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Virtual Private Network Layout

- A proof of the tree conjecture on a ring network
- Leen Stougie
- Eindhoven University of Technology (TUE)
- CWI, Amsterdam
- http//www.win.tue.nl/math/bs/spor/2004-15.pdf

Input to the VPN problem

- Undirected graph G(V,E)
- Subset of the vertices Wµ V (terminals)
- Communication bounds on the terminals b(i) for

all i2 W - Unit capacity costs on the edges c(e) for all e2

E

Communication bounds and scenarios

- b(i) is bound on total of incoming and outgoing

communication of node v (symmetric VPN) - A valid demand scenario is symmetric matrix

D(dik)ik2 W with dii0 satisfying - dik 0 8 i,k 2 W and k2 W dik b(i) 8

i2 W - D is the set of all valid scenarios

VPN Robust optimization

- Select for each pair i,k2 W a path for

communication - Reserve enough capacity on the edges E
- All demand in every valid communication scenario

D2D can be routed on the selected paths - The total cost of reserving capacity is minimum
- The paths are to be selected before seeing any

communication scenario

Routing variations of VPN

- SPR (Single path routing)
- For each pair i,k2 W exactly one path Pikµ E
- TTR (Terminal tree routing)
- SPR with for each i2 W, k2 WPik is a tree in G
- TR (Tree routing)
- SPR with i,k2 W Pik is a tree in G
- MPR (Multi-path routing)
- For each pair i,k2 W for each path P between i

and k, specify fraction of communication using P

Relation between the variations

- Lemma
- OPT(MPR) OPT(SPR) OPT(TTR) OPT(TR)
- Proof
- SPR is the MPR problem with the extra restriction

that all fractions must be 0 or 1. - The other inequalities are similarly trivial.

The open VPN-problem

- Conjecture 1
- SPR 2 P (polynomially solvable)
- Conjecture 2
- OPT(SPR)OPT(TR)
- Conjecture 3
- OPT(MPR)OPT(TR)

What do we know about VPN

- TR 2 P
- Kumar et al. 2002
- OPT(TR) OPT(TTR)
- Gupta et al. 2001
- OPT(TR) 2OPT(MPR)
- Gupta et al. 2001
- MPR 2 P
- Erlebach and Ruegg 2004, Altin et al. 2004,

Hurkens et al. 2004

The asymmetric VPN

- b(v) outgoing communication bound
- b-(v) incoming communication bound
- TR is NP-hard
- Gupta et al. 2001
- TR 2 P if v2 Wb-(v) v2 Wb(v)
- Italiano et al. 2002
- MPR 2 P
- Erlebach and Ruegg 2004, Altin et al. 2004,

Hurkens et al. 2004 - Constant Aprroximation ratios for SPR
- Gupta et al. 2001, Eisenbrandt et al. 2005

(randomized)

Conjecture 3 is true

- If G is a tree (trivial)
- If G is K4
- If G is a cycle !!!!
- If G is a 1-sum of graphs for which Conjecture 3

is true

Path-formulation of VPN

- Pik set of paths in G between i and k
- P set of all paths in G
- For each path p in G we define xp
- For all i and k 2 W, p2 Pikxp1
- SPR xp2 0,1 8p2 P
- MPR 0 xp 1 8p2 P

The capacity problem

- Given selected paths given values for x(p)
- Problem find capacities on edges z(e) 8 e2 E
- ep1 if e2P and 0 otherwise

Dual of the capacity finding problem

Path-formulation of SPR

- SPR Find x(p) minimizing e2 Eceze

Path-formulation of MPR

- MPR SPR with x(p) 0 i.o. x(p)2 0,1

Dual of the Path-formulation of MPR

- Dual-MPR

MPR and TR

- OPT(MPR) OPT(TR)
- Weak duality any feasible (, ) has

ik OPT(MPR) - Conjecture 3 OPT(MPR)OPT(TR)
- Conjecture 3 OPT(TR)Optimal solution value of

the dual of MPR

Optimal solution of TR (1)

- Notation b(U)v2 U b(v)
- Take tree T
- Each e2 T is cut in T splitting V in L(e) and

R(e) - Direct e to minimum of b(L(e)) and b(R(e))
- There is a unique vertex r with indegree 0, root
- Cost of T eminb(Le),b(Re) c(e)
- The minimum cost tree with r as the root is the

shortest path tree from r in G w.r.t. length

function c - OPT(TR) can be found in polynomial time

Optimal solution of TR (2)

- Let dG(u,v) the distance between u and v in G

w.r.t. length function c - The cost of optimal tree T is given by
- v b(v) dT(r,v)
- for some root vertex r.
- Moreover, it is bounded from below by
- v b(v) dG(r,v).
- Clearly it is bounded from above by
- v b(v) dT(u,v) forall u2V
- Compute shortest path tree rooted at u for all

u2V and select - the one with minumum cost solves OPT(TR) in

polynomial time

Conjecture 3 true for the cycle

- Lemma If Conjecture 3 is true for any cycle

with - - WV
- b(v)1 8 v2 V
- V is even
- Then Conjecture 3 is true for any cycle
- Theorem Conjecture 3 is true for any even cycle

with the above three properties

The even cycle (1)

- Vertices 0,1,2,...,2n-1
- Edges e1,e2,...,e2n
- Cost of tree by deleting edge ek
- (using eminb(Le),b(Re) c(e))
- We show there exist a dual solution with value

equal to - minek

The even cycle (2)

- MPR-dual restricitions for even cycle with b(v)1
- Only two possible paths between each pair of

vertices

The even cycle (3)The Tool Lemma

- The Tool Lemma
- - Let G(V,E) even circuit
- - b 1.
- - F µ E, F
- Then there exist E! R, not equal 0, and K

such that - support()µF
- 8 f2 F KC(f)mine2 E C(e )
- There is a dual solution (, ) with value K for

the MPR-dual problem with cost function

The even cycle (4)Part of Proof of Tool Lemma

- Proof By induction on F
- F1 (easy) Fek
- Take k1 and i0 8i k
- Clearly, mine2 EC(e )C(ek )0
- A feasible dual solution with value 0 is eih0,

ih0 8e2 E 8i,h 2 V

The even cycle (5)Part of Proof of Tool Lemma

- Proof (continued) Fgt1
- Case (i) There is a k such that ek2 F and its

opposite edge ekn2 F - (in figure read eka and eknb)

- Choose kkn1 and i0 8 i k,nk
- ) C(e)n 8e2 E
- Choose
- Verify that ijn

The even cycle (6)

- Theorem Let G(V,E) be an even circuit, c E!R

and b(v)1 8v2 V. Then the cost of an optimal

tree solution equals the value of an optimal dual

solution. - Proof An inductive primal-dual argument using

the Tool Lemma. - (By request on the blackboard)

Postlude

- OPT(MPR)OPT(TR) for any graph
- SPR polynomially solvable for any graph
- Proof for the cycle is complicated!
- Is there an easier proof for the cycle
- The crucial insight
- Complexity of the non-robust MPR-problem is also

open!

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