Refael Hassin

1
Approximation algorithms for quickest spanning
tree problems

• Refael Hassin
• Tel-Aviv University
• and
• Asaf Levin
• The Technion
• The Hebrew University

2
Problems definitions

• G(V,E) undirected multigraph.
• root is a special vertex.
• An edge e has length l(e) and capacity c(e).
• Denote
• For a path P the transmission time of P is

3
Transmission time - motivation

• Sending a unit of information in a single link
  with capacity c(e) is composed of the following
• It takes 1/c(e) time unit until the last bit
  enters the link.
• It takes another l(e) time unit until the last
  bit gets to the destination.
• Sending a unit of information in a path P where
  the bottleneck link has capacity c(e) is done in
  rate 1/c(e) and therefore it takes t(P)

4
Problems definitions (cont)

• The quickest radius spanning tree problem is to
  minimize
• The quickest diameter spanning tree problem is to
  minimize

5
Related problem the quickest path problem

• Given u,v the goal is to find a path P between u
  and v such that t(P) is minimized (introduced by
  Moore 1976, and discussed by other as well).
• Have applications in transportation and
  communication networks.
• Well-known to have a Polynomial time algorithm.
• No previous study of our pair of problems.

6
Our results

• The quickest radius spanning tree problem
• A 2-approximation algorithm
• For any egt0, unless PNP there is no 2-e
  approximation algorithm.
• The quickest diameter spanning tree problem
• A 3/2-approximation algorithm
• For any egt0, unless PNP there is no (3/2)-e
  approximation algorithm.
• Our results are best possible !!!

7
The union of the quickest paths is not always a
tree
root
2
3
1
l(e)1,r(e)2
l(e)0, r(e)7/2
8
Quickest path from root to 1
root
2
3
1
9
Quickest path from root to 2
root
2
3
1
10
Quickest path from root to 3
root
2
3
1
11
The union of the quickest paths
root
2
3
1
12
Conclusion

• The restriction in our problems to a spanning
  tree makes some difficulties.
• The motivation for this restriction is from
  routing protocols that use a tree architecture.

13
A 2-approximation for the quickest radius
spanning tree

• For every u compute the quickest path from root
  to u, and denote it by QP(root,u).
• Return the shortest path tree T in G from root
  according to the length function l

14
Approximation ratios proof
15
Sketch of hardness proof for the quickest radius
spanning tree problem

• The proof is composed of three steps
• A simple reduction from SAT that shows that
  unless PNP, there is no approximation algorithm
  with approximation ratio better than 3/2.
• We prove a lower bound of 2 on the approximation
  ratio of any algorithm that returns a spanning
  tree whose edges belong to the union of the
  quickest paths (of all vertices).
• Combining the ideas of the previous steps we get
  a complicated reduction from SAT that shows that
  unless PNP, there is no approximation algorithm
  with approximation ratio better than 2.

16
Simple reduction gadget sketch
l0, r1
root
leaf
l0.5, r0
17
A 3/2-approximation for the quickest diameter
spanning tree

• For every do
• Compute a minimum diameter spanning tree Tr in
  the graph that contains the edges with capacity
  at least 1/r.
• Return the minimum cost solution

18
Approximation ratio analysis

• Lemma Let then

19
Analysis (cont.)