39 research outputs found
Approximating Minimum Bounded Degree Spanning Trees to within One of Optimal
In the Minimum Bounded Degree Spanning Tree problem, we are given an undirected graph G=(V,E) with a degree upper bound Bv on each vertex v∈V, and the task is to find a spanning tree of minimum cost that satisfies all the degree bounds. Let OPT be the cost of an optimal solution to this problem. In this paper, we present a polynomial time algorithm which returns a spanning tree T of cost at most OPT and dT(v)≤Bv+1 for all v, where dT(v) denotes the degree of v in T. This generalizes a result of Fürer and Raghavachari [1994] to weighted graphs, and settles a conjecture of Goemans [2006] affirmatively. The algorithm generalizes when each vertex v has a degree lower bound Av and a degree upper bound Bv, and returns a spanning tree with cost at most OPT and Av−1≤dT(v) ≤ Bv+1 for all v ∈ V. This is essentially the best possible. The main technique used is an extension of the iterative rounding method introduced by Jain [2001] for the design of approximation algorithms
Improved Algorithm for Degree Bounded Survivable Network Design Problem
We consider the Degree-Bounded Survivable Network Design Problem: the
objective is to find a minimum cost subgraph satisfying the given connectivity
requirements as well as the degree bounds on the vertices. If we denote the
upper bound on the degree of a vertex v by b(v), then we present an algorithm
that finds a solution whose cost is at most twice the cost of the optimal
solution while the degree of a degree constrained vertex v is at most 2b(v) +
2. This improves upon the results of Lau and Singh and that of Lau, Naor,
Salavatipour and Singh
Near-linear Time Algorithm for Approximate Minimum Degree Spanning Trees
Given a graph , we wish to compute a spanning tree whose maximum
vertex degree, i.e. tree degree, is as small as possible. Computing the exact
optimal solution is known to be NP-hard, since it generalizes the Hamiltonian
path problem. For the approximation version of this problem, a
time algorithm that computes a spanning tree of degree at most is
previously known [F\"urer \& Raghavachari 1994]; here denotes the
minimum tree degree of all the spanning trees. In this paper we give the first
near-linear time approximation algorithm for this problem. Specifically
speaking, we propose an time algorithm that
computes a spanning tree with tree degree for any constant .
Thus, when , we can achieve approximate solutions with
constant approximate ratio arbitrarily close to 1 in near-linear time.Comment: 17 page
Matroidal Degree-Bounded Minimum Spanning Trees
We consider the minimum spanning tree (MST) problem under the restriction
that for every vertex v, the edges of the tree that are adjacent to v satisfy a
given family of constraints. A famous example thereof is the classical
degree-constrained MST problem, where for every vertex v, a simple upper bound
on the degree is imposed. Iterative rounding/relaxation algorithms became the
tool of choice for degree-bounded network design problems. A cornerstone for
this development was the work of Singh and Lau, who showed for the
degree-bounded MST problem how to find a spanning tree violating each degree
bound by at most one unit and with cost at most the cost of an optimal solution
that respects the degree bounds.
However, current iterative rounding approaches face several limits when
dealing with more general degree constraints. In particular, when several
constraints are imposed on the edges adjacent to a vertex v, as for example
when a partition of the edges adjacent to v is given and only a fixed number of
elements can be chosen out of each set of the partition, current approaches
might violate each of the constraints by a constant, instead of violating all
constraints together by at most a constant number of edges. Furthermore, it is
also not clear how previous iterative rounding approaches can be used for
degree constraints where some edges are in a super-constant number of
constraints.
We extend iterative rounding/relaxation approaches both on a conceptual level
as well as aspects involving their analysis to address these limitations. This
leads to an efficient algorithm for the degree-constrained MST problem where
for every vertex v, the edges adjacent to v have to be independent in a given
matroid. The algorithm returns a spanning tree T of cost at most OPT, such that
for every vertex v, it suffices to remove at most 8 edges from T to satisfy the
matroidal degree constraint at v
k-Trails: Recognition, Complexity, and Approximations
The notion of degree-constrained spanning hierarchies, also called k-trails,
was recently introduced in the context of network routing problems. They
describe graphs that are homomorphic images of connected graphs of degree at
most k. First results highlight several interesting advantages of k-trails
compared to previous routing approaches. However, so far, only little is known
regarding computational aspects of k-trails.
In this work we aim to fill this gap by presenting how k-trails can be
analyzed using techniques from algorithmic matroid theory. Exploiting this
connection, we resolve several open questions about k-trails. In particular, we
show that one can recognize efficiently whether a graph is a k-trail.
Furthermore, we show that deciding whether a graph contains a k-trail is
NP-complete; however, every graph that contains a k-trail is a (k+1)-trail.
Moreover, further leveraging the connection to matroids, we consider the
problem of finding a minimum weight k-trail contained in a graph G. We show
that one can efficiently find a (2k-1)-trail contained in G whose weight is no
more than the cheapest k-trail contained in G, even when allowing negative
weights.
The above results settle several open questions raised by Molnar, Newman, and
Sebo
Approximability of Connected Factors
Finding a d-regular spanning subgraph (or d-factor) of a graph is easy by
Tutte's reduction to the matching problem. By the same reduction, it is easy to
find a minimal or maximal d-factor of a graph. However, if we require that the
d-factor is connected, these problems become NP-hard - finding a minimal
connected 2-factor is just the traveling salesman problem (TSP).
Given a complete graph with edge weights that satisfy the triangle
inequality, we consider the problem of finding a minimal connected -factor.
We give a 3-approximation for all and improve this to an
(r+1)-approximation for even d, where r is the approximation ratio of the TSP.
This yields a 2.5-approximation for even d. The same algorithm yields an
(r+1)-approximation for the directed version of the problem, where r is the
approximation ratio of the asymmetric TSP. We also show that none of these
minimization problems can be approximated better than the corresponding TSP.
Finally, for the decision problem of deciding whether a given graph contains
a connected d-factor, we extend known hardness results.Comment: To appear in the proceedings of WAOA 201