On the linear extension complexity of stable set polytopes for perfect graphs
نویسندگان
چکیده
We study the linear extension complexity of stable set polytopes of perfect graphs. We make use of known structural results permitting to decompose perfect graphs into basic perfect graphs by means of two graph operations: 2-joins and skew partitions. Exploiting the link between extension complexity and the nonnegative rank of an associated slack matrix, we investigate the behavior of the extension complexity under these graph operations. We show bounds for the extension complexity of the stable set polytope of a perfect graph G depending linearly on the size of G and involving the depth of a decomposition tree of G in terms of basic perfect graphs. © 2018 Elsevier Ltd. All rights reserved.
منابع مشابه
The Generalized Stable Set Problem for Perfect Bidirected Graphs
Bidirected graphs are a generalization of undirected graphs. For bidirected graphs, we can consider a problem whichi is a natural extension of the maximum weighted stable set problem for undirected graphs. Here we call this problem the generalized stable set problem. It is well known that the maximum weighted stable set problem is solvable in polynomial time for perfect undirected graphs. Perfe...
متن کاملSimple Extensions of Polytopes
We introduce the simple extension complexity of a polytope P as the smallest number of facets of any simple (i.e., non-degenerate in the sense of linear programming) polytope which can be projected onto P . We devise a combinatorial method to establish lower bounds on the simple extension complexity and show for several polytopes that they have large simple extension complexities. These example...
متن کاملLinear Programming, the Simplex Algorithm and Simple Polytopes
In the first part of the paper we survey some far reaching applications of the basis facts of linear programming to the combinatorial theory of simple polytopes. In the second part we discuss some recent developments concurring the simplex algorithm. We describe sub-exponential randomized pivot roles and upper bounds on the diameter of graphs of polytopes.
متن کاملAlmost all webs are not rank-perfect
Graphs with circular symmetry, called webs, are relevant w.r.t. describing the stable set polytopes of two larger graph classes, quasi-line graphs [8,12] and claw-free graphs [7,8]. Providing a decent linear description of the stable set polytopes of claw-free graphs is a long-standing problem [9]. Ben Rebea conjectured a description for quasi-line graphs, see [12]; Chudnovsky and Seymour [2] v...
متن کاملAlmost all webs with odd clique number 5 are not rank - perfect
Graphs with circular symmetry, called webs, are relevant w.r.t. describing the stable set polytopes of two larger graph classes, quasi-line graphs [7,11] and claw-free graphs [6,7]. Providing a decent linear description of the stable set polytopes of claw-free graphs is a long-standing problem [8]. However, even the problem of finding all facets of stable set polytopes of webs is open. So far, ...
متن کامل