Transcript Robust Combinatorial Optimization with Exponential Scenarios

Robust Network Design with
Exponential Scenarios
By:
Rohit Khandekar
Guy Kortsarz
Vahab Mirrokni
Mohammad Salavatipour
Outline
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Robust vs. Stochastic Optimization
Robust Two-stage Network Design Problems
Related Work: Stochastic vs. Robust
Our Results
Algorithm for Robust Steiner Tree
Conclusion
Optimization Against Uncertainty
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Stochastic Optimization.
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Optimize the expected cost given the
probability distribution on the scenarios
Playing against a randomized uncertainty.
Robust Optimization.
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Optimize for the worst case scenarios.
Playing against an adversary.
Two-stage Optimization
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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.
Exponential vs. Polynomial
Scenarios.
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Polynomial Scenarios: explicit Scenarios.
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Stochastic Optimization: The probability of each
scenario is explicitly given.
Robust Optimization: All scenarios are specified.
Exponential Scenarios: implicit Scenarios
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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.
2-stage Robust Steiner Tree
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Given:
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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’)
2-stage Robust Steiner Tree (Exponential Scenarios)
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Given:
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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’)
Other Robust 2-stage Problems
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Robust Network Design:
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Robust Steiner Forest: at most k pairs show up.
Robust Facility Location.
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Other Robust Covering Problems:
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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.
Robust Set Cover.
Robust Vertex Cover.
Robust two-stage Min-cut.
Robust Min Cut Problem
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Robust 2-stage Min-cut:
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Given:
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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’).
Outline
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Robust vs. Stochastic Optimization
Robust Two-stage Network Design Problems
Related Work: Stochastic vs. Robust
Our Results
Algorithm for Robust Steiner Tree
Conclusion
Related Work
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Stochastic Two-stage Optimization:
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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:
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Swamy and Shmoys (FOCS05): sample average
approximation.
Related Work
Robust Two-stage Optimization:
Initiated by Ben-Tal, Gorashko, Guslitzer, and
Nemirovski.
 Polynomial Number of Scenarios:
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 Introduced
by Dhamhere, Goyal, Ravi, and Singh
(FOCS05): Facility Location & Steiner Tree
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Constant-factor approximation algorithms
 Improved
by Golovin, Goyal, and Ravi
(STACS06): Min Cut.
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Constant-factor approximation algorithm for Min-cut
(Is it NP-Hard?)
Exponential Scenarios: Known Results
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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?
Our Results
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Constant-factor for robust Steiner tree and
robust Facility Location.
Thm. 5.5-approx for two-stage robust Steiner
Tree
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Thm. 10-approx for robust facility location.
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Combinatorial Algorithm
Combinatorial Algorithm
Thm. 3-approx for robust Steiner Forest on
Trees.
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At most k pairs of nodes show up to be connected.
LP-based Algorithm
Our Results
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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.
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implies quasi-polynomial-time algorithms for NP.
Thm. Robust two-stage Min-cut is APX-hard.
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Even with uniform inflation factor.
Even with one source and three sinks.
NP-hardness was posed as an open problem (by
Golovin, Goyal, Ravi).
Robust Steiner Tree Algorithm
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Let OPT = OPT1 + q OPT2.
OPT1: Optimum in the first Stage.
OPT2: Optimum in the second Stage.
Algorithm A:
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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.
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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.
Algorithm (Cont’)
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Find-Centers: Find centers c1, c2, …, ck and assign
nodes to these centers s.t.
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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:
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Set of centers U= V(G) and C=empty; and i = 0.
i = i+1.
Select an arbitrary node c1 in U and add it to C.
Remove all nodes in distance · rOPT2 / k from U.
If U is nonempty, go back to step 2,
otherwise terminate.
Analysis
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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.
Conclusion
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Constant-factor Approximation Algorithms for
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Hardness Result
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2-stage robust Steiner tree
2-stage robust facility location
2-stage robust Steiner forest (on trees)
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:
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http://people.csail.mit.edu/mirrokni/ESA08.pdf