1. Approximation Algorithms for Stochastic Combinatorial Optimization Part I: Multistage problems Anupam Gupta Carnegie Mellon University

2. stochastic optimization Question: How to model uncertainty in the inputs?  data may not yet be available  obtaining exact data is difficult/expensive/time-consuming Goal: make (near)-optimal decisions given some predictions (probability distribution on potential inputs). Studied since the 1950s, and for good reason: many practical applications…

3. Approximation Algorithms Recent development of approximation algorithms for NP-hard stochastic optimization problems. Several different models, several different techniques. I will give an overview of some of the models/results/ideas in the two talks (today and on Thursday).

4. two representative formulations for stochastic Steiner tree

5. the Steiner tree problem Input: a metric space a root vertex r a subset R of terminals Output: a tree T connecting R to r of minimum length/cost. Facts: NP-hard and APX-hard MST is a 2-approximation cost(MST(R [ r)) ≤ 2 OPT(R) [Byrka et al. STOC ’10] give a 1.39-approximation

6. model I : “two-stage” Steiner tree The Model: Instead of one set R, we are given probability distribution ¼ over subsets of nodes. E.g., each node v independently belongs to R with probability p v Or, may be explicitly defined over a small set of “scenarios” p A = 0.6p B = 0.25p C = 0.15

7. model I : “two-stage” Steiner tree Stage I (“Monday”) Pick some set of edges E M at cost M (e) for each edge e Stage II (“Tuesday”) Random set R is drawn from ¼ Pick some edges E T,R so that E M [ E T,R connects R to root but cost changes to cost T,R (e) Objective Function: cost M (E M ) + E ¼ [ cost T,R (E T,R ) ] p A = 0.6p B = 0.25p C = 0.15

8. approximation algorithm Objective Function: cost M (E M ) + E ¼ [ cost T,R (E T,R ) ] Optimum: Sets E M * and E T,R * which achieve expected cost Z* A c-approximation: Find sets E M and E T,R that achieve expected cost c.Z* for some small factor c. p A = 0.6p B = 0.25p C = 0.15

9. model II : average-case online The Model: Instead of being given set R up-front, the nodes in R arrive one-by-one. Node must be connected to the root as soon as it arrives. However, the requests not drawn by an adversary, but randomly from some known stochastic process. E.g., each node v independently chosen with probability p v

10. model II : average-case online Measure of Goodness: Usual measure is competitive ratio So we consider or E ¾ [ cost of algorithm A on ¾ ] E ¾ [ OPT(set ¾ ) ] E [ cost of algorithm A on ¾ ] OPT(set ¾ ) max ¾ cost of algorithm A on ¾ OPT(set ¾ ) E¾E¾

11. a tale of two models… (At least) two ways of modeling Steiner tree problems become harder in the two-stage model (if no inflation, we can just solve the problem on Tuesday.) problems become easier in the online stochastic model (we know that the “adversary” is just a stochastic process.) Let us consider these two models in some detail now…

12. roadmap the two-stage model  eg: stochastic vertex cover (using LPs)  how to solve two-stage stochastic LPs  the sample average approximation method the stochastic online model  eg: stochastic Steiner tree (using sampling)

13. models with recourse The problem instance is revealed in “stages”  initially we perform some anticipatory actions  at each stage, more information released  we may take some more recourse actions at this point Initially, given “guesses” about final problem instance (i.e., given probability distribution ¼ over problem instances) Want to minimize: Cost(Initial actions) + E ¼ [ cost of recourse actions ]

14. representations of ¼  “Explicit scenarios” model  Complete listing of the sample space  “Black box” access to probability distribution  generates an independent random sample from ¼  Also, independent decisions  Each vertex v appears with probability p v indep. of others.

