Imagine that you are head of testing department of a software house. In your department, there

are several testing staff who design test$-$cases for different software modules. A module holds

some $$ number of lines of code and each test case may cover some of the code lines of the module.

For example, let our very simple module contains $10$ lines of code $($i$.$e$.=10)$ and we have $=$

$6$ number of different test cases prepared for the testing the module. Table $1$ shows an example

line numbers covered by each of $6$ different example test cases.

Table $1.$ Example test$-$case IDs and their code line coverage information.

Test$-$Case Covered Line Numbers

$1\{3,4,6,8\}$

$2\{1,2,4,9,10\}$

$3$

$\{5,6,8,10\}$

$4\{2,5,8,9\}$

$5$

$\{1,4,6,7,9,10\}$

$6$

$\{1,2,3,5,8\}$

As the head of the testing department, you wish to minimize the number of applied test cases per

module in order to save testing time. On the other hand, the selected minimum number of test

cases among $$ alternatives are required to cover all the lines of module under test together in

order to not to risk some of the module lines not being tested. In other words, the selected

minimum number of test cases should cover all the code lines of the module collectively. For

example for the above problem instance, you have to select the test cases $5$ and $6$ that covers

$2$

all the $10$ module lines together. In your solution, assume that each line of the module under test

is covered by at least one test case.

In order to solve the problem, you decide to develop a software tool that uses Transform &

Conquer algorithm design technique. According to the approach, first you need to convert given

problem instance into an Integer Linear Programming $($ILP$)$ problem then solve the constructed ILP

problem by using a back$-$end solver $($namely IBM$-$Cplex Solver$)$ on the fly. Finally, display the

obtained result in a meaningful way. See Figure $1$ below.

Figure $1.$ A simple architectural view for your test case selection software.

SAMPLE INPUT:

$10//$ Number of lines of code of the module

$6//$ Number of test cases

$3468//$ Covered lines by test case number #$1$

$124910//$ Covered lines by test case number #$2$

$56810//$ Covered lines by test case number #$3$

$2589//$ Covered lines by test case number #$4$

$1467910//$ Covered lines by test case number #$5$

$12358//$ Covered lines by test case number #$6$

SAMPLE OUTPUT:

IDs of the minimum number of selected test cases that covers all the module lines: $56$Test Case Selection Problem $(100$ pts$)$:

Imagine that you are head of testing deportment of a software house. In your department, there

are several tenting staff who desige test$-$caws for different software modules. A module holds

same a number of lines of code and each test case may cover some of the code lines of the module.

For example, let our wery simple module contains $10$ lines of code $($i$.$e$.n=10)$ and we hawe $k=$

$6$ number of differemt test cases prepared for the testing the module. Table $1$ shows an example

line numbers cowred by each of $6$ different enample test caser.

Table $1.$ Erample test$-$case IDs and their code lise coverage informantion.

As the head of the testing department, you wish to minimise the number of applied test cases per

modale in order to sawe testing time. On the other hand, the selected minimum number of test

cawes among $k$ altematives are required to cover all the lines of module under test together in

order to not to risk same of the module lines not being tested. in other words, the selected

minimum number of test cases should cover all the code lines of the module colloctively. For

enample for the above problem ingtance, you have to select the test cases $T_s$ and $T_k$ that cowers

all the $10$ modale lines together. V your solation, assame that each line of the module under test

is covered by at least one test case.

In oeder to solve the problem, you decide to develop a software tool that uses Tranaform $S.$

Conquer algorithm design technigue. According to the approach, first you need to corrvert gives

problem instance into an leteger Linear Programming $($LP$)$ peoblem then solve the constructed $11$ P

problem by using a back$-$end solver $($namely ISM$-$Celer Solver$)$ on the fly. Finally, display the

obtained result in a meaningful way. See Figure $1$ below.

Figure $1$ A simple architectural view for your test case selection softeare.

SAMPLE NPUT:

SAMPLE GUTEIT.

IDs of the minimum number of selected test cases that cowers all the module lines: $56$