# Forests, frames, and games: Algorithms for matroid sums and applications

@article{Gabow2005ForestsFA, title={Forests, frames, and games: Algorithms for matroid sums and applications}, author={H. Gabow and H. H. Westermann}, journal={Algorithmica}, year={2005}, volume={7}, pages={465-497} }

This paper presents improved algorithms for matroid-partitioning problems, such as finding a maximum cardinality set of edges of a graph that can be partitioned intok forests, and finding as many disjoint spanning trees as possible. The notion of a clump in a matroid sum is introduced, and efficient algorithms for clumps are presented. Applications of these algorithms are given to problems arising in the study of the structural rigidity of graphs, the Shannon switching game, and others.

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#### References

SHOWING 1-10 OF 42 REFERENCES

Efficient algorithms for matroid sums

- Mathematics
- 1988

Matroid theory is the theory of independent sets in a finite universe. The term "independent" is borrowed from linear algebra as is most of the terminology. Many combinatorial problems can be modeled… Expand

A Linear-Time Algorithm for Finding a Minimum Spanning Pseudoforest

- Mathematics, Computer Science
- Inf. Process. Lett.
- 1988

It is shown that a minimum cost spanning pseudoforest of a graph with n vertices and m edges can be found in O(m+n) time, which implies that aminimum spanning tree can be find in O (m) time for graphs with girth at least log(i)n for some constant i. Expand

Efficient Algorithms for Graphic Matroid Intersection and Parity (Extended Abstract)

- Mathematics, Computer Science
- ICALP
- 1985

Improved algorithms for other problems are obtained, including maintaining a minimum spanning tree on a planar graph subject to changing edge costs, and finding shortest pairs of disjoint paths in a network. Expand

A Note on Finding Minimum-Cost Edge-Disjoint Spanning Trees

- Mathematics, Computer Science
- Math. Oper. Res.
- 1985

This work considers the problem of finding a set of k edge-disjoint spanning trees in G of minimum total edge cost and presents an implementation of the matroid greedy algorithm that runs in O ( m log m + k 2 n 2 ) time. Expand

Matroid theory and its applications in electric network theory and in statics

- Mathematics
- 1989

I. The topics of this book The concept of a matroid has been known for more than five decades. Whitney (1935) introduced it as a common generalization of graphs and matrices. In the last two decades,… Expand

Maximally Distant Trees and Principal Partition of a Linear Graph

- Mathematics
- 1969

A pair of trees of a graph is said to be maximally distant if the distance between these trees is maximum in the graph. Necessary and sufficient conditions for maximally distant trees are presented.… Expand

NETWORK-FLOW ALGORITHMS FOR LOWER-TRUNCATED TRANSVERSAL POLYMATROIDS

- Mathematics
- 1983

In this paper, we introduce lower-truncated transversal polymatroids, and develop efficient algorithms of network-flow type for those polymatroids. The lower-truncated transversal polymatroid… Expand

Use of matroid theory in operations research, circuits and systems theory

- Mathematics
- 1981

Abstract Recent trends in application of matroid theory to systems problems in operations research and other fields of engineering science are surveyed. Most of the significant applications have been… Expand

Algorithms for Two Bottleneck Optimization Problems

- Mathematics, Computer Science
- J. Algorithms
- 1988

A bottleneck optimization problem on a graph with edge costs is the problem of finding a subgraph of a certain kind that minimizes the maximum edge cost in the subgraph, and a fast algorithms for two bottleneck optimization problems are proposed. Expand

The principal minors of a matroid

- Mathematics
- 1971

Abstract Kishi and Kajitani introduced the concepts of the principal partition of a graph and maximally distant forest pairs . These concepts lead to the determination of the topological degrees of… Expand