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

• 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

number partitioning

jobshop scheduling

magic square

n-queens

bin packing knight’s tour

Ramsey number

Orthogonal Latin Squares

Crystal Maze

graph colouring

m-queens

Team allocation

Crossword PuzzleRound-robin scheduling

Meeting scheduling problem

75 integer sequence

sudoku

What a

re th

e decision varia

bles?

Are there any symmetries?

Is there a variable/value ordering heuristic?

Is there another model?

We modelled some problems

How does search go?

Redundant constr

aints?

Dual or Hidden Variable encoding?

• 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

We modelled some problems

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

Search againLocal 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 againLimited Discrepancy Search

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

dual and hidden variable encoding

change of representation

Crossword puzzle, turn constraints into variablesInterchange values and variablesZero/one?SAT encoding of Graph Colouring

Levels of 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

consistency

Mechanics of choco

Mechanics of choco

homework

Ramsey & max

Local Search demo

m-queens

75 integers

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

summary

practice

number partitioning

jobshop scheduling

magic square

n-queens

bin packing

knight’s tour

Meeting Scheduling Orthogonal Latin Squares

Crystal Maze

graph colouring

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?

Modelling & Solving Problems

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