Virtual Private Network Layout

1. 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

2. 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

3. 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 dii=0 satisfying dik¸ 0 8 i,k 2 W and k2 Wdik· b(i) 8 i2 W • D is the set of all valid scenarios

4. 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

5. 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

6. 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.

7. The open VPN-problem • Conjecture 1: SPR 2 P (polynomially solvable) • Conjecture 2: OPT(SPR)=OPT(TR) • Conjecture 3: OPT(MPR)=OPT(TR)

8. 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

9. 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)

10. 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

11. 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 Pikxp=1 • SPR: xp2 {0,1} 8p2 P • MPR: 0· xp· 1 8p2 P

12. The capacity problem • Given selected paths: given values for x(p) • Problem: find capacities on edges z(e) 8 e2 E • ep=1 if e2P and 0 otherwise

13. Dual of the capacity finding problem

14. Path-formulation of SPR • SPR: Find x(p) minimizing e2 Eceze

15. Path-formulation of MPR • MPR: SPR with x(p)¸ 0 i.o. x(p)2 {0,1}

16. Dual of the Path-formulation of MPR Dual-MPR

17. 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

18. 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: emin{b(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

19. 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

20. Conjecture 3 true for the cycle Lemma: If Conjecture 3 is true for any cycle with: - W=V • b(v)=18 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

21. The even cycle (1) • Vertices 0,1,2,...,2n-1 • Edges e1,e2,...,e2n • Cost of tree by deleting edge ek: • (using emin{b(Le),b(Re)} c(e)) • We show there exist a dual solution with value equal to minek

22. The even cycle (2) • MPR-dual restricitions for even cycle with b(v)=1 • Only two possible paths between each pair of vertices

23. 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: K=C(f;)=mine2 E C(e; ) • There is a dual solution (, ) with value K for the MPR-dual problem with cost function 

24. The even cycle (4)Part of Proof of Tool Lemma • Proof:By induction on |F| • |F|=1 (easy): F={ek} • Take k=1 and i=0 8i k • Clearly, mine2 EC(e; )=C(ek; )=0 • A feasible dual solution with value 0 is eih=0, ih=0 8e2 E 8i,h 2 V

25. The even cycle (5)Part of Proof of Tool Lemma • Proof (continued): |F|>1 • Case (i): There is a k such that ek2 F and its opposite edge ek+n2 F • (in figure read ek=a and ek+n=b)

26. Choose k=k+n=1 and i=0 8 i k,n+k ) C(e;)=n 8e2 E Choose Verify that ij=n

27. 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)

28. 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!