19 research outputs found
Optimal General Matchings
Given a graph and for each vertex a subset of the
set , where denotes the degree of vertex
in the graph , a -factor of is any set such that
for each vertex , where denotes the number of
edges of incident to . The general factor problem asks the existence of
a -factor in a given graph. A set is said to have a {\em gap of
length} if there exists a natural number such that and . Without any restrictions the
general factor problem is NP-complete. However, if no set contains a gap
of length greater than , then the problem can be solved in polynomial time
and Cornuejols \cite{Cor} presented an algorithm for finding a -factor, if
it exists. In this paper we consider a weighted version of the general factor
problem, in which each edge has a nonnegative weight and we are interested in
finding a -factor of maximum (or minimum) weight. In particular, this
version comprises the minimum/maximum cardinality variant of the general factor
problem, where we want to find a -factor having a minimum/maximum number of
edges.
We present an algorithm for the maximum/minimum weight -factor for the
case when no set contains a gap of length greater than . This also
yields the first polynomial time algorithm for the maximum/minimum cardinality
-factor for this case
A Comparison between the Zero Forcing Number and the Strong Metric Dimension of Graphs
The \emph{zero forcing number}, , of a graph is the minimum
cardinality of a set of black vertices (whereas vertices in are
colored white) such that is turned black after finitely many
applications of "the color-change rule": a white vertex is converted black if
it is the only white neighbor of a black vertex. The \emph{strong metric
dimension}, , of a graph is the minimum among cardinalities of all
strong resolving sets: is a \emph{strong resolving set} of
if for any , there exists an such that either
lies on an geodesic or lies on an geodesic. In this paper, we
prove that for a connected graph , where is
the cycle rank of . Further, we prove the sharp bound
when is a tree or a unicyclic graph, and we characterize trees
attaining . It is easy to see that can be
arbitrarily large for a tree ; we prove that and
show that the bound is sharp.Comment: 8 pages, 5 figure
Approximation hardness of Travelling Salesman via weighted amplifiers
The expander graph constructions and their variants are the main tool used in gap preserving reductions to prove approximation lower bounds of combinatorial optimisation problems. In this paper we introduce the weighted amplifiers and weighted low occurrence of Constraint Satisfaction problems as intermediate steps in the NP-hard gap reductions. Allowing the weights in intermediate problems is rather natural for the edge-weighted problems as Travelling Salesman or Steiner Tree. We demonstrate the technique for Travelling Salesman and use the parametrised weighted amplifiers in the gap reductions to allow more flexibility in fine-tuning their expanding parameters. The purpose of this paper is to point out effectiveness of these ideas, rather than to optimise the expanderâs parameters. Nevertheless, we show that already slight improvement of known expander values modestly improve the current best approximation hardness value for TSP from 123/122 ([9]) to 117/116 . This provides a new motivation for study of expanding properties of random graphs in order to improve approximation lower bounds of TSP and other edge-weighted optimisation problems
Approximation algorithms for general cluster routing problem
Graph routing problems have been investigated extensively in operations
research, computer science and engineering due to their ubiquity and vast
applications. In this paper, we study constant approximation algorithms for
some variations of the general cluster routing problem. In this problem, we are
given an edge-weighted complete undirected graph whose vertex set
is partitioned into clusters We are also given a subset
of and a subset of The weight function satisfies the
triangle inequality. The goal is to find a minimum cost walk that visits
each vertex in only once, traverses every edge in at least once and
for every all vertices of are traversed consecutively.Comment: In COCOON 202
On Integer Multiflows and Metric Packings in Matroids
7> ! R+ is a metric if m(e) m(C \Gamma feg) for all circuits C of M and all elements e of C. \Delta is a family of metrics if for every binary matroid M , \Delta(M ) is a set of metrics defined on E(M ). We shall consider the family \Delta A = \Delta A (M) of metrics m : E(M) ! A. A metric m is bipartite if m(C) is even for all circuits C of M . The extreme rays of cone (\Delta A (M)) are called primitive. Let \Delta be a family of metrics, and<F2