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CSP: Examples

CSP: Examples. Industrial applications: scheduling, resource allocation, product configuration, etc. AI: Logic inference, temporal reasoning, NLP, etc. Puzzles: Sudoku & Minesweeper. 2,4,6,9. 3,5,7. <. <. <. <. <. 3,5,7. 5,6,7,8. 1,6,11. =. =. . <. <. 1,2,10. 8,9,11.

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CSP: Examples

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  1. CSP: Examples • Industrial applications: scheduling, resource allocation, product configuration, etc. • AI: Logic inference, temporal reasoning, NLP, etc. • Puzzles: Sudoku & Minesweeper

  2. 2,4,6,9 3,5,7 < < < < < 3,5,7 5,6,7,8 1,6,11 = =  < < 1,2,10 8,9,11 Constraint propagation • Removes from the problem values (or combinations of values) that are inconsistent with the constraints • Does not eliminate any solution 

  3. Sudoku as a CSP • Each cell is a variable (decision) with the domain [1..9] (choices) • Two models: Binary, 810 AllDiff binary constraints Non-binary, 27 AllDiff constraints of arity 9 Joint work with C. Reeson

  4. Propagation algorithms: demo • Generalized AC (GAC) • Arc Consistency (AC) • GAC on AllDiff[Régin, 94] • Arcs that do not appear in any matching that saturates the variables correspond to variable-value pairs that cannot appear in any solution • GAC on AllDiffis poly time c1 1 c2 2 c3 3 c4 4 c5 5 c6 6 c7 7 c8 8 c9 9

  5. Minesweeper as a CSP demo • Variables are the cells • Domains are {0,1} (i.e., safe or mined) • One constraint for each cell with a number (arity 1...8) Exactly two mines: 0000011 0000101 0000110, etc. Exactly three mines: 0000111 0001101 0001110, etc. Joint work with R. Woodward, K. Bayer & J. Snyder

  6. Geospatial reasoning Joint work with K. Bayer, M. Michalowski & C.A. Knoblock (USC) Google Maps Yahoo Maps Actual location Microsoft Live Local (as of November 2006)

  7. Building Identification (BID) problem • Layout: streets and buildings • Phone book • Complete/incomplete • Assumption: all addresses in phone book correspond to a building in the layout S1 S2 B2 B1 B3 B4 = Building S3 = Corner building S1#1, S1#4, S1#8, S2#7, S2#8, S3#1, S3#2, S3#3, S3#15, … B6 B7 B10 Si = Street B5 B8 B9

  8. Basic (address numbering) rules No two buildings can have the same address Ordering Numbers increase/decrease along a street Parity Numbers on a given side of a street are odd/even Ordering Parity B1 B1 < B2 < B3 B3 Odd Even B2 B4

  9. Additional information Landmarks Gridlines 1600 Pennsylvania Avenue S1 #138 S1 #208 B1 B2 B1 B2 S1

  10. Query • Given an address, what buildings could it be? Given a building, what addresses could it have? S1 S2 S1#1,S1#4, S1#8,S2#7, S2#8,S3#1, S3#2,S3#3, S3#15 B2 B1 B3 B4 = Building S3 = Corner building S1#1, S3#1, S3#15 B6 B7 B10 Si = Street B5 B8 B9

  11. Parity constraints Ordering constraints Corner constraints Phone-book constraints Optional: grid constraints CSP model B2 B1 B5 B3 B4 IncreasingEast S2 B1 B2 B1c S1 OddOnNorth

  12. Example constraint network S1 S2 S1#1,S1#4, S1#8,S2#7, S2#8,S3#1, S3#2,S3#3, S3#15 B2 B1 B3 B4 = Building S3 = Corner building B6 B7 B10 Si = Street B5 B8 B9

  13. GTAAP: Task • Hiring & managing GTAs as instructors + graders • Given • A set of courses • A set of graduate teaching assistants • A set of constraints that specify allowable assignments • Find a consistent & satisfactory assignment • Consistent: assignment breaks no (hard) constraints • Satisfactory: assignment maximizes • number of courses covered • happiness of the GTAs • Often, number of hired GTAs is insufficient

  14. Motivation • Context • “Most difficult duty of a department chair” [Reichenbach, 2000] • Assignments done manually, countless reviews, persistent inconsistencies • Unhappy instructors, unhappy GTAs, unhappy students • Observation • Computers are good at maintaining consistency • Humans are good at balancing tradeoffs • Our solution • An online, constraint-based system • With interactive & automated search mechanisms

  15. Outline • Task & Motivation • System Architecture & Interfaces • Scientific aspects • Problem Modeling • Problem Solving • Comparing & Characterizing Solvers • Motivation revisited & Conclusions

