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Solving the StationRepackingProblem
Faez Ahmed
Introduction
Initial Analysis
Meta-AlgorithmicTechniques
Problem-SpecificEnhancements
Caching
Conclusion
Solving the Station Repacking Problem
Faez Ahmed
CMSC 828M, Fall 2016University of Maryland, College Park
October 25, 2016
Solving the StationRepackingProblem
Faez Ahmed
Introduction
Initial Analysis
Meta-AlgorithmicTechniques
Problem-SpecificEnhancements
Caching
Conclusion
Introduction
Reverse AuctionTelevision broadcasters are paidto free the radio spectrum, whichis reallocated totelecommunication for objectiveslike profit and social good
Solving the StationRepackingProblem
Faez Ahmed
Introduction
Initial Analysis
Meta-AlgorithmicTechniques
Problem-SpecificEnhancements
Caching
Conclusion
Background
FCC chose auction design based on a descending clock.Participating station relinquish broadcast rights, with thisprice offer falling for a given station as long as it remainsrepackable.Many repacking problems need to be solved and every failurecorresponds to a lost opportunity to lower a price offer.
Solving the StationRepackingProblem
Faez Ahmed
Introduction
Initial Analysis
Meta-AlgorithmicTechniques
Problem-SpecificEnhancements
Caching
Conclusion
The Station Repacking Problem
I Each US television station s ∈ S is assigned a channelcs ∈ C
I Forbidden station channel pairs (s, c), (s’, c’) provided,leading to interference graph
I The effect of the auction will be to remove somebroadcasters from the airwaves completely, and toreassign channels to the remaining stations from areduced set.
The station repacking problem is then the task of finding arepacking that assigns each station a channel from itsdomain that satisfies the interference constraints.
Solving the StationRepackingProblem
Faez Ahmed
Introduction
Initial Analysis
Meta-AlgorithmicTechniques
Problem-SpecificEnhancements
Caching
Conclusion
Solving the StationRepackingProblem
Faez Ahmed
Introduction
Initial Analysis
Meta-AlgorithmicTechniques
Problem-SpecificEnhancements
Caching
Conclusion
Why is this hard?
Problem is NP-complete. Still can be solved effectively inpractice
I Descending clock has partial solutions available
I Three classes so problem can be decomposed- UHFband
I Interference graph with 2173 stations: co-channelconstraint and adjacent channel constraint
Saying that repacking problem is feasible has higher cost.Why?Good news: They have training (100,572), test (10000) andvalidation (1000) data from simulations.
Solving the StationRepackingProblem
Faez Ahmed
Introduction
Initial Analysis
Meta-AlgorithmicTechniques
Problem-SpecificEnhancements
Caching
Conclusion
Why is this hard?
Problem is NP-complete. Still can be solved effectively inpractice
I Descending clock has partial solutions available
I Three classes so problem can be decomposed- UHFband
I Interference graph with 2173 stations: co-channelconstraint and adjacent channel constraint
Saying that repacking problem is feasible has higher cost.Why?
Good news: They have training (100,572), test (10000) andvalidation (1000) data from simulations.
Solving the StationRepackingProblem
Faez Ahmed
Introduction
Initial Analysis
Meta-AlgorithmicTechniques
Problem-SpecificEnhancements
Caching
Conclusion
Why is this hard?
Problem is NP-complete. Still can be solved effectively inpractice
I Descending clock has partial solutions available
I Three classes so problem can be decomposed- UHFband
I Interference graph with 2173 stations: co-channelconstraint and adjacent channel constraint
Saying that repacking problem is feasible has higher cost.Why?Good news: They have training (100,572), test (10000) andvalidation (1000) data from simulations.
Solving the StationRepackingProblem
Faez Ahmed
Introduction
Initial Analysis
Meta-AlgorithmicTechniques
Problem-SpecificEnhancements
Caching
Conclusion
Initial AnalysisMixed Integer Programming xs,c variable used andconstraint that each station is assigned to exactly onechannel, and interference constraints.MIP used (CPLEX and Gurobi) and results were not good.
Figure: ECDF of runtimes for default MIP and SAT solvers. The bars show fraction of SAT andUNSAT instances binned by their (fastest) runtime. Although present, unsatisfiable instances form aninsignificant portion of instances solved
Solving the StationRepackingProblem
Faez Ahmed
Introduction
Initial Analysis
Meta-AlgorithmicTechniques
Problem-SpecificEnhancements
Caching
Conclusion
Propositional satisfiability problem
SAT 101: Given a set of propositional formulas, decidewhether or not there is an assignment to the variables in theformula such that the formula is satisfied
Then posed it as a SAT problem with boolean variable andthree kinds of clauses —each station is assigned at least onechannel, each station is assigned at most one channel andinterference constraints are respected.
