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

A. Frank

A. Hoffman

A. Schrijver

B. Chazelle

B. Korte

F. Grandoni

J. Könemann

K. Chaudhuri

P.N. Klein

R. Ravi

S. Arora

T. Király

V. Bilo

Z. Nutov

English

arXiv

Crossref

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

Bansal, N Nikhil

Khandekar, R

Könemann, J

Nagarajan, V

Peis, B

Repository TU/e

On generalizations of network design problems with degree bounds

Bansal, N.

Khandekar, R.

Könemann, J.

Nagarajan, V.

Peis, B.

Pure OAI Repository

The problem of designing efficient networks with degree-bound constraints has received a lot of attention recently. In this paper, we study several generalizations of this fundamental problem. Our generalizations are of the following two types:
- Generalize constraints on vertex-degree to arbitrary subsets of edges.
- Generalize the underlying network design problem to other combinatorial optimization problems like polymatroid intersection and lattice polyhedra.
We present several algorithmic results and lower bounds for these problems. At a high level, our algorithms are based on the iterative rounding/relaxation technique introduced in the context of degree bounded network design by Lau et al. and Singh-Lau. However many new ideas are required to apply this technique to the problems we consider. Our main results are:
-We consider the minimum crossing spanning tree problem in the case that the ""degree constraints"" have a laminar structure (this generalizes the well-known bounded degree MST). We provide a (1,b+O(log n)) bicriteria approximation for this problem, that improves over earlier results.
- We introduce the minimum crossing polymatroid intersection problem, and give a (2,2b+Delta-1) bicriteria approximation (where Delta is the maximum number of degree-constraints that  an element is part of). In the special case of bounded-degree arborescence (here Delta=1), this improves the previously best known (2,2b+2) bound to (2,2b).
- We also introduce the minimum crossing lattice polyhedra problem, and obtain a (1,b+2*Delta-1) bicriteria approximation under certain condition. This result provides a unified framework and common generalization of various problems studied previously, such as degree bounded matroids

Bansal, Nikhil

Khandekar, Rohit

Könemann, Jochen

Nagarajan, Viswanath

Peis, Britta

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http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.221.1246

On Generalizations of Network Design Problems with Degree Bounds

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