## COMP9414: Artificial Intelligence Assignment 1: Constraint Optimization

This assignment concerns developing optimal solutions to a scheduling problem inspired by the scenario of hosting a number of visitors to an organization such as a university department. Each visitor must have a number of meetings, however there are both constraints on availability (of rooms and hosts), and preferences of each visitor for the days and times of each meeting. Some of the constraints are “hard” constraints (cannot be violated in any solution), while the preferences are “soft” constraints (can be satisfied to more or less degree). Each soft constraint has a cost function giving the “penalty” for scheduling the meeting at a given time (lower costs are preferred). The aim is to schedule all the required meetings so that the sum total of all the penalties is minimized, and all the constraints are satisfied.

More technically, this assignment is an example of a constraint optimization problem, a problem that has constraints like a standard Constraint Satisfaction Problem (CSP), but also costs as- sociated with each solution. For this assignment, you will implement a greedy algorithm to find optimal solutions to visitor hosting problems that are specified and read in from a file. However, unlike the greedy search algorithm described in the lectures on search, this greedy algorithm has the property that it is guaranteed to find an optimal solution for any problem (if a solution exists).

You must use the AIPython code for constraint satisfaction and search to develop a greedy search method that uses costs to guide the search, as in heuristic search. The search will use a priority queue ordered by the values of the heuristic function that give a cost for each node in the search. The heuristic function for use in this assignment is defined below. The nodes in the search are CSPs, i.e. each state is a CSP with variables, domains and the same constraints (and a cost estimate). The transitions in the state space implement domain splitting subject to arc consistency. A goal state is an assignment of values to all variables that satisfies all the constraints.

A CSP for this assignment is a set of variables representing meetings, binary constraints on pairs of meetings, and unary constraints (hard or soft) on meetings. The domains are all working hours in one week, and meetings are all assumed to be 1 hour duration. Days are represented (in the input and output) as strings ‘mon’, ‘tue’, ‘wed’, ‘thu’ and ‘fri’, and times are represented as strings ‘9am’, ‘10am’, ‘11am’, ‘12pm’, ‘1pm’, ‘2pm’, ‘3pm’ and ‘4pm’. The only possible values are a combination of a day and time, e.g. ‘mon 9am’. Each meeting name is a string (with no spaces), and each constraint is hard or soft.

The possible constraints are as follows:

# binary constraints

constraint, ⟨m1⟩ before ⟨m2⟩

constraint, ⟨m1⟩ same-day ⟨m2⟩

constraint, ⟨m1⟩ one-day-between ⟨m2⟩

constraint,⟨m1⟩one-hour-between⟨m2⟩ #1hourbetweenendm1andstartm2

# 1 full day between m1 and m2

# hard domain constraints

domain, ⟨m⟩, ⟨day⟩, hard

domain, ⟨m⟩, ⟨time⟩, hard

domain, ⟨m⟩, ⟨day⟩ ⟨time⟩-⟨day⟩ ⟨time⟩, hard # day-time range domain, ⟨m⟩, morning, hard # finishes at or before 12pm domain, ⟨m⟩, afternoon, hard # starts on or after 12pm

domain, ⟨m⟩, before ⟨day⟩, hard domain, ⟨m⟩, before ⟨time⟩, hard domain, ⟨m⟩, before ⟨day⟩ ⟨time⟩, hard domain, ⟨m⟩, after ⟨day⟩, hard domain, ⟨m⟩, after ⟨time⟩, hard domain, ⟨m⟩, after ⟨day⟩ ⟨time⟩, hard

# soft domain constraints domain, ⟨m⟩, early-week, soft domain, ⟨m⟩, late-week, soft domain, ⟨m⟩, early-morning, soft domain, ⟨m⟩, midday, soft domain, ⟨m⟩, late-afternoon, soft

Each soft constraint has a cost function, defining a “penalty” for only partially satisfying the constraint. For example, a soft constraint that a meeting be early-week is satisfied if the meeting is on Tuesday, but with a cost (of 1, as defined below). Costs are always integers. The cost functions are defined as follows:

early-week(d, t): the number of days from mon to d (0 if d = mon) late-week(d, t): the number of days from d to fri (0 if d = fri) early-morning(d, t): the number of hours from 9am to t

midday(d, t): the number of hours from 12pm to t late-afternoon(d, t): the number of hours from t to 4pm

Finally, to define the cost of a solution (that may only partially satisfy the soft constraints), add the costs associated with each soft constraint. Let V be the set of variables and C the set of constraints. Suppose a soft constraint c applies to variable v and let (day(v), time(v)) be the value assigned to v in a solution S. For example, c might be early-morning which applies to variable m1 (v) and the value assigned to m1 might be mon 10am. Let costc be the cost function for the constraint (defined above). Then:

cost(S) = ?c∈C costc(day(v), time(v))

### Heuristic

In this assignment, you will implement greedy search using a priority queue to order nodes based on a heuristic function h. This function must take an arbitrary CSP and return an estimate of the distance from any state S to a solution. So, in contrast to a solution, each variable v is associated with a set of possible values (the current domain).

