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Approximation Algorithms for Stochastic Optimization

Approximation Algorithms for Stochastic Optimization. Chaitanya Swamy Caltech and U. Waterloo Joint work with David Shmoys Cornell University. Stochastic Optimization. Way of modeling uncertainty .

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Approximation Algorithms for Stochastic Optimization

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  1. Approximation Algorithms for Stochastic Optimization Chaitanya Swamy Caltech and U. Waterloo Joint work with David Shmoys Cornell University

  2. Stochastic Optimization • Way of modeling uncertainty. • Exact data is unavailable or expensive – data is uncertain, specified by a probability distribution. Want to make the best decisions given this uncertainty in the data. • Applications in logistics, transportation models, financial instruments, network design, production planning, … • Dates back to 1950’s and the work of Dantzig.

  3. Stochastic Recourse Models Given : Probability distribution over inputs. Stage I : Make some advance decisions – plan ahead or hedge against uncertainty. Uncertainty evolves through various stages. Learn new information in each stage. Can take recourse actions in each stage – canaugment earlier solution paying a recourse cost. Choose initial (stage I) decisions to minimize (stage I cost) + (expected recourse cost).

  4. 2-stage problem º 2 decision points k-stage problem º k decision points stage I 0.2 0.4 0.3 stage II 0.5 scenarios in stage k stage I 0.2 0.02 0.3 0.1 stage IIscenarios

  5. 2-stage problem º 2 decision points k-stage problem º k decision points stage I stage I 0.2 0.2 0.4 0.3 stage II 0.02 0.3 0.1 0.5 stage IIscenarios scenarios in stage k Choose stage I decisions to minimize expected total cost = (stage I cost) + Eall scenarios [cost of stages 2 … k].

  6. Stage I: Open some facilities in advance; pay cost fifor facility i. Stage I cost = ∑(i opened) fi. stage I facility 2-Stage Stochastic Facility Location Distribution over clients gives the set of clients to serve. facility client set D

  7. Actual scenarioA = { clients to serve}, materializes. Stage II: Can open more facilities to serve clients in A; pay cost fiA to open facility i. Assign clients in A to facilities. Stage II cost = ∑ fiA + (cost of serving clients in A). i opened in scenario A 2-Stage Stochastic Facility Location Distribution over clients gives the set of clients to serve. Stage I: Open some facilities in advance; pay cost fifor facility i. Stage I cost = ∑(i opened) fi. facility stage I facility

  8. Want to decide which facilities to open in stage I. Goal: Minimize Total Cost = (stage I cost)+ EA ÍD[stage II cost for A]. • How is the probability distribution specified? • A short (polynomial) list of possible scenarios • Independent probabilities that each client exists • A black box that can be sampled.

  9. Approximation Algorithm • Hard to solve the problem exactly. • Even special cases are #P-hard. • Settle for approximate solutions. Give polytime algorithm that always finds near-optimal solutions. • A is a a-approximation algorithm if, • A runs in polynomial time. • A(I) ≤ a.OPT(I) on all instances I, • ais called the approximation ratio of A.

  10. Overview of Previous Work • polynomial scenario model: Dye, Stougie & Tomasgard; • Ravi & Sinha; Immorlica, Karger, Minkoff & Mirrokni. • Immorlica et al.: also consider independent activation model proportional costs: (stage II cost) = l(stage I cost), • e.g., fiA = l.fifor each facility i,in each scenario A. • Gupta, Pál, Ravi & Sinha (GPRS04): black-box model but also with proportional costs. • Shmoys, S (SS04): black-box model with arbitrary costs. • approximation scheme for 2-stage LPs + rounding procedure “reduces” stochastic problems to their deterministic versions. • for some problems improve upon previous results.

  11. Boosted Sampling (GPRS04) • Proportional costs: (stage II cost) = l(stage I cost) • Note: l is same as s in previous talk. • Sample l times from distribution • Use “suitable” algorithm to solve deterministic instance consisting of sampled scenarios (e.g., all sampled clients) – determines stage I decisions • Analysis relies on the existence of cost-shares that can be used to share the stage I cost among sampled scenarios.

  12. Shmoys, S ’04 vs. Boosted sampling Both work in the black-box model: arbitrary distributions. • Can handle arbitrary costs in the two stages. • LP rounding: give an algorithm to solve the stochastic LP. • Need many more samples to solve stochastic LP. Need proportional costs: (stage II cost) = l(stage I cost) l can depend on scenario. Primal-dual approach: cost-shares obtained by exploiting structure via primal-dual schema. Need only l samples.

