1,174 research outputs found
Reconstructing a Simple Polytope from its Graph
Blind and Mani (1987) proved that the entire combinatorial structure (the
vertex-facet incidences) of a simple convex polytope is determined by its
abstract graph. Their proof is not constructive. Kalai (1988) found a short,
elegant, and algorithmic proof of that result. However, his algorithm has
always exponential running time. We show that the problem to reconstruct the
vertex-facet incidences of a simple polytope P from its graph can be formulated
as a combinatorial optimization problem that is strongly dual to the problem of
finding an abstract objective function on P (i.e., a shelling order of the
facets of the dual polytope of P). Thereby, we derive polynomial certificates
for both the vertex-facet incidences as well as for the abstract objective
functions in terms of the graph of P. The paper is a variation on joint work
with Michael Joswig and Friederike Koerner (2001).Comment: 14 page
Fractional Perfect b-Matching Polytopes. I: General Theory
The fractional perfect b-matching polytope of an undirected graph G is the
polytope of all assignments of nonnegative real numbers to the edges of G such
that the sum of the numbers over all edges incident to any vertex v is a
prescribed nonnegative number b_v. General theorems which provide conditions
for nonemptiness, give a formula for the dimension, and characterize the
vertices, edges and face lattices of such polytopes are obtained. Many of these
results are expressed in terms of certain spanning subgraphs of G which are
associated with subsets or elements of the polytope. For example, it is shown
that an element u of the fractional perfect b-matching polytope of G is a
vertex of the polytope if and only if each component of the graph of u either
is acyclic or else contains exactly one cycle with that cycle having odd
length, where the graph of u is defined to be the spanning subgraph of G whose
edges are those at which u is positive.Comment: 37 page
Acyclic orientations with path constraints
Many well-known combinatorial optimization problems can be stated over the
set of acyclic orientations of an undirected graph. For example, acyclic
orientations with certain diameter constraints are closely related to the
optimal solutions of the vertex coloring and frequency assignment problems. In
this paper we introduce a linear programming formulation of acyclic
orientations with path constraints, and discuss its use in the solution of the
vertex coloring problem and some versions of the frequency assignment problem.
A study of the polytope associated with the formulation is presented, including
proofs of which constraints of the formulation are facet-defining and the
introduction of new classes of valid inequalities
Weighted graphs defining facets: a connection between stable set and linear ordering polytopes
A graph is alpha-critical if its stability number increases whenever an edge
is removed from its edge set. The class of alpha-critical graphs has several
nice structural properties, most of them related to their defect which is the
number of vertices minus two times the stability number. In particular, a
remarkable result of Lov\'asz (1978) is the finite basis theorem for
alpha-critical graphs of a fixed defect. The class of alpha-critical graphs is
also of interest for at least two topics of polyhedral studies. First,
Chv\'atal (1975) shows that each alpha-critical graph induces a rank inequality
which is facet-defining for its stable set polytope. Investigating a weighted
generalization, Lipt\'ak and Lov\'asz (2000, 2001) introduce critical
facet-graphs (which again produce facet-defining inequalities for their stable
set polytopes) and they establish a finite basis theorem. Second, Koppen (1995)
describes a construction that delivers from any alpha-critical graph a
facet-defining inequality for the linear ordering polytope. Doignon, Fiorini
and Joret (2006) handle the weighted case and thus define facet-defining
graphs. Here we investigate relationships between the two weighted
generalizations of alpha-critical graphs. We show that facet-defining graphs
(for the linear ordering polytope) are obtainable from 1-critical facet-graphs
(linked with stable set polytopes). We then use this connection to derive
various results on facet-defining graphs, the most prominent one being derived
from Lipt\'ak and Lov\'asz's finite basis theorem for critical facet-graphs. At
the end of the paper we offer an alternative proof of Lov\'asz's finite basis
theorem for alpha-critical graphs
- …