They compared 18 SAT solvers and found DCCA (localsearch algorithm for satisfiability) as best performing SATsolver.
Solving the StationRepackingProblem
Faez Ahmed
Introduction
Initial Analysis
Meta-AlgorithmicTechniques
Problem-SpecificEnhancements
Caching
Conclusion
Meta-Algorithmic Techniques
I Algorithm configuration —Setting design decisionsexposed as parameters to optimize an algorithmsaverage performance. They used SequentialModel-based Algorithm Configuration algorithm.
I Algorithm portfolios —Selects a small set of algorithmswith complementary performance on problems ofinterest to run in parallel
Solving the StationRepackingProblem
Faez Ahmed
Introduction
Initial Analysis
Meta-AlgorithmicTechniques
Problem-SpecificEnhancements
Caching
Conclusion
Hydra
I Combine Algorithm configuration and Algorithmportfolios.
I Identify sets of complementary solvers from highlyparameterized design spaces via algorithm configuration,by greedily adding configurations that make the greatestpossible marginal contribution to an existing portfolio.
I Best solver was DCCA and clasp solver added 4%performance
Solving the StationRepackingProblem
Faez Ahmed
Introduction
Initial Analysis
Meta-AlgorithmicTechniques
Problem-SpecificEnhancements
Caching
Conclusion
Problem-Specific Enhancements
A feasible, partial assignment γ− is available.Local Augmenting- Keep the entire graph fixed exceptneighbours of s+ to find feasible solution. Assign s+ to achannel and update the channel assignments of the stationsin s+ neighborhood, holding the rest of γ− fixed.DCCA-preSAT 78.5%Starting Assignment for Local Search Solvers DCCAwork by searching a space of complete assignments andseeking a feasible point, typically following gradients tominimize an objective function that counts violatedconstraints. DCCA+ 85.4% instances
Solving the StationRepackingProblem
Faez Ahmed
Introduction
Initial Analysis
Meta-AlgorithmicTechniques
Problem-SpecificEnhancements
Caching
Conclusion
Figure: ECDF of runtimes of the SAT solvers we include in ourfinal portfolio. The bars show fraction of SAT and UNSATinstances binned by their (fastest) runtime
Solving the StationRepackingProblem
Faez Ahmed
Introduction
Initial Analysis
Meta-AlgorithmicTechniques
Problem-SpecificEnhancements
Caching
Conclusion
Problem Simplification
Graph Decomposition Identify disconnected componentsand solve each separately, giving smaller instances to SATsolvers. If we identify a single component as infeasible, wecan immediately declare the entire problem infeasibleUnder-constrained Station Removal Heuristic ofcomparing a stations available channels to its number ofneighboring stations.
Solving the StationRepackingProblem
Faez Ahmed
Introduction
Initial Analysis
Meta-AlgorithmicTechniques
Problem-SpecificEnhancements
Caching
Conclusion
Hail Hydra
First two rounds of Hydra, identified DCCA-preSAT andDCCA+ solvers. Then clasp-h1 and clasp-h2 were identified.
Solving the StationRepackingProblem
Faez Ahmed
Introduction
Initial Analysis
Meta-AlgorithmicTechniques
Problem-SpecificEnhancements
Caching
Conclusion
Containment Caching
1. Check whether the feasible cache contains a superset ofS, in which case the original problem is feasible.
2. Check whether a subset of S belongs to the infeasiblecache, in which case the original problem is infeasible.
3. Simplify and decompose the given instance and queryeach component in the feasible cache.
Solving the StationRepackingProblem
Faez Ahmed
Introduction
Initial Analysis
Meta-AlgorithmicTechniques
Problem-SpecificEnhancements
Caching
Conclusion
Results
Figure: Time saved per cache hit. Each point represents a cachehit on a particular key: the x-axis represents the number of timesthe corresponding key was hit, while the y-axis represents theamount of time each individual cache hit saved
Solving the StationRepackingProblem
Faez Ahmed
Introduction
Initial Analysis
Meta-AlgorithmicTechniques
Problem-SpecificEnhancements
Caching
Conclusion
Results
Solving the StationRepackingProblem
Faez Ahmed
Introduction
Initial Analysis
Meta-AlgorithmicTechniques
Problem-SpecificEnhancements
Caching
Conclusion
Key Contributions
I Combine state-of-the-art SAT solvers andmeta-algorithmic techniques
I SATFC 2.0 (solving 99.0% of test instances in under 0.2seconds, and 99.6% in under a minute.
I Containment cache performance improvement withmore historical data
I Improvement over SATFC, which was used by FCC in2014
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