Download presentation

Presentation is loading. Please wait.

Published byJaden Dunlap Modified over 4 years ago

1
Constraint Programming Peter van Beek University of Waterloo

2
Applications n Reasoning tasks: ä abductive, diagnostic, temporal, spatial n Cognitive tasks: ä machine vision, natural language processing n Combinatorial tasks: ä scheduling, sequencing, planning

3
Constraint Programming n CP = solve problems by specifying constraints on acceptable solutions n Why CP? ä constraints often a natural part of problems ä once problem is modeled using constraints, wide selection of solution techniques available

4
Constraint-based problem solving n Model problem ä specify in terms of constraints on acceptable solutions ä define variables (denotations) and domains ä define constraints in some language n Solve model ä define search space / choose algorithm –incremental assignment / backtracking search –complete assignments / stochastic search ä design/choose heuristics n Verify and analyze solution

5
n Model problem ä specify in terms of constraints on acceptable solutions ä define variables (denotations) and domains ä define constraints in some language n Solve model ä define search space / choose algorithm –incremental assignment / backtracking search –complete assignments / stochastic search ä design/choose heuristics n Verify and analyze solution Constraint-based problem solving Constraint Satisfaction Problem

6
Constraint satisfaction problem n A CSP is defined by ä a set of variables ä a domain of values for each variable ä a set of constraints between variables n A solution is ä an assignment of a value to each variable that satisfies the constraints

7
n Model problem ä specify in terms of constraints on acceptable solutions ä define variables (denotations) and domains ä define constraints in some language n Solve model ä define search space / choose algorithm –incremental assignment / backtracking search –complete assignments / stochastic search ä design/choose heuristics n Verify and analyze solution Constraint-based problem solving

8
Example: Assembly line sequencing n What order should the cars be manufactured? n Constraints: ä even distributions ä changes in colors ä run length constraints

9
Example: Scheduling What is the earliest that they can all set of for school? Four students, Algy, Bertie, Charlie, and Digby share a flat. Four newspapers are delivered. Each student reads the newspapers in a particular order and for a specified amount of time. Algy arises at 8:30, Bertie and Charlie at 8:45, Digby at 9:30.

10
Schedule Sun 8 am91011 Express Guardian FT

11
Example: Graph coloring Given k colors, does there exist a coloring of the nodes such that adjacent nodes are assigned different colors

12
Example: 3-coloring Variables: v 1, v 2, v 3, v 4, v 5 Domains: {1, 2, 3} Constraints: v i v j if v i and v j are adjacent v2v2 v3v3 v1v1 v5v5 v4v4

13
Example: 3-coloring One solution: v 1 1 v 2 2 v 3 2 v 4 1 v 5 3 v2v2 v3v3 v1v1 v5v5 v4v4

14
Example: Boolean satisfiability Given a Boolean formula, does there exist a satisfying assignment (an assignment of true or false to each variable such that the formula evaluates to true)

15
Example: 3-SAT Variables: x 1, x 2, x 3, x 4, x 5 Domains: {True, False} Constraints: ( x 1 x 2 x 4 ), ( x 2 x 4 x 5 ), ( x 3 x 4 x 5 ) ( x 1 x 2 x 4 ) ( x 2 x 4 x 5 ) ( x 3 x 4 x 5 )

16
Example: 3-SAT One solution: x 1 False x 2 False x 3 False x 4 True x 5 False ( x 1 x 2 x 4 ) ( x 2 x 4 x 5 ) ( x 3 x 4 x 5 )

17
Example: n-queens Place n-queens on an n n board so that no pair of queens attacks each other

18
Example: 4-queens 4 3 2 1 x1 x1 x2x2 x3 x3 x4 x4 Variables: x 1, x 2, x 3, x 4 Domains: {1, 2, 3, 4} Constraints: x i x j and | x i - x j | | i - j |

19
Example: 4-queens One solution: x 1 2 x 2 4 x 3 1 x 4 3 4 3 2 1 x1 x1 x2x2 x3 x3 x4 x4 Q Q Q Q

20
Search tree for 4-queens x1 x1 x2x2 x3 x3 x4 x4 1234 (1,1,1,1)(4,4,4,4)(2,4,1,3)(3,1,4,2)

21
Specification of forward checking Invariant:1 i c, c j n, ( x j, x i ) is arc-consistent x1 x1 x c-1 xc xc x c+1 xn xn currentpastfuture

22
Forward checking {2}{5} Q Q Q 1 1 11 1 1 1 1 1 2 2 2 22 2 3 33 33 {3} {1,4,6} {1,3,4} {3,4,6} x1 x1 x5 x5 x6x6 x3 x3 x4 x4 x2x2 432156 x1 x1 x5 x5 x6x6 x3 x3 x4 x4 x2x2 1

23
Enforcing arc-consistency {a, b, c} < xi xi xj xj

24
Enforcing path-consistency < {a, b, c} xi xi xj xj xk xk < <

25
Search graph for 4-queens (1,1,1,1) (1,1,4,2) (2,4,1,3) (3,1,4,2) (1,1,1,2) (1,1,1,3) (1,1,1,4) (1,4,1,3) (4,1,4,2) 0 0 4 6 1 3 4 1 1

26
Stochastic search Q Q Q x1 x1 Q Initial assignment x2 x2 x3 x3 x4 x4 112 Pick queen in conflict Q Q Q x1 x1 Q x2 x2 x3 x3 x4 x4 Move to min. conflicts Q Q Q x1 x1 Q x2 x2 x3 x3 x4 x4 02 Pick queen in conflict Q Q Q x1 x1 Q x2 x2 x3 x3 x4 x4 2

27
Tractability NP NP-Complete P (SAT, TSP, ILP, CSP, …)

28
Reducibility NP-Complete 3-SAT ILP CSP binary CSP (0,1)-ILP

29
CSP, binary CSP, SAT, 3-SAT, ILP,... n Model and solve in one of these languages n Model in one language, translate into another to solve Options

Similar presentations

Presentation is loading. Please wait....

OK

Chapter 5: Constraint Satisfaction ICS 171 Fall 2006.

Chapter 5: Constraint Satisfaction ICS 171 Fall 2006.

© 2018 SlidePlayer.com Inc.

All rights reserved.

To make this website work, we log user data and share it with processors. To use this website, you must agree to our Privacy Policy, including cookie policy.

Ads by Google

Ppt on nazism and hitler in germany Ppt on acid-base titration experiment Ppt on written business communication Ppt on thyroid function test Ppt on case study of company Play ppt on ipad 2 Ppt on diode as rectifier diodes Ppt on wireless power transmission system Ppt on causes of land pollution Ppt on cross sectional analysis of financial statement information