Graph Optimization - Lab 1 - ORLAB - Operations Research

Graph Optimization - Lab 1ORLAB - Operations Research Laboratory

Stefano Gualandi

October 24, 2011

Stefano Gualandi Graph Optimization - Lab 1

What you do during lab sessions

Algorithms Solution(s)

Modeling Interpretation

Math Model

Data

Solver

Problem Strategy


Check this out: http://challenge.roadef.org/2012/en/


http://challenge.roadef.org/2012/en/

What you do during lab sessions

Algorithms Solution(s)


Math Model

Data

Solver

Problem Strategy


Check this out: http://challenge.roadef.org/2012/en/


http://challenge.roadef.org/2012/en/

WARNINGS

I Warning 1: Lab sessions complement exercise sessions

I Warning 2: read carefully the error messages!I Common pitfalls:

I unbounded variablesI missing constraintsI missunderstood constraintsI wrong quantification in the summation of constraintsI wrong objective function


A short introduction to AMPL

In AMPL, models, data, and scripts are written in three separatetext (ASCII) files:

I FILENAME.mod: it contains the model of the problem, i.e.,variables, constraints, and objective function;

I FILENAME.dat: it contains the data specifing an instance ofthe problem, i.e., the coefficients of the constraints andobjective function

I FILENAME.run: it contains script commands that typicallyselect the solver, set parameters, solve the problem, anddisplay the solution in a format of your choice. It can be usedto implement basic algorithms.

As editor, you can use Notepad, Emacs (with syntax highlight), orGusek (based on SciiTech).


AMPL keywords

I set: defines set of elements

I param: defines constant parameters like constants, vectors, ormatrices

I var: defines continuous variables. They can be defined alsoas

I integerI binary

I sum: defines a linear sum

I subject to: defines a family of constraints

I minimize or maximize: defines the objective function

Check this out: http://www.ampl.com/FAQ/index.html


http://www.ampl.com/FAQ/index.html

AMPL support for scripting

I let ...: change the value of a set or parameter

I for {set expression} ...: iterate over the elements of aset

I if <condition> then ... else ...: check conditional

I repeat { ... } while (condition): loop while a givencondition is verifies

I repeat { ... } until (condition): loop until a givencondition is verifies

I Uniform(a,b): draw a random number in the interval [a...b]

I Uniform01(): draw a random number in the interval [0...1]

Check this out: http://www.ampl.com/NEW/newlang.html


http://www.ampl.com/NEW/newlang.html

Invoking AMPL commands

You have two options:

1. Use the AMPL shell, and type the following commands (goodfor learning the basic steps):

I ampl: option solver cplex;I ampl: model YOUR PATH/YOUR FILE.mod;I ampl: data YOUR PATH/YOUR FILE.dat;I ampl: solve;I ampl: display x;

2. Use the script command file by DOS or LINUX shell (good forrapid math modeling)

I % ampl YOUR FILE.mod YOUR FILE.dat YOUR FILE.run;


Example 1: Cabled Networks


Example 2: Wireless Networks

https://nodes.wlan-si.net/network/topology/Stefano Gualandi Graph Optimization - Lab 1

https://nodes.wlan-si.net/network/topology/

Dealing with Directed Graphs

param n > 0; # Number of nodes

set N := 1..n; # Set of nodes

set A within N cross N; # Set of arcs

# Forward and Backward star of each node

set FS {i in N} := setof {j in N: (i,j) in A} (i,j);

set BS {i in N} := setof {j in N: (j,i) in A} (j,i);

# In and Out degree of each vertex

param in_degree {i in N} := card(BS[i]);

param out_degree {i in N} := card(FS[i]);

# Example of node weight vector

param w {N} default 1;

# Example of arc capacity vector

param u {A} default 1;


Graph examples

data;

param n := 5;

param: A : u :=1 2 21 3 44 3 22 3 55 1 32 5 2;


Print a Directed Graph

printf "\nFORWARD STARS:\n";

for {i in N} {

printf "node %d: ", i;

for {(j,l) in FS[i]}

printf "(%d,%d)[u=%d] ", j, l, u[j,l];

printf "\n";

}

printf "\nBACKWARD STARS:\n";

for {i in N} {

printf "node %d: ", i;

for {(j,l) in BS[i]}

printf "(%d,%d)[u=%d] ", j, l, u[j,l];

printf "\n";

}

printf "\nDEGREES \n";

display out_degree, in_degree;

Command line:

ampl directed-graphs.mod digraph-1.dat print-digraph.run


Warm up (or Weak up!)

Start with the exercise 1 on the handouts


Exercise 3: kruskal.run

repeat {

let MinCostEdges := {(i,j) in F: c[i,j] == min{(u,v) in F} c[u,v]};

let F := F diff MinCostEdges;

for {(i,j) in MinCostEdges} {

if cc[i] == 0 then {

if cc[j] == 0 then { # Case 1: neither vertex is in the tree.

let cc[i] := max{v in N} cc[v] + 1;

let cc[j] := cc[i]; let T := T union {(i,j)};

} else { # Case 2: j is in the tree, but i isn’t.

let cc[i] := cc[j]; let T := T union {(i,j)};

}

} else {

if cc[j] == 0 then { # Case 3: i is in the tree, but j isn’t.

let cc[j] := cc[i]; let T := T union {(i,j)};

} else if cc[i] <> cc[j] then { # Case 4: i and j are in different connected components

for {v in N: cc[v] == cc[j]} let cc[v] := cc[i];

let T := T union {(i,j)};

} else printf "Adding edge (%d,%d) would create a cycle.\n\n",i,j;

}

}

} while (card(T) < n - 1);

printf "A minimum MST has cost %d.\n",sum{(i,j) in T} c[i,j];


Challenge 2: FarmVille



You begin the game with a 12 by 12 piece of land. That gives you144 one by one squares in which to plant crops. To simplify theexercise, we have the following assumptions

1. Restrict my attention to the four crops initially available(strawberries, eggplant, wheat, and soybeans);

2. Assume crops will be harvested as soon as they are ripe (thatis, ignore wilting);

3. Disregard the random bonuses and gifts that show up everynow and then.

The cost column includes the cost of plowing the land (15 coins)plus the cost of purchasing the seeds. Because the harvest times ofall four crops are multiples of four hours, we can think in terms of4-hour periods.



You initially have 200 coins, one ripe and one half-grown square ofstrawberries, one ripe and one half-grown square of eggplant. Byharvesting the two ripe squares, your fortune grows to 323 coins.We also go ahead and clean (delete) the two half-grown squares sothat we can begin with a clean slate.

Question: What is the largest amount of cash one can obtain in 6days? (need to rest on the 7th day).


Links and email

I Lab sessions web site:http:

//www-dimat.unipv.it/~gualandi/graph-optimization/

I E-mail:stefano.gualandi [at] unipv.it


http://www-dimat.unipv.it/~gualandi/graph-optimization/

http://www-dimat.unipv.it/~gualandi/graph-optimization/

Graph Optimization - Lab 1 - ORLAB - Operations Research

Documents