  13. Stochastic Set Cover (SSC) Universe U = {e1, …, en }, subsets S1, S2, …, SmÍ U, set S has weight wS. Deterministic problem: Pick a minimum weight collection of sets that covers each element. • Stochastic version: Set of elements to be covered is given by a probability distribution. • choose some sets initially paying wS for set S • subset A Í U to be covered is revealed • can pick additional sets paying wSAfor set S. • Minimize(w-cost of sets picked in stage I)+ • EA ÍU [wA-cost of new sets picked for scenario A].

  14. A Linear Program for SSC For simplicity, consider wSA = WS for every scenario A. wS : stage I weight of set S pA : probability of scenario A Í U. xS : indicates if set S is picked in stage I. yA,S: indicates if set S is picked in scenario A. Minimize ∑SwSxS +∑AÍU pA ∑S WSyA,S subject to, ∑S:eÎS xS + ∑S:eÎS yA,S ≥ 1for each A Í U, eÎA xS, yA,S ≥ 0 for each S, A. Exponential number of variables and exponential number of constraints.

  15. A Rounding Theorem Assume LP can be solved in polynomial time. Suppose for the deterministic problem, we have an a-approximation algorithm wrt. the LP relaxation, i.e., A such that A(I)≤a.(optimal LP solutionfor I) for every instance I. e.g., “the greedy algorithm” for set cover is a log n-approximation algorithm wrt. LP relaxation. Theorem: Can use such ana-approx. algorithm to get a 2a-approximation algorithm for stochastic set cover.

  16. Rounding the LP Assume LP can be solved in polynomial time. Suppose we have an a-approximation algorithm wrt. the LP relaxation for the deterministic problem. Let (x,y) : optimal solution with cost OPT. ∑S:eÎS xS + ∑S:eÎS yA,S ≥ 1for each A Í U, eÎA Þ for every element e, either ∑S:eÎS xS ≥ ½ OR in each scenario A : eÎA, ∑S:eÎS yA,S ≥ ½. Let E = {e : ∑S:eÎS xS ≥ ½}. So (2x) is afractional set coverfor the set E Þ can round to get aninteger set coverSfor E of cost ∑SÎSwS ≤ a(∑S 2wSxS) . Sis the first stage decision.

  17. A Consider any scenario A. Elements in A Ç E are covered. For every e Î A\E, it must be that ∑S:eÎS yA,S ≥ ½. So (2yA) is afractional set coverfor A\E Þ can round to get a set coverof W-cost≤a(∑S 2WSyA,S) . Rounding (contd.) Sets Set in S Elements Element in E Using thisto augmentSin scenario A,expected cost ≤ ∑SÎSwS+ 2a.∑ AÍU pA (∑S WSyA,S) ≤ 2a.OPT.

  18. Rounding (contd.) • An a-approx. algorithm for deterministic problem gives a 2a-approximation guarantee for stochastic problem. • In the polynomial-scenario model, gives simple polytime approximation algorithms for covering problems. • 2log n-approximation for SSC. • 4-approximation for stochastic vertex cover. • 4-approximation for stochastic multicut on trees. • Ravi & Sinha gave a log n-approximation algorithm for SSC, 2-approximation algorithm for stochastic vertex cover in the polynomial-scenario model.

  19. Rounding the LP Assume LP can be solved in polynomial time. Suppose we have an a-approximation algorithm wrt. the LP relaxation for the deterministic problem. Let (x,y) : optimal solution with cost OPT. ∑S:eÎS xS + ∑S:eÎS yA,S ≥ 1for each A Í U, eÎA Þ for every element e, either ∑S:eÎS xS ≥ ½ OR in each scenario A : eÎA, ∑S:eÎS yA,S ≥ ½. Let E = {e : ∑S:eÎS xS ≥ ½}. So (2x) is afractional set coverfor the set E Þ can round to get aninteger set coverSof cost ∑SÎSwS ≤ a(∑S 2wSxS) . Sis the first stage decision.

  20. A Compact Convex Program pA : probability of scenario A Í U. xS : indicates if set S is picked in stage I. Minimize h(x) = ∑SwSxS +∑AÍU pAfA(x) s.t. xS ≥ 0 for each S (SSC-P) where fA(x) = min. ∑S WSyA,S s.t. ∑S:eÎS yA,S ≥ 1– ∑S:eÎS xSfor each eÎA yA,S ≥ 0 for each S. Equivalent to earlier LP. Each fA(x) is convex, so h(x) is a convex function.