15. example I : vertex cover vertex cover = set of vertices that hit all edges. Finding a minimum cost vertex cover is NP-hard. 2-approx:several algorithms easy one: solve the linear program relaxation and round

16. Boolean variable x(v) = 1 iff vertex v chosen in the vertex cover minimize  v c(v) x(v) subject to x(v) + x(w) ≥ 1for each edge (v,w) in edge set E and x’s are in {0,1} integer-program formulation

17. stochastic vertex cover Explicit scenario model: M scenarios explicitly listed. Edge set E k appears with prob. p k Vertex costs c(v) on Monday, c k (v) on Tuesday if scenario k appears. Pick V 0 on Monday, V k on Tuesday such that (V 0 [ V k ) covers E k. Minimize c(V 0 ) + E k [ c k (V k ) ] p 1 = 0.1p 2 = 0.6p 3 = 0.3

18. Boolean variable x(v) = 1 iff v chosen on Monday, y k (v) = 1 iff v chosen on Tuesday if scenario k realized minimize  v c(v) x(v) +  k p k [  v c k (v) y k (v) ] subject to [ x(v) + y k (v) ] + [ x(w) + y k (w) ] ≥ 1for each k, edge (v,w) in E k and x’s, y’s are Boolean integer-program formulation

19. minimize  v c(v) x(v) +  k p k [  v c k (v) y k (v) ] subject to [ x(v) + y k (v) ] + [ x(w) + y k (w) ] ≥ 1for each k, edge (v,w) in E k Now choose V 0 = { v | x(v) ≥ ¼ }, and V k = { v | y k (v) ≥ ¼ } Note: if we have explicit multi-stage solution with k stages, gives 2k approximation We are increasing variables by factor of 4  we get a 4-approximation linear-program relaxation

20. solving the LP and rounding  This idea useful for many stochastic problems  Set cover, Facility location, some cut problems  Tricky when the sample space is exponentially large  exponential number of variables and constraints  natural (non-trivial) approaches have run-times depending on the variance of the problem…  Shmoys and Swamy approach:  consider this doubly-exponential vertex cover LP in black-box model  can approximate it arbitrarily well, smaller run-times.  solution has exponential size, but we need only polynomially-large parts of it at a time.

21. roadmap the two-stage model  eg: stochastic vertex cover (using LPs)  how to solve two-stage stochastic LPs  the sample average approximation method the stochastic online model  eg: stochastic Steiner tree (using sampling)

22. Approximation Algorithms for Stochastic Problems Part II: Online (Stochastic) Problems Anupam Gupta Carnegie Mellon University

23. the Steiner tree problem Input: a metric space a root vertex r a subset R of terminals Output: a tree T connecting R to r of minimum length/cost. Facts: NP-hard and APX-hard MST is a 2-approximation cost(MST(R [ r)) ≤ 2 OPT(R) [Byrka et al. STOC ’10] give a 1.39-approximation

24. model II : average-case online Requests: generated by a given stochastic process, have to be connected to the root. Measure of Goodness: Want to minimize E ¾ [ cost of algorithm A on ¾ ] E ¾ [ OPT(set ¾ ) ]

25. recap: the standard online model

26. The Greedy Algorithm Given: metric space, root vertex r, requests come online

27. measure of goodness Competitive ratio of algorithm A: maxCost(algorithm A on sequence ¾ ) ¾ Cost(optimum Steiner tree on ¾ )

28. The Greedy Algorithm [Imase Waxman ’91] The greedy algorithm is O(log k) competitive for sequences of length k.

29. The Greedy Algorithm [Imase Waxman ’91] The greedy algorithm is O(log k) competitive for sequences of length k. And this is tight.

30. The Greedy Algorithm [Imase Waxman ’91] The greedy algorithm is O(log k) competitive for sequences of length k. And this is tight.

31. stochastic online model

32. random arrivals Suppose demands are nodes in V drawn uniformly at random, independently of previous demands. uniformity: not important could have (given) probability ¼ : V  [0,1] independence: important, lower bounds otherwise Measure of goodness: E ¾ [ cost of algorithm A on ¾ ] E ¾ [ OPT(set ¾ ) ] Assume: know the length k of the sequence

33. greed is (still) bad [Imase Waxman ’91] The greedy algorithm is £ (log k) competitive for sequences of length k. Tight example holds also for (uniformly) random demands.

