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CS880: Approximations Algorithms Scribe: Dave Andrzejewski Lecturer: Shuchi Chawla Topic: Embedding metrics into trees Date: 3/30/
This lecture introduces the idea of embedding a metric into a tree, and applies this technique to the development of an approximation algorithm for the Multicommodity ”Buy at Bulk” Network Design problem.
22.1 Multicommodity Buy-At-Bulk Network Design
22.1.1 Problem formulation
GIVEN:
- a graph G = (V, E)
- edge lengths `e
- pairs of demand vertices (si, ti)
- quantities qi to be sent si → ti
- a concave cost function f (ce) for ”buying” capacity ce on edge e
DO: find
- a set of paths Pi from si to ti such that
p∈Pi p^ ≥^ qi^ for each^ i
- a set of edge capacity purchases ce such that
{p|e∈p} p^ ≤^ ce
such that the total cost
e f^ (ce)`e^ is minimized.
The cost of purchasing capcity ce on edge e is defined as f (ce)`e. This means that the cost of purchasing edge capacity is linearly related to the length of that edge, which will be important for our analysis. Also note that the cost function f is concave, and shared by all edges. This makes our formulation the ”uniform” case. If each edge were allowed to have a different cost function fe, it would be the non-uniform case, which is much harder, and was not known to have any sub-polynomial approximation until recently, when a poly-log approximation was discovered [1].
22.1.2 Algorithm design
Our first observation is that this problem would be greatly simplified for the special case where G is a tree, because each si → ti path would be unique.
Furthermore, we recall that the total cost is linear in the edge lengths `e. This means that if we can find a low-distortion embedding from our graph G to some tree T , it will be relatively simple to analyze the impact of the distortion on our cost function.
22.1.3 Tree embeddings
To analyze potential embeddings, we must first ask which graph structures would be most difficult to embed into a tree. A natural first thought is to consider a complete graph. However, note that we could simply place any single node as the hub, and have all other nodes only be connected to the root as spokes. In the simplified case of a a graph with uniform edge costs, this clearly achieves an expansion factor of ρ = 2 (Figure 22.1.1).
The actual worst-case would be a graph which is simply a single large cycle of all n nodes (n-cycle). In this case, we can create a tree by simply removing any single edge. However, the distance between the 2 endpoints of that edge has now expanded by a factor of ρ = Ω(n) (Figure 22.1.1). It can be shown, in fact, that no embedding of the n-cycle into trees has distortion o(n).
T
G T
G
Figure 22.1.1: Embedding example graphs into trees.
- Solve uniform buy-at-bulk network design on T
- Foreach e = (u, v) ∈ T , find shortest (u, v) path p in G and install capacity cTe on all edges in p
- Map all pTi to their corresponding paths in G
This procedure clearly recovers a feasible solution on G, since all paths and capacities in the valid T solution are feasibly mapped to G.
What is the cost of our converted solution?
Claim 22.1.4 E[costT ] ≤ βOP T
Proof: Translate OP TG to some solution in T by mapping each edge (u, v) in G to the unique path between u and v in T. Then
E[OP TT ] ≤ E[costT ] ≤
(u,v)∈Eg
f (cOP Te )dT (u, v) (22.1.1)
(u,v)∈Eg
f (cOP Te )βdG(u, v) (22.1.2)
= β
(u,v)∈Eg
f (cOP Te )`e (22.1.3)
= βOP T (22.1.4)
Claim 22.1.5 Given a solution of cost X in T , our solution in G will have cost ≤ X.
Proof: costG =
(u,v)∈ET
f (cTe )dG(u, v) ≤
(u,v)∈ET
f (cTe )dT (u, v) = X
These claims prove the theorem.
Note that this analysis relied on the fact that our objective function is linear in lengths `e = dG(u, v).
These strategies were originally developed by Bartal, who derived O(log^2 n) and O(log n log log n) probabilistic embeddings of graphs into distributions over trees [4] [5]. Fakcharoenphol, Rao, and Talwar later improved these results to a O(log n) probabilistic embedding, which was also shown to be tight [3].
22.1.6 O(log n)-probabilistic embedding
How can be get an O(log n)-probabilistic embedding of general graphs into trees? The basic idea is to do a hierarchical clustering on all vertices in G (Figure 22.1.2).
Figure 22.1.2: Hierarchical clustering by partitioning.
Under this scheme, all vertices of the original graph G are leaf nodes in our tree T. Each cluster in our hierarchical clustering then corresponds to a sub-tree in T. That is, all interior nodes of T are artifacts of our clustering scheme and were not originally present in G.
Starting from a graph with diameter ∆, we want our probabilistic embedding to have the property that dT (x, y) ≥ dG(x, y)∀x, y.
We can achieve this by building our clusters such that the diameter of the initial root cluster is ∆, the diameter of each child cluster is ∆/2, and so on.
PSfrag replacements
Figure 22.1.3: Tree representation of our hierarchical clustering.
Next, we need a partitioning scheme in order to build each level of our hierarchical clustering.
Definition 22.1.6 A β low-diameter low-distortion partitioning with parameter δ is a partition of V into {V 1 , V 2 , ..., Vk } such that
- diam(Vi) ≤ δ ∀i
References
[1] C. Chekuri, M. T. Hajiaghayi, G. Kortsarz, M. R. Salavatipour. Approximation Algorithms for Non-Uniform Buy-at-Bulk Network Design. FOCS 2006, 677-686.
[2] Richard Durbin, Sean Eddy, Anders Krogh, and Graeme Mitchison. Biological Sequence Anal- ysis: Probabilistic Models of Proteins and Nucleic Acids. Cambridge University Press, 1998.
[3] Jittat Fakcharoenphol, Satish Rao, Kunal Talwar. A Tight Bound on Approximating Arbitrary Metrics by Tree Metrics. STOC 2003, 448-455.
[4] Yair Bartal. Probabilistic Approximations of Metric Spaces and Its Algorithmic Applications. FOCS 1996, 184-193.
[5] Yair Bartal. On approximating arbitrary metrices by tree metrics. STOC 1998, 161-168.
[6] V. Vazirani. Approximation Algorithms. Springer, 2001.