  21. The General Strategy 1. Get a (1+e)-optimal fractional first-stage solution (x) by solving the convex program. 2. Convert fractional solution (x) to integer solution • decouple the two stages near-optimally • use a-approx. algorithm for the deterministic problem to solve subproblems. Obtain a c.a-approximation algorithm for the stochastic integer problem. Many applications: set cover, vertex cover, facility location, multicut on trees, …

  22. Solving the Convex Program Minimize h(x) subject to xÎP. h(.) : convex Ellipsoid method P y • Need a procedure that at any point y, • if yÏP, returns a violated inequality • which shows that yÏP

  23. Solving the Convex Program Minimize h(x) subject to xÎP. h(.) : convex Ellipsoid method P • Need a procedure that at any point y, • if yÏP, returns a violated inequality • which shows that yÏP • if yÎP, computes the subgradient • of h(.) at y • dÎÂm is a subgradientof h(.) at u, if "v, h(v)-h(u) ≥ d.(v-u). • Given such a procedure, ellipsoid runs in polytime and returns points x1, x2, …, xkÎP such that mini=1…k h(xi) is close to OPT. h(x) ≤ h(y) y d Computing subgradients is hard. Evaluating h(.) is hard.

  24. Solving the Convex Program Minimize h(x) subject to xÎP. h(.) : convex Ellipsoid method P • Need a procedure that at any point y, • if yÏP, returns a violated inequality • which shows that yÏP • if yÎP, computes an approximate • subgradient of h(.) at y • d'ÎÂm is an e-subgradient at u, • if "vÎP, h(v)-h(u) ≥ d'.(v-u) – e.h(u). • Given such a procedure, can compute point xÎP such that • h(x) ≤OPT/(1-e) + r without ever evaluating h(.)! h(x) ≤ h(y) y d' Can compute e-subgradients by sampling.

  25. Putting it all together Get solution x with h(x) close to OPT. Sample initially to detect if OPT is large – this allows one to get a (1+e).OPT guarantee. Theorem: (SSC-P) can be solved to within a factor of(1+e)in polynomial time, with high probability. Gives a(2log n+e)-approximation algorithm forthe stochastic set cover problem.

  26. A Solvable Class of Stochastic LPs Minimize h(x) = w.x+∑AÍU pAfA(x) s.t. x ÎPÍÂm where fA(x) = min. wA.yA + cA.rA s.t. DArA + TAyA ≥ jA – TAx yA Î Âm, rA Î Ân, yA, rA ≥ 0. ≥ 0 Theorem: Can get a (1+e)-optimal solution for this class of stochastic programs in polynomial time. Includes covering problems (e.g., set cover, network design, multicut), facility location problems, multicommodity flow.

  27. Moral of the Story • Even though the stochastic LP relaxation has exponentially many variables and constraints, we can still obtain near-optimal fractional first-stage decisions • Fractional first-stage decisions are sufficient to decouple the two stages near-optimally • Many applications: set cover, vertex cover, facility locn., multicommodity flow, multicut on trees, … • But we have to solve convex program with many samples (not just l)!

  28. Sample Average Approximation • Sample Average Approximation (SAA) method: • Sample initially N times from scenario distribution • Solve 2-stage problem estimating pA with frequency of occurrence of scenario A • How large should N be to ensure that an optimal solution to sampled problem is a (1+e)-optimal solution to original problem? • Kleywegt, Shapiro & Homem De-Mello (KSH01): • bound N by variance of a certain quantity – need not be polynomially bounded even for our class of programs. • S, Shmoys ’05 : • show using e-subgradients that for our class, N can be poly-bounded. • Charikar, Chekuri & Pál ’05: • give another proof that for a class of 2-stage problems, N can be poly-bounded.

  29. Multi-stage Problems k-stage problem º k decision points Given : Distribution over inputs. Stage I : Make some advance decisions – hedge against uncertainty. Uncertainty evolves in various stages. Learn new information in each stage. Can take recourse actions in each stage – canaugment earlier solution paying a recourse cost. stage I 0.2 0.4 0.3 stage II 0.5 scenarios in stage k Choose stage I decisions to minimize expected total cost = (stage I cost) + Eall scenarios [cost of stages 2 … k].

  30. Multi-stage Problems Fix k = number of stages. LP-rounding: S, Shmoys ’05 • Ellipsoid-based algorithm extends • SAA method also works black-box model, arbitrary costs Rounding procedure of SS04 can be easily adapted: lose an O(k)-factor over the deterministic guarantee • O(k)-approx. for k-stage vertex cover, facility location, multicut on trees; k.log n-approx. for k-stage set cover Gupta, Pál, Ravi & Sinha ’05: boosted sampling extends but with outcome-dependent proportional costs • 2k-approx. for k-stage Steiner tree (also Hayrapetyan, S & Tardos) • factors exponential in k for k-stage vertex cover, facility location Computing e-subgradients is significantly harder, need several new ideas

  31. Open Questions • Combinatorial algorithms in the black box model and with general costs. What about strongly polynomial algorithms? • Incorporating “risk” into stochastic models. • Obtaining approximation factors independent of k for k-stage problems. Integrality gap for covering problems does not increase. Munagala has obtained a 2-approx. for k-stage VC. • Is there a larger class of doubly exponential LPs that one can solve with (more general) techniques?

  32. Thank You.

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