989 research outputs found
Quadratic diameter bounds for dual network flow polyhedra
Both the combinatorial and the circuit diameters of polyhedra are of interest
to the theory of linear programming for their intimate connection to a
best-case performance of linear programming algorithms.
We study the diameters of dual network flow polyhedra associated to -flows
on directed graphs and prove quadratic upper bounds for both of them:
the minimum of and for the combinatorial
diameter, and for the circuit diameter. The latter
strengthens the cubic bound implied by a result in [De Loera, Hemmecke, Lee;
2014].
Previously, bounds on these diameters have only been known for bipartite
graphs. The situation is much more involved for general graphs. In particular,
we construct a family of dual network flow polyhedra with members that violate
the circuit diameter bound for bipartite graphs by an arbitrary additive
constant. Further, it provides examples of circuit diameter
Robust capacitated trees and networks with uniform demands
We are interested in the design of robust (or resilient) capacitated rooted
Steiner networks in case of terminals with uniform demands. Formally, we are
given a graph, capacity and cost functions on the edges, a root, a subset of
nodes called terminals, and a bound k on the number of edge failures. We first
study the problem where k = 1 and the network that we want to design must be a
tree covering the root and the terminals: we give complexity results and
propose models to optimize both the cost of the tree and the number of
terminals disconnected from the root in the worst case of an edge failure,
while respecting the capacity constraints on the edges. Second, we consider the
problem of computing a minimum-cost survivable network, i.e., a network that
covers the root and terminals even after the removal of any k edges, while
still respecting the capacity constraints on the edges. We also consider the
possibility of protecting a given number of edges. We propose three different
formulations: a cut-set based formulation, a flow based one, and a bilevel one
(with an attacker and a defender). We propose algorithms to solve each
formulation and compare their efficiency
Polyhedra with few 3-cuts are hamiltonian
In 1956, Tutte showed that every planar 4-connected graph is hamiltonian. In
this article, we will generalize this result and prove that polyhedra with at
most three 3-cuts are hamiltonian. In 2002 Jackson and Yu have shown this
result for the subclass of triangulations. We also prove that polyhedra with at
most four 3-cuts have a hamiltonian path. It is well known that for each non-hamiltonian polyhedra with 3-cuts exist. We give computational
results on lower bounds on the order of a possible non-hamiltonian polyhedron
for the remaining open cases of polyhedra with four or five 3-cuts.Comment: 21 pages; changed titl
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
The Salesman's Improved Tours for Fundamental Classes
Finding the exact integrality gap for the LP relaxation of the
metric Travelling Salesman Problem (TSP) has been an open problem for over
thirty years, with little progress made. It is known that , and a famous conjecture states . For this problem,
essentially two "fundamental" classes of instances have been proposed. This
fundamental property means that in order to show that the integrality gap is at
most for all instances of metric TSP, it is sufficient to show it only
for the instances in the fundamental class. However, despite the importance and
the simplicity of such classes, no apparent effort has been deployed for
improving the integrality gap bounds for them. In this paper we take a natural
first step in this endeavour, and consider the -integer points of one such
class. We successfully improve the upper bound for the integrality gap from
to for a superclass of these points, as well as prove a lower
bound of for the superclass. Our methods involve innovative applications
of tools from combinatorial optimization which have the potential to be more
broadly applied
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