CPM!
Just do it!
JChoco
We jumped in, and took it for a spin
Some counting problems
Meeting scheduling problem … you got your hands dirty
JChoco
Meeting scheduling problem … you got your hands dirty
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Model and solve
Reflect on alternative models
Symmetries
Heuristics
Pre-processing
Equitable graph colouring
• with side constraints
A surprisingly rich problem!
Opening up the hood: part 1, search
Opening up the hood: part 1, search
So how does CP work?
BT: Chronological BackTracking
Thrashing!
FC: forward checking
Thrashing!
CBJ: Conflict-directed BackJumping
Thrashing!
Opening up the hood: part 2, arc-consistency
Opening up the hood: part 2, arc-consistency
So how does CP work?
Arc-consistency
Definition (aka 2-consistency)
Covered 1-2-3-Consistency
Algorithms
AC3
AC4/6/2001
AC5
Is AC a decision procedure?
modelling
We modelled some problems
Meeting scheduling problem
Team allocation
number partitioning
m-queens
75 integer sequence
magic square
sudoku
jobshop scheduling
Ramsey number
bin packing
Crossword Puzzle
Orthogonal Latin Squares
Crystal Maze
knight’s tour
Round-robin scheduling
n-queens
graph colouring
We modelled some problems
Dual or Hidden Variable encoding?
We modelled some problems
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Variables & their domains
Constraints
Identifying the decision variables
Size of the model
• number of variables, number of constraints
• Use big-O
Size of the search space (worst case)
• Use big-O
Symmetries
• And how to eliminate if possible
Heuristics
Alternative models
• Swap the perspective variable/values?
• Replace constraints with variables?
• Did this in crossword puzzle
• Zero/one possible?
• Did this in team allocation and others
Channeling between models
• See above
phase transition phenomena
Where are the hard problems?
Constrainedness? Who cares?
In optimisation, as a heuristic, in solving problems
heuristics
Variable and value ordering heuristics
Static, topological properties
Fail-first, and it’s realisation
Constrainedness
Dual viewpoint heuristics
Promise
Regret
Domain Specific (example JSSP slack-based, Warnsdorff)
Specialised constraints
specialised constraints
• max
• sub-tour elimination
• Knight’s tour
•…
• the glass box
local search
Local search (aka neighbourhood search)
HC, SA, TS, GLS, GA, ACO, …
Why we need them
escape local optima
Problems in using them
incompleteness, move operators,
evaluation/fitness functions,
tuning parameters,
Problems in using them in CP
aka meta-heuristics
Search again
Limited Discrepancy Search
Search again
lds
• motivation for LDS
• when might we use it?
• when should we not use it?
• what happens when problem is unsat?
Parallel search
• Amdahl’s law
• How much speed up can we get?
• Satisfiable v Unsatisfiable problems
• Optimisation
• Superlinear?
• BitSet parallelism
• How to split up work
• Static
• Embarrassingly
• Dynamic work stealing/donation
change of representation
change of representation
dual and hidden variable encoding
Crossword puzzle, turn constraints into variables
Interchange values and variables
Zero/one?
SAT encoding of Graph Colouring
Levels of consistency
consistency
• Node Consistency (NC)
• Arc-consistency (AC)
• Path Consistency (PC)
• Generalised arc-consistency (GAC)
• Bounds consistency
• Inverse Path Consistency (IPC aka PIC)
• Singleton Arc-consistency (SAC)
• … and others
Mechanics of choco
Mechanics of choco
homework
Ramsey & max
Local Search demo
m-queens
75 integers
summary
summary
theory
• csp <v,c,d> and it’s complexity
• search (bt, fc, cbj, …)
• thrashing
• arc-consistency
• levels of consistency
• heuristics
• local search & lds & parallel search
• dual & hidden variables
• sat
• phase transition phenomena & constrainedness
Modelling & Solving Problems
practice
number partitioning
magic square
jobshop scheduling
Crystal Maze
Meeting Scheduling
Orthogonal Latin Squares
n-queens
graph colouring
bin packing
knight’s tour
Modelling & Solving Problems
practice
• variables (enumerated/bound/setsVar) and their domains
• constraints (neq, allDiff, ifOnlyIf, …)
• decision variables
• what propagation might take place
• size of the encoding/model (how it scales with problem size)
• heuristics (dynamic/static, variable/value)
• size of the search/state space
• how search goes (example was knight’s tour)
• alternative models (dual? hidden? values as variables?)
• optimisation (with maximise/minimise, seqn of decision problems)
• dealing with conflicts (soft constraints & penalties)
• symmetry breaking (ramsey, bin packing, …)
• redundant constraints (magic square, MOLS …)
• what will make problems hard to solve and what will make them easy?
Where to next?
Where to next?
• smarter (explanation/interaction)
• local search (comet & beyond)
• exploiting multi-cores (Moore’s law ends July 2017)
• in propagation
• in search
• easier to use (Miguel & Frisch)
• don’t need a BSc, MSc, MRes, PhD
• a killer app
• constraint spread sheet?
• something on a phone?
• you