Pierre Flener (Uppsala)
Alan M. Frisch (York)
Brahim Hnich, Zeynep Kiziltan (Uppsala)
Ian Miguel, and Toby Walsh (York)


Constraint programs with one or more matrix of decision vars

Common patterns in such models


Which warehouses supply which stores?
 0/1 matrices

Open(warehouse)

Supply(warehouse,store)

Constraints



Each store has a warehouse: row sum on Supply=1
Warehouse capacity: column sum on Supply <= ci
Channelling from Supply to
Open


Combinatorial problems

BIBDs, magic squares, projective planes, …
 Design

Rack configuration, template and slab design, …

Scheduling

Classroom, social golfer, …

Assignment

Warehouse location, progressive party, …


Ease of problem statement

Side constraints, variable indexing, …

Improved constraint propagation

Symmetry breaking, indistinguishable values, linear models, …
We argue that matrix operations should become first class objects in constraint programming languages. MATLAB meet OPL?


Row or column sum

Weighted row/column sum

Single non-zero entry

Matrix sum

Scalar product

Channelling
This pretty much describes all the examples!
These constraints should be provided as language primitives?
Efficient and powerful propagators developed?


Steel mill slab design
 Nasty “colour” constraint

Stops it being simple knapsack problem

Channel into matrix model

Colour constraint easily and efficiently stated

Easy to combine models

Multiple models


Warehouse location

Either 1-d matrix,
Supply(store)=warehouse

Or 2-d matrix,
Supply(store,warehouse)=0/1
2-d matrix is purely linear so can use LP solver

Alan Frisch

Often rows or columns
(or both) are symmetric

All weeks (cols) can be permuted in a timetable

All slabs (rows) of same size can be permuted

Lex order rows/cols

See our talk in
Symmetry workshop


Values in problem can be indistinguishable

In progressive party problem:
Assign(guest,period)=host
But host boats of same size are indistinguishable

Channel into 0/1 matrix with extra dimension
Assign3(guest,period,host)=0/1

Value symmetry => variable symmetry


Use variables to index into arrays

E.g. channelling in progressive party problem
Assign3(guest,period,Assign(guest,period))=1 compared to
Assign3(guest,period,host)=1 iff Assign(guest,period)=host

Reduces number of constraints from cubic to quadratic
Hooker (and others) argue that such indexing is one of the significant advantages CP has over IP


Matrix models common

Common types of constraints posted on matrices

Row/column sum, symmetry breaking, channelling, …

Matrix operations should be made first-class objects in modelling languages

MATLAB, EXCEL, …