Solving the Station Repacking Problem...Faez Ahmed Introduction Initial Analysis Meta-Algorithmic Techniques Problem-Speci c Enhancements Caching Conclusion Figure:ECDF of runtimes

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


Faez Ahmed

Introduction

Initial Analysis



Caching

Conclusion

Introduction

Reverse AuctionTelevision broadcasters are paidto free the radio spectrum, whichis reallocated totelecommunication for objectiveslike profit and social good


Faez Ahmed

Introduction

Initial Analysis



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.


Faez Ahmed

Introduction

Initial Analysis



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.


Faez Ahmed

Introduction

Initial Analysis



Caching

Conclusion


Faez Ahmed

Introduction

Initial Analysis



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.


Faez Ahmed

Introduction

Initial Analysis



Caching

Conclusion

Why is this hard?





Saying that repacking problem is feasible has higher cost.Why?

Good news: They have training (100,572), test (10000) andvalidation (1000) data from simulations.


Faez Ahmed

Introduction

Initial Analysis



Caching

Conclusion

Why is this hard?





Saying that repacking problem is feasible has higher cost.Why?Good news: They have training (100,572), test (10000) andvalidation (1000) data from simulations.


Faez Ahmed

Introduction

Initial Analysis



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


Faez Ahmed

Introduction

Initial Analysis



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.


Faez Ahmed

Introduction

Initial Analysis



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


Faez Ahmed

Introduction

Initial Analysis



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


Faez Ahmed

Introduction

Initial Analysis



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


Faez Ahmed

Introduction

Initial Analysis



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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


Faez Ahmed

Introduction

Initial Analysis



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.


Faez Ahmed

Introduction

Initial Analysis



Caching

Conclusion

Hail Hydra

First two rounds of Hydra, identified DCCA-preSAT andDCCA+ solvers. Then clasp-h1 and clasp-h2 were identified.


Faez Ahmed

Introduction

Initial Analysis



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.


Faez Ahmed

Introduction

Initial Analysis



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


Faez Ahmed

Introduction

Initial Analysis



Caching

Conclusion

Results


Faez Ahmed

Introduction

Initial Analysis



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

Documents

Solving the Station Repacking Problem...Faez Ahmed Introduction Initial Analysis Meta-Algorithmic Techniques Problem-Speci c Enhancements Caching Conclusion Figure:ECDF of runtimes