# Covering Graphs Using Trees and Stars

@inproceedings{Even2003CoveringGU, title={Covering Graphs Using Trees and Stars}, author={Guy Even and Naveen Garg and Jochen K{\"o}nemann and Ramamoorthi Ravi and Amitabh Sinha}, booktitle={RANDOM-APPROX}, year={2003} }

A tree cover of a graph G is defined as a collection of trees such that their union includes all the vertices of G. The cost of a tree cover is the weight of the maximum weight tree in the tree cover. Given a positive integer k, the k-tree cover problem is to compute a minimum cost tree cover which has no more than k trees. Star covers are defined analogously. Additionally, we may also be provided with a set of k vertices which are to serve as roots of the trees or stars. In this paper, we…

## 25 Citations

### Approximation to the Minimum Rooted Star Cover Problem

- MathematicsTAMC
- 2007

This paper presents a constant ratio approximation algorithm for finding a minimum cardinality set of rooted stars, that covers all vertices in V such that the length of each rooted star is at most D.

### Improved Approximation Algorithms for the Min-max Tree Cover and Bounded Tree Cover Problems

- Computer Science, MathematicsAlgorithmica
- 2012

Improved approximation algorithms for the Min-Max Tree Cover and Bounded Tree Cover problems are provided improving the 3-approximation bound in Arkin et al. (J. Algorithms 59:1–18, 2006).

### Improved approximation algorithms for Min-Max Tree Cover, Bounded Tree Cover, Shallow-Light and Buy-at-Bulk k-Steiner Tree, and (k, 2)-Subgraph

- Computer Science, Mathematics
- 2011

This thesis provides improved approximation algorithms for the Min-Max k-Tree Cover, Bounded Tree Cover and Shallow-Light k-Steiner Tree, (k, 2)-subgraph problems and shows the approximation algorithm for BBkST implies approximation factors for some other network design problems.

### Min-max cover of a graph with a small number of parts

- Computer Science, MathematicsDiscret. Optim.
- 2015

### Approximations for minimum and min-max vehicle routing problems

- Mathematics, Computer ScienceJ. Algorithms
- 2006

### D S ] 3 D ec 2 01 9 Approximating Star Cover Problems

- Physics, Mathematics
- 2019

Given a metric space (F ∪ C, d), we consider star covers of C with balanced loads. A star is a pair (f, Cf ) where f ∈ F and Cf ⊆ C, and the load of a star is ∑ c∈Cf d(f, c). In minimum load k-star…

### Approximation Algorithms for Min-Max Cycle Cover Problems

- Computer ScienceIEEE Transactions on Computers
- 2015

Improved approximation algorithms for the vehicle routing problem, which has wide application backgrounds and has been paid lots of attentions in past decades, are studied by devising improved approximate algorithms for it and its variants.

### A PPROXIMATION A LGORITHMS FOR SOME M IN - MAX V EHICLE R OUTING P ROBLEMS

- Computer Science, Mathematics
- 2013

This thesis considers min-max vehicle routing problems, in which the maximum cost incurred by the subgraph corresponding to each vehicle is to be minimized, and study two types of covering problems and present new or improved approximation algorithms for them.

### A Sublogarithmic Approximation for Highway and Tollbooth Pricing

- Computer ScienceICALP
- 2010

The main result is a deterministic algorithm for the tollbooth problem on trees whose approximation ratio is O(log m/log logm), where m denotes the number of edges in the underlying graph.

### Approximating Star Cover Problems

- MathematicsAPPROX-RANDOM
- 2020

New bicriteria approximations for the two problems using novel rounding algorithms for their standard LP relaxations are obtained.

## References

SHOWING 1-10 OF 19 REFERENCES

### Approximation Algorithms for Min-sum p-clustering

- Computer Science, MathematicsDiscret. Appl. Math.
- 1998

### Clustering to minimize the sum of cluster diameters

- Computer Science, MathematicsSTOC '01
- 2001

This work presents a primal-dual based constant factor approximation algorithm that achieves a logarithmic approximation which also applies when the distance function is asymmetric and an incremental clustering algorithm that maintains a solution whose cost is at most a constant factors times that of optimal with a constant factor blowup in the number of clusters.

### The k-traveling repairman problem

- Mathematics, Computer ScienceSODA '03
- 2003

An 8.497α-approximation algorithm is given for this generalization of the metric traveling repairman problem, also known as the minimum latency problem, to multiple repairmen, where α denotes the best achievable approximation factor for the problem of finding the least cost rooted tree spanning i vertices (i-MST) problem.

### Approximation algorithms for some routing problems

- Business, Computer Science17th Annual Symposium on Foundations of Computer Science (sfcs 1976)
- 1976

Several polynomial time approximation algorithms for some NP-complete routing problems are presented, and the worst-case ratios of the cost of the obtained route to that of an optimal are determined.…

### Approximating min-sum k-clustering in metric spaces

- Computer Science, MathematicsSTOC '01
- 2001

The first polynomial time non-trivial approximation algorithm for the min-sum k-clustering problem in general metric spaces is given, based on embedding of metric spaces into hierarchically separated trees.

### An approximation algorithm for the generalized assignment problem

- BusinessMath. Program.
- 1993

The generalized assignment problem can be viewed as the following problem of scheduling parallel machines with costs. Each job is to be processed by exactly one machine; processing jobj on machinei…

### Improved approximation algorithms for a capacitated facility location problem

- BusinessSODA '99
- 1999

This work builds on recent results on approximation algorithms for facility location problems that rely on rounding the fractional solution to a linear programming relaxation by improving on the 5.69-approximation algorithm of [7].

### Approximation algorithms for facility location problems

- Business
- 2004

This thesis improves a long line of previous results and gives a 1.52-approximation algorithm for the uncapacitated facility location problem and considers several important generalizations of the unccapacitated problem, including: (1) Capacitate facility location: Each facility can serve only a certain amount of clients.

### Approximation algorithms for facility location problems (extended abstract)

- Computer ScienceSTOC '97
- 1997

A polynomial-time algorithm is given that finds a solution of cost within a factor of 3.16 of the optimal for the uncapacitated facility location, which is the first constant performance guarantee known for this problem.