Robust Network Design with Exponential Scenarios

1. Robust Network Design with Exponential Scenarios By: Rohit Khandekar Guy Kortsarz Vahab Mirrokni Mohammad Salavatipour

2. Outline • Robust vs. Stochastic Optimization • Robust Two-stage Network Design Problems • Related Work: Stochastic vs. Robust • Our Results • Algorithm for Robust Steiner Tree • Conclusion

3. Optimization Against Uncertainty • Stochastic Optimization. • Optimize the expected cost given the probability distribution on the scenarios • Playing against a randomized uncertainty. • Robust Optimization. • Optimize for the worst case scenarios. • Playing against an adversary.

4. Two-stage Optimization • Construct a partial solution in stage one. • Wait for a real scenario to show up. • Complete the solution and pay more if you buy things in the second step. • Goal: Decide what to buy in advance and what to do in step two for each scenario.

5. Exponential vs. Polynomial Scenarios. • Polynomial Scenarios: explicit Scenarios. • Stochastic Optimization: The probability of each scenario is explicitly given. • Robust Optimization: All scenarios are specified. • Exponential Scenarios: implicit Scenarios • Stochastic Optimization: Implicit probability distribution is given, e.g., each client t will show up with probability pt. • Robust Optimization: Family of scenarios are defined implicitly, e.g., an upper bound on the number of clients is given, i.e., at most k clients will show up.

6. 2-stage Robust Steiner Tree • Given: • Edge-weighted graph G(V, E)each edge e with cost ce. • Nodes of G correspond to clients. • Scenarios T1, T2, …, Tp which are the subsets of nodes one of which we will need to connect at the end. • An inflation factor q for the increased cost in the second step. • Output: Buy a subset E1of edges of G in advance. • Adversary will choose (the worst) scenario Ti and we need to buy another subset of edges E’ at a larger cost qce for each edge e such that E1 U E’ connects all nodes in Ti • Goal: minimize total cost = c(E1) + q c(E’)

7. 2-stage Robust Steiner Tree (Exponential Scenarios) • Given: • edge-weighted graph G(V, E);a set of terminals T • A parameter k:k terminals (clients) will show up. • Inflation factor q: edge costs increase in second step. • Output: Buy a subset E1in stage one; k terminals are given in stage 2 and we need to buy another subset of edges E’ at cost qce for each edge e s.t.E1 U E’ connects all the k terminals. • Goal: minimize total cost = c(E1) + q c(E’)

8. Other Robust 2-stage Problems • Robust Network Design: • Robust Steiner Forest: at most kpairs show up. • Robust Facility Location. • At most k clients will show up. We buy a set of facilities in advance. • Adversary chooses the worst k clients for us to cover. • We should open new facilities at larger cost & minimize the total opening cost + connection costs. • Other Robust Covering Problems: • Robust Set Cover. • Robust Vertex Cover. • Robust two-stage Min-cut.

9. Robust Min Cut Problem • Robust 2-stage Min-cut: • Given: • An edge-weighted graph G(V, E)each edge e with cost c_e. • Pairs of nodes of G correspond to clients. • An inflation factor q for the increased cost in the second step. • Output: Buy a subset E_1of edges of G in advance. • Adversary will choose (the worst) pair (s_i, t_i)and we need to buy another subset of edges E’ at a larger cost qc_e for each edge e such that E_1 U E’disconnects s_iandt_i. • Goal: Choose a subset E_1 with the minimum total cost (Total Cost = cost of E_1 + q times cost of E’).

10. Outline • Robust vs. Stochastic Optimization • Robust Two-stage Network Design Problems • Related Work: Stochastic vs. Robust • Our Results • Algorithm for Robust Steiner Tree • Conclusion

11. Related Work • Stochastic Two-stage Optimization: • Dye, Stougie, and Tomasgard: Two-stage matching problems • Considered by Immorlica, Karger, Minkoff, and Mirrokni [IKMM] (SODA03) and Ravi and Sinha (IPCO03) (Two-stage covering problems). • IKMM considered the exponential scenarios: each client shows up with probability p. • Improved by Gupta, Pal, Ravi, and Sinha (STOC03) using Boosted Sampling: constant-factor approximation for Steiner tree. Also later considered the black box model. • Swamy and Shmoys (FOCS04): O(log n)-approximation algorithm for two-stage stochastic set cover via solving an exponential linear program. • Multi-stage Optimization: • Swamy and Shmoys (FOCS05): sample average approximation.

