Virtual Private Network Layout

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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

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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

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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

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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

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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

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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.

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The open VPN-problem

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

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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

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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)

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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

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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

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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

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Dual of the capacity finding problem
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Path-formulation of SPR

• SPR Find x(p) minimizing ?e2 Eceze

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Path-formulation of MPR

• MPR SPR with x(p) 0 i.o. x(p)2 0,1

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Dual of the Path-formulation of MPR

• Dual-MPR

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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

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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

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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

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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

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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

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The even cycle (2)

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

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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 ?

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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

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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)

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• Choose ?k?kn1 and ?i0 8 i? k,nk
• ) C(e?)n 8e2 E
• Choose
• Verify that ? ?ijn

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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)

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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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