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 E1 of 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 E1 in 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 k pairs 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.

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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_1 of 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_i and t_i.
• Goal Choose a subset E_1 with the minimum total
  cost (Total Cost cost of E_1 q times cost of
  E).

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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.

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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?)

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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?

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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

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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).

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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, , 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 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.

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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 Cempty and i
  0.
• 2) i i1.
• 3) Select an arbitrary node c1 in U and add it to
  C.
• 4) Remove all nodes in distance rOPT2 / k from
  U.
• 5) If U is nonempty, go back to step 2,
• otherwise terminate.

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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 qlt3.51, Run a trivial algorithm (not buy
  anything in the first stage),
• otherwise, Run Algorithm A.

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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