321 research outputs found
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
On Generalizations of Network Design Problems with Degree Bounds
Iterative rounding and relaxation have arguably become the method of choice
in dealing with unconstrained and constrained network design problems. In this
paper we extend the scope of the iterative relaxation method in two directions:
(1) by handling more complex degree constraints in the minimum spanning tree
problem (namely, laminar crossing spanning tree), and (2) by incorporating
`degree bounds' in other combinatorial optimization problems such as matroid
intersection and lattice polyhedra. We give new or improved approximation
algorithms, hardness results, and integrality gaps for these problems.Comment: v2, 24 pages, 4 figure
On a generalization of iterated and randomized rounding
We give a general method for rounding linear programs that combines the
commonly used iterated rounding and randomized rounding techniques. In
particular, we show that whenever iterated rounding can be applied to a problem
with some slack, there is a randomized procedure that returns an integral
solution that satisfies the guarantees of iterated rounding and also has
concentration properties. We use this to give new results for several classic
problems where iterated rounding has been useful
Approximating Minimum-Cost k-Node Connected Subgraphs via Independence-Free Graphs
We present a 6-approximation algorithm for the minimum-cost -node
connected spanning subgraph problem, assuming that the number of nodes is at
least . We apply a combinatorial preprocessing, based on the
Frank-Tardos algorithm for -outconnectivity, to transform any input into an
instance such that the iterative rounding method gives a 2-approximation
guarantee. This is the first constant-factor approximation algorithm even in
the asymptotic setting of the problem, that is, the restriction to instances
where the number of nodes is lower bounded by a function of .Comment: 20 pages, 1 figure, 28 reference
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
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
Survivable paths in multilayer networks
We consider the problem of protection in multilayer networks. In single-layer networks, a pair of disjoint paths can be used to provide protection for a source-destination pair. However, this approach cannot be directly applied to layered networks where disjoint paths may not always exist. In this paper, we take a new approach which is based on finding a set of paths that may not be disjoint but together will survive any single physical link failure. We consider the problem of finding the minimum number of survivable paths. In particular, we focus on two versions of this problem: one where the length of a path is restricted, and the other where the number of paths sharing a fiber is restricted. We prove that in general, finding the minimum survivable path set is NP-hard, whereas both of the restricted versions of the problem can be solved in polynomial time. We formulate the problems as Integer Linear Programs (ILPs), and use these formulations to develop heuristics and approximation algorithms.National Science Foundation (U.S.) (NSF grant CNS-0830961)National Science Foundation (U.S.) (NSF grant CNS-1017800)United States. Defense Threat Reduction Agency (grant HDTRA-09-1-005)United States. Defense Threat Reduction Agency (grant HDTRA1-07-1-0004
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