The heuristic estimates the cost of the best possible solution reachable from a given state S by assuming each variable can be assigned the value which minimizes the total cost of the soft constraints applying to that variable. For example, the cost of a meeting with two soft constraints, early-week and early-morning, is 2 if the meeting is assigned Tuesday (cost 1) at 10am (cost 1).

The heuristic function adds these minimal costs over the set of all variables. Let S be a CSP with variables V and let the domain of v be dom(v). Suppose Cv are the constraints that apply to variable v. Then:

h(S) = ?v∈V min(day(v),time(v))∈dom(v)(?c∈Cv costc(day(v), time(v))) Implementation

Use the Python code for generic search algorithms in searchGeneric.py. This code includes a class Searcher with a method search that implements depth-first search using a list (treated as a stack) to solve any search problem (as defined in searchProblem.py). For this assignment, modify the AStarSearcher class that extends Searcher and makes use of a priority queue to store the frontier of the search. Order the nodes in the priority queue based on the cost of the nodes calculated using the heuristic function.

Use the Python code in cspProblem.py, which defines a CSP with variables, domains and con- straints. Add costs to CSPs by extending this class to include a cost and a heuristic function h to calculate the cost. Also use the code in cspConsistency.py. This code implements the transitions in the state space necessary to solve the CSP. The code includes a class Search with AC from CSP that calls a method for domain splitting. Every time a CSP problem is split, the resulting CSPs are made arc consistent (if possible). Rather than extending this class, write a new class Search with AC from Cost CSP that has the same methods but implements domain splitting over constraint optimization problems.

Put your main code in one Python file called cspOptimizer.py. You should also submit any other files from AIPython needed to run your program (see below). Your program should read input from a file passed as an argument and print output to standard output.

Sample Input

All input will be a sequence of lines defining a number of meetings, binary constraints and domain constraints, in that order. Comment lines (starting with a ‘#’ character) may also appear in the file, and your program should be able to process and discard such lines. All input files can be assumed to be of the correct format – there is no need for any error checking of the input file.

Below is an example of the input form and meaning. Note that you will have to submit at least three input test files with your assignment. These test files should include one or more comments to specify what scenario is being tested.

# two meetings with one binary constraint and the same domain constraints meeting, m1

meeting, m2

# one binary constraint

constraint, m1 before m2

# domain constraints

domain, m1, mon, hard

domain, m2, mon, hard

domain, m1, early-morning, soft domain, m2, early-morning, soft

Sample Output

Print the output to standard output as a series of lines, giving a day and time for each meeting (in the order the meetings were defined). If the problem has no solution, print ‘No solution’. When

there are multiple optimal solutions, your program should produce one of them. Important: For auto-marking, make sure there are no extra spaces at the ends of lines, and no extra line at the end of the output. Set all display options in the AIPython code to 0.

The output corresponding to the above input is as follows:

m1:mon 9am

m2:mon 10am

cost:1

### Submission

• Submit all your files using the following command (this includes relevant AIPython code): give cs9414 ass1 cspOptimizer.py search*.py csp*.py display.py *.txt

• Your submission should include:

– Your .py source file(s) including any AIPython files needed to run your code

– A series of .txt files (at least three) that you have used as input files to test your system (each including comments to indicate the scenarios tested), and the corresponding .txt output files (call these input1.txt, output1.txt, input2.txt, output2.txt, etc.)

• When your files are submitted, a test will be done to ensure that your Python files run on the CSE machine (take note of any error messages printed out)

• Check that your submission has been received using the command: cs9414 classrun -check ass1

Assessment

Marks for this assignment are allocated as follows: • Correctness (auto-marked): 10 marks

• Programming style: 5 marks

Late penalty: 3 marks per day or part-day late off the mark obtainable for up to 3 (calendar) days after the due date

Assessment Criteria

• Correctness: Assessed on standard input tests, using calls of the form: python3 cspOptimizer.py input1.txt > output1.txt

• Programming style: Understandable class and variable names, easy to understand code, good reuse of AIPython code, adequate comments, suitable test files

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