Discrete Optimization Under Uncertainty Sahil singla Institute for Advanced Study and Princeton University Oct 2 nd 2019 Example How to Sell a Diamond Sell One Diamond potential buyers with values ID: 770262
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Discrete OptimizationUnder Uncertainty Sahil singla Institute for Advanced Study and Princeton University Oct 2nd, 2019
Example: How to Sell a Diamond? Sell One Diamond: potential buyers with values Goal: Maximize the accepted buyer’s value Similar ExamplesSelling a house, ad-slots on webFinding a secretary, marriage partner Problem easy if all values known : What if uncertainty in buyer values? 2 Discrete Optimization Uncertainty
Given a Finite Universe : Given an Objective Function : Given Constraints : Select in polytime a solution set toExamples:Maximization : k elements, Matching, Max-cut, Knapsack, (Bin) Packing, Orienteering, Welfare Max, SAT, Increasing Subseq, Throughput Scheduling, Submod over MatroidMinimization: MST, (Set) Covering, Steiner Network, Traveling Salesman Problem, Facility Location, Scheduling, Graph Coloring, Multi-cut, Clustering, Discrepancy 3 What is Discrete Optimization? E.g., size at most #diamonds E.g , sum of values E.g , n buyers What if Input is Uncertain?
Why Consider Uncertainty? Certainty is “expensive”, Lack of future knowledge, “Private” informationTwo Approaches to Uncertainty today: Stochastic Optimization: Priors (distributions) on all uncertain parametersComputation is bottleneck Online Decision Making: No idea about (some of) the parameters Information is bottleneckComputation also a bottleneck 4 How to Model Uncertainty?
OUTLINE Introduction: Discrete Optimization under UncertaintyStochastic Optimization: Adaptivity Gaps Online Decision Making: Competitive Ratio 5
Model 1: Selling a Diamond Given Independent Bernoulli Distributions on Values Constraint: Given , let Adaptively find of size ≤k Objective: Maximize Expected Value Difficult to Find: Exponential sized Want : A polytime (approximate) algorithm 6 height
Non-Adaptive Algorithms Select a fixed set Benefits : Easier to Represent : Just output the set Find : is submodular 7 Concern : Large Adaptivity How large can GAP get?
Results and Open Questions 8 Thm [GN S ’16,’17, B S Z’19] : Adaptivity Gap Constraints = Downward-Closed Function = Monotone Submodular Submodular : If then E.g. , Max function or Sum of top Open Question : Is Adaptivity Gap ? Constraints = Downward-Closed Function = Monotone Subadditive Subadditive : For all , ADAP NON-ADAP ALGO GAP
OUTLINE Introduction: Discrete Optimization under UncertaintyStochastic Optimization: Adaptivity Gaps Online Decision Making: Competitive Ratio 9
Model 2: Selling a Diamond Buyer values unknown Buyers arrive uniformly and reveal Irrevocably decide if accepting Objective: Maximize expected value : How to compare different algorithms? Competitive Ratio of Alg: Difficult to find optimal even with exponential time Want : A polytime (approximate) algorithm 10 time Notice, always
Secretary Problems Why competitive ratio : Ignore till , select next if largest till now Multiple-Secretary Problem: Sell diamonds to maximize sum 11 Thm [ Dynkin’63 ] : Competitive Ratio Open Question : Is Competitive Ratio ? Thm [ Kleinberg’04 ] : Competitive Ratio Matroid-Secretary Problem: Given a matroid on buyers, sell to maximize sum We know is
Conclusion Study Discrete Optimization Problems under UncertaintyComparison: Open Problems: Is adaptivity gap for subadditive functions?Is competitive ratio for matroid secretary? 12 Stochastic Optimization All distributions given Computation bottleneckShow small Adaptivity GapsOnline Decision Making Some parameters fully unknown Information/computation bottleneckShow small Competitive Ratio Questions ?
Further SLIDES 13
Model 1: Selling a Diamond Given Indep Probability Distributions on Values Given Probing Prices: Adaptively find Constraint: Given Price Budget (knapsack) Objective: Maximize Expected Value 14 ADAP NON-ADAP ALGO GAP GAP
Model 2: Selling a Diamond Sell One Diamond: Multiple potential buyersBuyers Arrive and Make a Take-it-or-Leave-it Bid Decide Immediately and Irrevocably When Should you Accept the Bid?Goal:Maximize value of the accepted bid Cannot go back to a declined bidSimilar ExamplesSelling ad-slots Finding a secretary/marriage partner 15 time
Non-Adaptive Algorithms Select a fixed set in the beginning to Benefits : Easier to Represent : Just output the setFind: is submodular Example : Can probe & take w.p . each takes w.p . 16 0.5 0.5 NO YES Concern : Large Adaptivity How large can GAP get?
Secretary Problems Why competitive ratio : Ignore till , select next if largest till now 17 Thm [ Dynkin’63 ] : Competitive Ratio Open Question : Is Competitive Ratio ? Matroid-Secretary Problem: Given a matroid on buyers, sell to maximize sum We know is