34. however, we can prove… For stochastic online Steiner tree, “augmented greedy” with known length k achieves E ¾ [ cost of algorithm A on ¾ ] E ¾ [ OPT(set ¾ ) ] ≤ 4

35. Augmented greedy 1.Sample k vertices A = {a 1, a 2, …, a k } independently. 2.Build an MST T 0 on these vertices A [ root r. 3.When actual demand points x t (for 1 · t · k) arrives, greedily connect x t to the tree T t-1

36. Augmented greedy

37. 1.Sample k vertices A = {a 1, a 2, …, a k } independently. 2.Build an MST T 0 on these vertices A [ root r. 3.When actual demand points x t (for 1 · t · k) arrives, greedily connect x t to the tree T t-1

38. Proof for augmented greedy Let S = {x 1, x 2, …, x k } Claim 1: E[ cost(T 0 ) ] ≤ 2 £ E[ OPT(S) ] Claim 2: E[ cost of k augmentations in Step 3 ] ≤ E[ cost(T 0 ) ] 1.Sample k vertices A = {a 1, a 2, …, a k } independently. 2.Build an MST T 0 on these vertices A [ root r. 3.When actual demand points x t (for 1 · t · k) arrives, greedily connect x t to the tree T t-1 1.Sample k vertices A = {a 1, a 2, …, a k } independently. 2.Build an MST T 0 on these vertices A [ root r. 3.When actual demand points x t (for 1 · t · k) arrives, greedily connect x t to the tree T t-1  Ratio of expectations ≤ 4

39. Proof for augmented greedy Let S = {x 1, x 2, …, x k } Claim 2: E S,A [ augmentation cost ] ≤ E A [ MST(A [ r) ] Claim 2a: E S,A [   x 2 S d(x, A [ r) ] ≤ E A [ MST(A [ r) ] Claim 2b: E x,A [ d(x, A [ r) ] ≤ (1/k) E A [ MST(A [ r) ] 1.Sample k vertices A = {a 1, a 2, …, a k } independently. 2.Build an MST T 0 on these vertices A [ root r. 3.When actual demand points x t (for 1 · t · k) arrives, greedily connect x t to the tree T t-1 1.Sample k vertices A = {a 1, a 2, …, a k } independently. 2.Build an MST T 0 on these vertices A [ root r. 3.When actual demand points x t (for 1 · t · k) arrives, greedily connect x t to the tree T t-1

40. Proof for augmented greedy Claim 2b: E x,A [ d(x, A [ r) ] ≤ (1/k) E A [ MST(A [ r) ] Consider the MST(A [ x [ r)

41. Proof for augmented greedy Claim 2b: E x,A [ d(x, A [ r) ] ≤ (1/k) E A [ MST(A [ r) ] = E[ distance from one random point to (k random points [ r) ] ≥ (1/k) * k * E y, A-y [ distance(y, (A-y) [ r) ] ≥ E[ distance from one random point to (k-1 random points [ r) ] ≥ E[ distance from one random point to (k random points [ r) ]

42. Proof for augmented greedy Let S = {x 1, x 2, …, x k } Claim 1: E[ cost(T 0 ) ] ≤ 2 £ E[ OPT(S) ] Claim 2: E[ cost of k augmentations in Step 3 ] ≤ E[ cost(T 0 ) ] 1.Sample k vertices A = {a 1, a 2, …, a k } independently. 2.Build an MST T 0 on these vertices A [ root r. 3.When actual demand points x t (for 1 · t · k) arrives, greedily connect x t to the tree T t-1 1.Sample k vertices A = {a 1, a 2, …, a k } independently. 2.Build an MST T 0 on these vertices A [ root r. 3.When actual demand points x t (for 1 · t · k) arrives, greedily connect x t to the tree T t-1  Ratio of expectations ≤ 4