M. Laurent, A Generalization of antiwebs to independence systems and their canonical facets, Math. Programming 45, (1989), 97-108.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....the well known Steinitz problem for polytopes. In Section 4 we investigate rank facets of the feasible subsystem polytope. In particular, we focus attention on the rank inequalities arising from generalized antiwebs, which generalize cliques, odd holes and antiholes to general independence systems [33]. Finally, the appendix contains the proof of a result stated in Section 3 which completes the discussion but is not required in Section 4. Below we denote the ith row of the matrix A by a , 1 m; for S [m] m , AS denotes the S n matrix consisting of the rows ....

....if r(S) r(T ) r(S T ) for all T S, T #= #. For any set S E, S must be closed and nonseparable for the corresponding rank inequality to define a facet of P (I) These conditions generally are only necessary, but sufficient conditions can be stated using the following concept [33]. For S E, the critical graph GS (I) S, F ) is defined as follows: e, e # ) F , for e, e # S, if and only if there exists an independent set I such that I S, I = r(S) and e I , e # I , I e e # # I. It is shown in [33] that if S is a closed subset of E and the ....

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M. LAURENT, A generalization of antiwebs to independence systems and their canonical facets, Math. Programming 45 (1989), pp. 97--108.

....polytope, mainly based on the study of special configurations of the family of circuits. Among them are, to name a few, ffl the acyclic subdigraph polytope [GJR85b, Jn85] ffl the bipartite subgraph polytope [BGM85] ffl the planar subgraph polytope [JM93] We refer the reader to [EJR87, Lau89] and [BN89a, BN89b, CS89, NS89, Sas89] for the study of the facial structure of the independence system polytope in general. In Section 2, we introduce an extension of the node packing problem in hypergraphs, called transitive packing, by taking a kind of transitive elements into account. The ....

.... we introduce in the next section for the transitive packing polytope have been presented earlier for the independence system polytope; generalized cycle, generalized clique, and generalized antihole inequalities by Euler, Jnger, and Reinelt [EJR87] and generalized antiweb inequalities by Laurent [Lau89] It will turn out that our inequalities are more general even if we restrict ourselves to the independence system polytope. Nevertheless, in order to keep the terminology simple we will give them the same names and point out the restrictions that lead to the known inequalities, respectively. So ....

[Article contains additional citation context not shown here]

M. Laurent. A generalization of antiwebs to independence systems and their canonical facets. Mathematical Programming, 45:97 -- 108, 1989.

.... S and nonseparable if r(S) r(T ) r(S T ) for all T S; T 6= For any set S E, S must be closed and nonseparable for the corresponding rank inequality to de ne a facet of P (I) These conditions generally are only necessary, but sucient conditions can be stated using the following concept [21]. For S E, the critical graph GS (I) S; F ) is de ned as follows: e; e 0 ) 2 F , for e, e 0 2 S, if and only if there exists an independent set I such that I S, jI j = r(S) and e 2 I , e 0 = 2 I , I e e 0 2 I. It is shown in [21] that if S is a closed subset of E and the critical ....

....can be stated using the following concept [21] For S E, the critical graph GS (I) S; F ) is de ned as follows: e; e 0 ) 2 F , for e, e 0 2 S, if and only if there exists an independent set I such that I S, jI j = r(S) and e 2 I , e 0 = 2 I , I e e 0 2 I. It is shown in [21] that if S is a closed subset of E and the critical graph GS (I) of I on S is connected, then the corresponding rank inequality induces a facet of the polytope P (I) See references in [15] 4.1 Rank facets of the FS polytope As PFS is an independence system polytope, it is full dimensional if ....

[Article contains additional citation context not shown here]

M. Laurent, A generalization of antiwebs to independence systems and their canonical facets, Mathematical Programming, 45 (1989), pp. 97-108.

.... Lee and Leung (1993b) Nemhauser and Park (1991) Covering, packing and partition: Balas and Padberg (1972) Padberg (1973,1977,1980) Nemhauser and Trotter (1974) Trotter (1975) Wolsey (1976b) Balas and Zemel (1977) Balas and Ho (1980) Balas and Ng (1989a,b) Cornu ejols and Sassano (1989) Laurent (1989), Nobili and Sassano (1989) Sassano (1989) Chopra and Rao (1993) Ferreira et al. 1996,1998) Muller and Schulz (1996) Cheng and Cunningham (1997) Cut polytopes: Barahona and Mahjoub (1986) Barahona et al. 1988) Conforti et al. 1990 91a,b) De Sousa and Laurent (1995) Deza et al. 1992) ....

M. Laurent (1989) "A generalization of antiwebs to independence systems and their canonical facets", Mathematical Programming 45 97--108.

.... (A) are precisely the same ffl There is a 1 Gamma 1 correspondence between nonzero vertices of P (A) and vertices of P (A) We define the function Phi : R V nB R V by Phi(x) x 1 Deltax Gamma1 . We remark that this function is analagous to the transformation given by Laurent ([13]) except that in this context, the function is applied directly to the polyhedron (i.e. to the polyhedron instead of its antiblocker) Note that Phi Gamma1 = Phi and that for any vectors x 1 ; x k , and scalars c 1 ; c k : if P i c i x i = 0, then P i (c i (1 Delta x i ....

M. Laurent, A Generalization of antiwebs to independence systems and their canonical facets, Math. Programming 45, (1989), 97-108.

....polytope. Among them are, just to name a few, the acyclic subdigraph polytope [GJR85b, Jun85] the bipartite subgraph polytope [BGM85] and the planar subgraph polytope [JM93] Only recently the thorough study of the facial structure of the independence system polytope in general has begun, see [EJR87, Lau89]. In Sect. 2, we introduce in extension of the node packing problem in hypergraphs the concept of transitive packing by taking a kind of transitive elements into account. The problems we consider can be described as max c x s. t. Ax pA Gamma 1l (1) xu 2 f0; 1g where A is now an arbitrary 0= ....

.... Subclasses of the classes of valid inequalities which we introduce have been presented earlier for the independence system polytope: generalized cycle, generalized clique, and generalized antihole inequalities by Euler, Junger, and Reinelt [EJR87] and generalized antiweb inequalities by Laurent [Lau89]. It will turn out that our inequalities are more general even if we restrict to the independence system polytope. Nevertheless, in order to keep the terminology simple we will give them the same names and point out the restrictions that lead to the known inequalities, respectively. 3.1 ....

[Article contains additional citation context not shown here]

M. Laurent. A generalization of antiwebs to independence systems and their canonical facets. Mathematical Programming, 45:97 -- 108, 1989.

.... and Leung (1993b) Nemhauser and Park (1991) Covering, packing and partition: Balas and Padberg (1972) Padberg (1973,1977,1980) Nemhauser and Trotter (1974) Trotter (1975) Wolsey (1976b) Balas and Zemel (1977) Balas and Ho (1980) Balas and Ng (1989a,b) Cornu ejols and Sassano (1989) Laurent (1989), Nobili and Sassano (1989) Sassano (1989) Chopra and Rao (1993) Cut polytopes: Barahona and Mahjoub (1986) Barahona et al. 1988) Conforti et al. 1990 91a,b) De Sousa and Laurent (1991) Deza et al. 1992) Deza and Laurent (1992a,b) Pulleyblank and Shepherd (1993) Balas et al. 1994b) ....

M. Laurent (1989) "A generalization of antiwebs to independence systems and their canonical facets", Mathematical Programming 45 97--108.

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M. Laurent, A Generalization of antiwebs to independence systems and their canonical facets, Math. Programming 45, (1989), 97-108.

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