1. Constraint Systems Laboratory February 18, 2009 CSP Modeling Examples CSP: Examples Industrial applications: scheduling, resource allocation, product configuration, etc. AI: Logic inference, temporal reasoning, NLP, etc. 1 Puzzles: Sudoku & Minesweeper

2. Constraint Systems Laboratory February 18, 2009 CSP Modeling Examples Constraint propagation Removes from the problem values (or combinations of values) that are inconsistent with the constraints Does not eliminate any solution 2 < < = = < < 1,2,10 1,6,11 2,4,6,9 3,5,7 5,6,7,8  8,9,11 < < < 

3. Constraint Systems Laboratory February 18, 2009 CSP Modeling Examples 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 3 Joint work with C. Reeson

4. Constraint Systems Laboratory February 18, 2009 CSP Modeling Examples 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 AllDiff is poly time Propagation algorithms: demo demo Arc Consistency (AC) 4 Generalized AC (GAC) c9c9 c1c1 c2c2 c3c3 c4c4 c5c5 c6c6 c7c7 c8c8 1 2 3 4 5 6 7 8 9

5. Constraint Systems Laboratory February 18, 2009 CSP Modeling Examples Minesweeper as a CSP demo demo 5 Exactly two mines: 0000011 0000101 0000110, etc. Exactly three mines: 0000111 0001101 0001110, etc. Variables are the cells Domains are {0,1} (i.e., safe or mined) One constraint for each cell with a number (arity 1...8) Joint work with R. Woodward, K. Bayer & J. Snyder

6. Constraint Systems Laboratory February 18, 2009 CSP Modeling Examples Geospatial reasoning Joint work with K. Bayer, M. Michalowski & C.A. Knoblock (USC) 6 Google Maps Yahoo Maps Actual location Microsoft Live Local (as of November 2006)

7. Constraint Systems Laboratory February 18, 2009 CSP Modeling Examples 7 Layout: streets and buildings Phone book –Complete/incomplete –Assumption: all addresses in phone book correspond to a building in the layout Building Identification (BID) problem B6 B8 B2 B4 B5 B3 B9 B10B7 B1 S1S2 S3 Si = Building = Corner building = Street S1#1, S1#4, S1#8, S2#7, S2#8, S3#1, S3#2, S3#3, S3#15, …

8. Constraint Systems Laboratory February 18, 2009 CSP Modeling Examples 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 B1 << B2B3 Odd Even Parity B1 B2 B3 B4

9. Constraint Systems Laboratory February 18, 2009 CSP Modeling Examples 9 Additional information Landmarks B1B2 1600 Pennsylvania Avenue Gridlines B1B2 S1 S1 #138S1 #208

10. Constraint Systems Laboratory February 18, 2009 CSP Modeling Examples 10 Query 1.Given an address, what buildings could it be? B6 B8 B2 B4 B5 B3 B9 B10B7 B1 S1S2 S3 Si = Building = Corner building = Street S1#1,S1#4, S1#8,S2#7, S2#8,S3#1, S3#2,S3#3, S3#15 S1#1, S3#1, S3#15 2.Given a building, what addresses could it have?

11. Constraint Systems Laboratory February 18, 2009 CSP Modeling Examples 11 CSP model S1 S2 B2 OddOnNorth B1 B1c B3 B4 B5 B1 B2 IncreasingEast

12. Constraint Systems Laboratory February 18, 2009 CSP Modeling Examples 12 Example constraint network B6 B8 B2 B4 B5 B3 B9 B10B7 B1 S1S2 S3 Si = Building = Corner building = Street S1#1,S1#4, S1#8,S2#7, S2#8,S3#1, S3#2,S3#3, S3#15

13. Constraint Systems Laboratory February 18, 2009 CSP Modeling Examples 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 1.number of courses covered 2.happiness of the GTAs Often, number of hired GTAs is insufficient

14. Constraint Systems Laboratory February 18, 2009 CSP Modeling Examples 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. Constraint Systems Laboratory February 18, 2009 CSP Modeling Examples Outline Task & Motivation System Architecture & Interfaces Scientific aspects –Problem Modeling –Problem Solving –Comparing & Characterizing Solvers Motivation revisited & Conclusions

16. Constraint Systems Laboratory February 18, 2009 CSP Modeling Examples System Architecture 1.Web-interface for applicants Password Protected Access for GTAs http://cse.unl.edu/~gta Cooperative, hybrid Search Strategies Other structured, semi-structured, or unstructured DBs In progress Visualization widgets Password Protected Access for Manager http://cse.unl.edu/~gta 2.Web-interface for manager  View / edit GTA records  Setup classes  Specify constraints  Enforce pre-assignments Local DB 3.A local relational database  Graphical selective queries Interactive Search Automated Search Heuristic BT Stochastic LS Multi-agent Search Randomized BT 4.Drivers for  Interactive assignments  Automated search algorithms

17. Constraint Systems Laboratory February 18, 2009 CSP Modeling Examples GTA interface: Preference Specification

18. Constraint Systems Laboratory February 18, 2009 CSP Modeling Examples Manager interface: TA Hiring & Load

19. Constraint Systems Laboratory February 18, 2009 CSP Modeling Examples Outline Task & Motivation System Architecture & Interfaces Scientific aspects –Problem Modeling –Problem Solving –Comparing & Characterizing Solvers Motivation revisited & Conclusions

20. Constraint Systems Laboratory February 18, 2009 CSP Modeling Examples 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. Constraint Systems Laboratory February 18, 2009 CSP Modeling Examples Outline Task & Motivation System Architecture & Interfaces Scientific aspects –Problem Modeling –Problem Solving –Comparing & Characterizing Solvers Motivation revisited & Conclusions

22. Constraint Systems Laboratory February 18, 2009 CSP Modeling Examples 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. Constraint Systems Laboratory February 18, 2009 CSP Modeling Examples Manager interface: Interactive Selection

24. Constraint Systems Laboratory Dual perspective Task-centered viewResource-centered view

25. Constraint Systems Laboratory February 18, 2009 CSP Modeling Examples 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 24 hr: 51 (26%) 1 min: 55 (20%) Max depth: 57 Shallowest level reached by BT after … Number of variables: 69 Depth of the tree: 69

26. Constraint Systems Laboratory February 18, 2009 CSP Modeling Examples 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. Constraint Systems Laboratory February 18, 2009 CSP Modeling Examples 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. Constraint Systems Laboratory February 18, 2009 CSP Modeling Examples 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. Constraint Systems Laboratory February 18, 2009 CSP Modeling Examples 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. Constraint Systems Laboratory February 18, 2009 CSP Modeling Examples 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. Constraint Systems Laboratory February 18, 2009 CSP Modeling Examples Outline Task & Motivation System Architecture & Interfaces Scientific aspects –Problem Modeling –Problem Solving –Comparing & Characterizing Solvers Motivation revisited & Conclusions

32. Constraint Systems Laboratory February 18, 2009 CSP Modeling Examples 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. Constraint Systems Laboratory February 18, 2009 CSP Modeling Examples Other sample projects 1.Graduate TA Assignment Project (GTAAP) Modeling, search, GUI 2.Temporal Reasoning Constraint propagation, search, graph theory 3.Symmetry detection Search, databases (computational) 4.Structural decompositions Databases (theory), tractability results 33

34. Constraint Systems Laboratory February 18, 2009 CSP Modeling Examples The Research Modeling & Reformulation Propagation algorithms Search algorithms Decomposition algorithms Symmetry identification & breaking 34