CPM!

It is rocket science

CPM

Just do it!

We jumped in and did it!

mantra

A constraint satisfaction problem (CSP) is a triple, (V,D,C) where V is a set of variables, each with a domain of values (in D) and a set of constraints C restricting the set of allowed values.The problem is then to find an assignment of values to variablesfrom their domains, such that the constraints are satisfied (sat) …or to show no such assignment exists (unsat)

Generally, this problem in NP-complete

looking under the hood search

looking under the hood search

Taking graph colouring as an example (simple & easy to visualise)

•chronological backtracking • backward checking• thrashing (slavishly repeating actions with same outcomes)

•chronological backtracking • forward checking• Also thrashes

•conflict directed backjumping• learning from dead-ends• check backwards or forwards

looking under the hood Arc consistency


viewing the CSP as a graph (the constraint graph)

node consistency (1-consistency) and arc consistency (2-consistency)


• AC3 for binary constraints (using revise)• Coarse grained ).( 3deO


• Fine grained, two pass algorithms• AC4, AC6, …, AC8 ).( 2deO


• Object oriented perspective (AC5)• Constraints as objects• Specialised revise function• Complexity then a function of revision


• AC2001 • A simple trick makes coarse grained optimal (when we have no semantics to exploit)


looking under the hood all different


An example of a specialised constraint•N-ary•Use of Hall sets•Example was sudoku


looking under the hood heuristics


Static, topological properties

Fail-first, and it’s realisation

Constrainedness

Dual viewpoint heuristics

Promise

Regret

Domain Specific (example JSSP slack-based)


A problemNumber Partitioning

• A variety of models• Static variable ordering heuristic• Symmetry breaking• Use of “scalar” constraint• Optimisation (minimise difference)• First experience of phase transition phenomena

A problemBin Packing

• A model presented• Modelling of bins using 0/1 variables and scalar• … and sum• Optimise as sequence of decision problems• Recognise decision variables• Effect of value ordering• First-fit decreasing heuristic• Symmetry breaking on bins

• lex or load?• Consider alternative model• Code & empirical study

A problem

Bin Packing N-Queens

Crystal Maze Job Shop Scheduling Problem

Graph Colouring

Number Partitioning

Banquet Planning

Meeting Scheduling Problem

Workforce Management

Stable Roommates

Super Tree ConstructionMatch Race Scheduling

Maximum Clique

A problem

Lecture Lecture

Lecture Lecture

Lecture

Lecture

Assessed Exercise

Assessed Exercise

Homework

For Information Only

For Information OnlyFor Information Only

Lecture

A problem

A problem

Phase Transition Phenomena Where are the hard problems?

• Ubiquitous • SAT, GCol, Number Partition, SAT, …

• Theory! (kappa)• Can use kappa to engineer heuristics!

Phase Transition Phenomena Where are the hard problems?

Discrepancy Search

• Limited Discrepancy Search … and why it might work• Improved LDS• Take discrepancies late or early?

Parallel CP (part 1)

Parallel CP (part 2)

Local Search

It’s search Jim, but not as we know it

Local Search

• TSP as a working example• Hill Climbing (HC)

• Cost function• Representation• Move operator

• HC gets stuck in local minima• Local search cannot prove optimality• Local search can be very very very fast

Local Search

Escaping local minima•Simulated annealing (SA)

• Needs an annealing schedule• Parameter need to be tuned

•Tabu Search (TS)• Tabu list and what to put in it

•Guided Local Search (GLS)•Genetic Algorithms (GA)

• Population based• Representation as chromosone• Cross over and mutation• Fitness function• Survival of the fittest

• Balance exploration with exploitation

5 lectures

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

So?

• Know about BT & FC and CBJ (high level)

• Know about levels of consistency and cost• NC, AC, PC, PIC, SAC, GAC

• Know about local search• HC, SA, TS, GA would suffice

• Discrepancy Search and Parallel Search• Why LDS? • Why parallel search?• What is best that can be had?

• Heuristics• What do they hope to achieve?• Can you suggest some?• Given a problem, what would be appropriate?

• …

So?Part 1 Preparing for the exam

• Model and Solve a given problem•Variables, domains, constraints, …•…, heuristics, symmetry break, alternative model

• Have a look at Ciaran’s slides on allDifferent

• If you were given a piece of choco3 code could you understand it?•“name that code”

• If you were given some variables, with domains, and constraints•Could you make the variables arc consistent?

• Have a look at the homework•Why not do the homework.?

• The last “Three Problems”•Ignore these. These were to give you insight to research in CP•This is a Masters course after all

• …


• Follow up the links on Barbara Smith’s AR33 notes


30 lectures were given

One of your challenges for the future is to “do more with less”.

You are now in a better position to do that.

Many of the really important, and unseen, computational challengesare hard: resource allocation, packing, production scheduling, workforcemanagement, routing problems, transportation problems, matching, rostering,timetabling, energy use, energy planning ….

Few people stop to think how these problems are actually addressed, if at all.

You have looked at some of these problems. You have tried to model and solve them, sometimesusing complete search (bt, fc, cbj, mac), quasi-complete (lds) and incomplete (local search),finding appropriate heuristics and trying to locate “the really hard problems”

Last Words

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CPM!