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

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

Example: warehouse location  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

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

Why matrix model?  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?

Common constraint types  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?

Ease of problem statement  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

Improved propagation  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

Symmetry breaking  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 Alan Frisch

Indistinguishable values  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

Variable indexing  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

Conclusions  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, …