Constraint Satisfaction Problems: Formulation

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Gilbert Walton
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1 Constraint Satisfaction Problems: Formulation Slides adapted from: 6.0 Tomas Lozano Perez and AIMA Stuart Russell & Peter Norvig Brian C. Williams 6.0- September 9 th, 00 Reading Assignments: Much of the material covered only in lecture slides. ĺ Read ALL Slides. AIMA Ch. 5 Constraint Satisfaction Problems (CSPs) AIMA Ch.. pp Visual Interpretation as solving a CSP Problem Set Problem covers constraints. Due Monday, October 6th.

2 Constraint Satisfaction Problems ueens Problem: Place queens on a x chessboard so that no How do we formulate? Chessboard positions ueen - or blank Two positions on a line (vertical, horizontal, diagonal) cannot both be Constraint Satisfaction Problem (CSP) A Constraint Satisfaction Problem is a triple <V,D,C>, where: V is a set of variables V i D is a set of variable domains, The domain of variable Vi is denoted D i C is a set of constraints on assignments to V Each constraint specifies a set of allowed variable values. Example: A,B in {,} C = {{<,><,>}} A CSP Solution: is any assignment to V, such that all constraints in C are satisfied.

3 Encodings Are Essential: ueens ueens Problem: Place queens on a x chessboard so that no How big is the encoding? Chessboard positions ueen - or blank Two positions on a line (vertical, horizontal, diagonal) cannot both be 5 Encodings Are Essential: ueens Place queens so that no What is a better formulation? Assume one queen per column. Determine what row each queen should be in.,,,, {,,, } i <> j On different rows i - j <> i-j Stay off the diagonals Example: C, = {(,) (,) (,) (,) (,) (,)} 6

4 Encodings Are Essential: ueens Place queens so that no,,,, {,,, } i <> j On different rows i - j <> i-j Stay off the diagonals Example: C, = {(,) (,) (,) (,) (,) (,)} What is C? 7 A general class of CSPs Binary CSP each constraint relates at most two variables Depict as a Constraint Graph Nodes are variables Arcs are binary constraints Unary constraint arc. Variable V i with values in domain D i Binary constraint arc Unary constraints just cut down domains 8

5 CSP Classic: Graph Coloring Pick colors for map regions, without coloring adjacent regions with the same color regions allowed colors adjacent regions must have different colors 9 Real World: Scheduling as a CSP Choose time for activities: observations on Hubble telescope jobs on machine tools terms to take required classes are activities activity 5 time sets of possible start times (or chunks of time). Activities that use the same resource cannot overlap in time. Preconditions are satisfied 0

6 Case Study: Course Scheduling Given: 0 courses (8.0, 8.0, ), and 0 terms (Fall, Spring,...., Spring 5). Find a legal schedule. Pre-requisites satisfied Courses offered on limited terms Limited number of courses taken per term (say ) Avoid time conflicts Note, traditional CSPs are not for expressing (soft) preferences e.g. minimize difficulty, balance subject areas, etc. Alternative choices for variables & values VARIABLES DOMAINS A. per Term All legal combinations of courses, (Fall ) (Spring ) (Fall ) (Spring )... all offered during that term. B. per Term-Slot subdivide each term into course slots: (Fall,) (Fall, ) (Fall, ) (Fall, ) All courses offered during that term C. per Course Terms or term-slots (Term-slots make it easier to express constraints on limited number of courses per term.)

7 Encoding Assume: = Courses, = term-slots : Prerequisite At least term before At least term after Courses offered only during certain terms For pairs of courses that must be ordered. Filter domain Term-slots not equal Limit # courses for all pairs of vars. Use term-slots only once Avoid time conflicts term not equal For course pairs offered at same or overlapping times Good News / Bad News Good News - Bad News - very general & interesting classes of problems includes NP-Hard (intractable) problems

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