  16. Password Protected Access for GTAs http://cse.unl.edu/~gta Password Protected Access for Manager http://cse.unl.edu/~gta • Web-interface for applicants • Web-interface for manager • View / edit GTA records • Setup classes • Specify constraints • Enforce pre-assignments Visualization widgets Local DB Other structured, semi-structured, or unstructured DBs Interactive Search Automated Search Heuristic BT Stochastic LS Multi-agent Search Randomized BT • A local relational database • Graphical selective queries Cooperative, hybrid Search Strategies • Drivers for • Interactive assignments • Automated search algorithms In progress System Architecture

  17. GTA interface: Preference Specification

  18. Manager interface: TA Hiring & Load

  19. Outline • Task & Motivation • System Architecture & Interfaces • Scientific aspects • Problem Modeling • Problem Solving • Comparing & Characterizing Solvers • Motivation revisited & Conclusions

  20. Constraint-based Model • Variables • Grading, conducting lectures, labs & recitations • Values • Hired GTAs (+ preference for each value in domain) • Constraints • Unary: ITA certification, enrollment, time conflict, non-zero preferences, etc. • Binary (Mutex): overlapping courses • Non-binary: same-TA, capacity, confinement • Objective • longest partial and consistent solution (primary criterion) • while maximizing GTAs’ preferences (secondary criterion)

  21. Outline • Task & Motivation • System Architecture & Interfaces • Scientific aspects • Problem Modeling • Problem Solving • Comparing & Characterizing Solvers • Motivation revisited & Conclusions

  22. Problem Solving • Interactive decision making • Seamlessly switching between perspectives • Propagates decisions (MAC) • Automated search algorithms • Heuristic backtrack search (BT) • Stochastic local search (LS) • Multi-agent search (ERA) • Randomized backtrack search (RDGR) • Future: Auction-based, GA, MIP, LD-search, etc. • On-going: Cooperative/hybrid strategies

  23. Manager interface: Interactive Selection

  24. Dual perspective Task-centered view Resource-centered view

  25. Shallowest level reached by BT after … Number of variables: 69 24 hr: 51 (26%) 1 min: 55 (20%) Max depth: 57 Depth of the tree: 69 Heuristic BT Search • Since we don’t know, a priori, whether instance is solvable, tight, or over-constrained • Modified basic backtrack mechanism to deal with this situation • We designed & tested various ordering heuristics: • Dynamic LD was consistently best • Branching factor relatively huge (30) • Causes thrashing, backtrack never reaches early variables

  26. Stochastic Local Search • Hill-climbing with min-conflict heuristic • Constraint propagation: • To handle non-binary constraints (e.g., high-arity capacity constraints) • Greedy: • Consistent assignments are not undone • Random walk to avoid local maxima • Random restarts to recover from local maxima

  27. Multi-Agent Search (ERA)[Liu et al. 02] • “Extremely” decentralized local search • Agents (variables) seek to occupy best positions (values) • Environment records constraint violation in each position of an agent given positions of other agents • Agents move, egoistically, between positions according to reactive Rules • Decisions are local • An agent can always kick other agents from a favorite position even when value of ‘global objective function’ is not improved • ERA appears immune to local optima • Lack of centralized control • Agents continue to kick each other • Deadlock appears in over-constrained problems

  28. Randomized BT Search • Random variable/value selection allows BT to visit a wider area of the search space [Gomes et al. 98] • Restarts to overcome thrashing • Walsh proposed RGR [Walsh 99] • Our strategy, RDGR, improves RGR with dynamic choice of cutoff values for the restart strategy [Guddeti & Choueiry 04]

  29. Optimizing solutions • Primary criterion: solution length • BT, LS, ERA, RGR, RDGR • Secondary criterion: preference values • BT, LS, RGR, RDGR • Criterion: • Average preference • Geometric mean • Maximum minimal preference

  30. More Solvers… • Interactive decision making • Automated search algorithms • BT, LS, ERA, RGR, RDGR. • Future: Auction-based, GA, MIP, LD-search, etc. • On-going: Cooperative / hybrid strategies

  31. Outline • Task & Motivation • System Architecture & Interfaces • Scientific aspects • Problem Modeling • Problem Solving • Comparing & Characterizing Solvers • Motivation revisited & Conclusions

  32. Conclusions • Integrated interactive & automated problem-solving strategies • Reduced the burden of the manager • Lead to quick development of ‘stable’ solutions • Our efforts • Helped the department • Trained students in CP techniques • Paved new avenues for research • Cooperative, hybrid search • Visualization of solution space

  33. Other sample projects • Graduate TA Assignment Project (GTAAP) • Modeling, search, GUI • Temporal Reasoning • Constraint propagation, search, graph theory • Symmetry detection • Search, databases (computational) • Structural decompositions • Databases (theory), tractability results

  34. The Research • Modeling & Reformulation • Propagation algorithms • Search algorithms • Decomposition algorithms • Symmetry identification & breaking

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