12. Related Work Robust Two-stage Optimization: • Initiated by Ben-Tal, Gorashko, Guslitzer, and Nemirovski. • Polynomial Number of Scenarios: • Introduced by Dhamhere, Goyal, Ravi, and Singh (FOCS05): Facility Location & Steiner Tree • Constant-factor approximation algorithms • Improved by Golovin, Goyal, and Ravi (STACS06): Min Cut. • Constant-factor approximation algorithm for Min-cut (Is it NP-Hard?)

13. Exponential Scenarios: Known Results • Feige, Jain, Mahdian, Mirrokni (IPCO 07) • LP-based Approximation Algorithms for Robust Covering: Exponential Size LP. • General Framework for covering problems: Online Competitive Algorithms  Two-stage Robust Approximation Algorithms. • constant-factor approximation for robust vertex. • O(log m log n)-approximation for robust set cover • O(log m)-approximation for robust metric facility location: Constant-factor approximation for facility location? • LP-based algorithm does not work for Robust Network Design: Robust Steiner Tree?, or Robust Steiner Forest?

14. Our Results • Constant-factor for robust Steiner tree and robust Facility Location. • Thm. 5.5-approx for two-stage robust Steiner Tree • Combinatorial Algorithm • Thm.10-approx for robust facility location. • Combinatorial Algorithm • Thm.3-approx for robust Steiner Forest on Trees. • At most k pairs of nodes show up to be connected. • LP-based Algorithm

15. Our Results • Hardness Results • Thm. Better than O(log1/2- n)-approximation factor for two-stage robust Steiner Forest with two inflation factors is hard, even if only each scenario is only one pair. • implies quasi-polynomial-time algorithms for NP. • Thm.Robust two-stage Min-cut is APX-hard. • Even with uniform inflation factor. • Even with one source and three sinks. • NP-hardness was posed as an open problem (by Golovin, Goyal, Ravi).

16. Robust Steiner Tree Algorithm • Let OPT = OPT1 + q OPT2. • OPT1: Optimum in the first Stage. • OPT2: Optimum in the second Stage. • Algorithm A: • A: first stage: • 1) Guess OPT2 [Binary search to find it]. • 2) Find-Centers: Find centers c1, c2, …, ckand assign nodes to these centers s.t. • Distance of each two centers is at least r OPT2 / k. • Each node is close to its assigned center is at most r OPT2 / k. • 3) Buy an approx optimal Steiner tree on c1, c2, …, ck. • B) Second Stage: Buy the shortest path from each client to the closest terminal.

17. Algorithm (Cont’) • Find-Centers: Find centers c1, c2, …, ck and assign nodes to these centers s.t. • Distance of each two centers is at least r OPT2 / k. • Each node is close to its center, dist is at most r OPT2 / k. • Find-Centers Algorithm: • 1) Set of centers U= V(G) and C=empty; and i = 0. • 2) i = i+1. • 3) Select an arbitrary node c1 in U and add it to C. • 4) Remove all nodes in distance· rOPT2 / kfrom U. • 5) If U is nonempty, go back to step 2, otherwise terminate.

18. Analysis • Second Stage: k clients, each with cost q(r OPT2 / k), thus q OPT2 • First Stage: • Lemma: The cost is at most (r/r-4)OPT1 + OPT2. • Proof: By contradiction  Otherwise the optimum solution pays more than q OPT2 in the second stage. Prove this by constructing the right mapping between the optimum and our solution (Details in the paper). • Final Algorithm: • If q<3.51, Run a trivial algorithm (not buy anything in the first stage), otherwise, Run Algorithm A.

19. Conclusion • Constant-factor Approximation Algorithms for • 2-stage robust Steiner tree • 2-stage robust facility location • 2-stage robust Steiner forest (on trees) • Hardness Result • 2-stage robust min-cut is APX-hard. • 2-stage robust Steiner forest (with 2 inflation factors) in not approximable better than O(log1/2-n). • Open Problems: Robust Multiway-Cut, Robust Steiner Forest, Robust Buy-at-Bulk Network Design. • Please find a revised version of the paper at: • http://people.csail.mit.edu/mirrokni/ESA08.pdf