schrijver, "Relaxations of vertex packing," Journal of combinatorial Theory, Series B 40 (1986) 330-343.

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