Set covering problem

Authors: Sherry Liang, Khalid Alanazi, Kumail Al Hamoud

Introduction

The set covering problem is a significant NP-hard problem in combinatorial optimization. In the set covering problem, two sets are given: a set U of elements and a set S of subsets of the set U. Each subset in S is associated with a predetermined cost, and the union of all the subsets covers the set U. This combinatorial problem then concerns finding the optimal number of subsets whose union covers the universal set while minimizing the total cost.1 The problem has many applications in the airline industry, and it was explored on an industrial scale as early as the 1970s.2

Problem formulation

The mathematical formulation of the set covering problem is define as follows. We define U = {Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u_i} , ….. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u_m} } as the universe of elements and S = {Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s_1} , ….,Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s_n} } as a collection of subsets such that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s_i} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \in} U and the union of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s_i} covers all elements in U (i.e. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cup} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s_i} = U ). Addionally, each set Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s_i} must cover at least one element of U and has associated cost Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c} that is larger than zero (i.e. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_i} > 0). The objective is to find the minimum cost sub-collection of sets X Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \in} S that covers all the elements in the universe U.

Integer linear program formulation

An integer linear program (ILP) model can be formulated for the minimum set covering problem as follows:

Decision variables

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y_i = \begin{cases} 1, & \text{if subset }i\text{ is selected} \\ 0, & \text{otherwise } \end{cases}}

Objective function

minimize Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{i=1}^n c_i y_i}

Constraints

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{i=1}^n a_{ij} y_i >= 1, \forall j= 1,....,m}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y_i \in \{0, 1\}, \forall i = 1,....,n}

The objective function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{i=1}^n c_i y_i} is defined to minimize the number of subset Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s_i} that cover all elements in the universe by minimizing its total cost. The first constraint implies that every element j in the universe U must be be covered where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_{ij}} =1 if j Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \in} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s_i} and 0 otherwise. The second constraint Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y_i \in \{0, 1\} } indicates that the decision variables are binary.

Set covering problems are significant NP-hard optimization problems, which means that for large size instances, integer linear program algorithms can be difficult to solve in polynomial time. Therefore, there exist approximation algorithms that can solve large scale problems in polynomial time with optimal or near-optimal solutions. One of the widely used approximation algorithms for set covering is the classical greedy algorithms.⁷

Greedy approximation algorithm

Greedy algorithms can be used to approximate for optimal or near-optimal solutions for large scale set covering instances in polynomial solvable time. The greedy heuristics applies iterative process that, at each stage, select the largest number of uncovered elements in the universe U, and delete the uncovered elements, until all elements are covered. Let T be the set that contain the covered elements, and U be the set that contain the elements of Y that still uncovered. At the beginning of the iteration, T is empty and all elements Y belong to U. We iteratively select the set of S that covers the largest number of elements in U and add it to the covered elements in T. An example of this algorithm is presented below.

Greedy algorithm for minimum set cover

Numerical Example

Let’s consider a simple example where we assign cameras at different locations. Each location covers some areas of stadiums, and our goal is to put the least amount of cameras such that all areas of stadiums are covered. We have stadium areas from 1 to 15, and possible camera locations from 1 to 8.

We are given that camera location 1 covers stadium areas {1,3,4,6,7}, camera location 2 covers stadium areas {4,7,8,12}, while the remaining camera locations and the stadium areas that the cameras can cover are given in table 1 below:

Table 1
Camera Location	Stadium Areas
1	1,3,4,6,7
2	4,7,8,12
3	2,5,9,11,13
4	1,2,14,15
5	3,6,10,12,14
6	8,14,15
7	1,2,6,11
8	1,2,4,6,8,12

To reformulate the problem, we can list the stadium areas and the camera locations that cover each stadium area. For instance, stadium area 1 can be covered with camera location {1,4,7,8}, stadium area 2 can be covered with camera location {3,4,7,8}, while the rest are given in table 2 below:

Table 2
Stadium Area	Camera Location
1	1,4,7,8
2	3,4,7,8
3	1,5
4	1,2,8
5	3
6	1,5,7,8
7	1,2
8	2,6,8
9	3
10	5
11	3,7
12	2,5,8
13	3
14	4,5,6
15	4,6

We can then represent the above information using binary values. If the stadium area Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i} can be covered with camera location Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j} , then we have Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y_{ij} = 1} . If not,Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y_{ij} = 0} . For instance, stadium area 1 is covered by camera location 1, so Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y_{11} = 1} , while stadium area 1 is not covered by camera location 2, so Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y_{12} = 0} . The binary variables Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y_{ij}} values are given in the table below:

Table 3
	1	2	3	4	5	6	7	8
1	1			1			1	1
2			1	1			1	1
3	1				1
4	1	1						1
5			1
6	1				1		1	1
7	1	1
8		1				1		1
9			1
10					1
11			1				1
12		1			1			1
13			1
14				1	1	1
15				1		1

We introduce another binary variable Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z_j} to indicate if a camera is installed at location Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j} . Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z_j = 1} if camera is installed at location Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j} , while Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z_j = 0} if not.

Our objective is to minimize Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{j=1}^8 z_j} . For each stadium, there’s a constraint that the stadium area Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i} has to be covered by at least one camera location. For instance, for stadium area 1, we have Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z_1 + z_4 + z_7 + z_8 \geqslant 1} , while for stadium 2, we have Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z_3 + z_4 + z_7 + z_8 \geqslant 1} . All the 15 constraints that corresponds to 15 stadium areas are listed below:

Constraints 1 to 15:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1. z_1 + z_4 + z_7 + z_8 \geqslant 1}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2. z_3 + z_4 + z_7 + z_8 \geqslant 1}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3. z_1 + z_5 \geqslant 1}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 4. z_1 + z_2 + z_8 \geqslant 1}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 5. z_3 \geqslant 1}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 6. z_1 + z_5 + z_7 + z_8 \geqslant 1}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 7. z_1 + z_2 \geqslant 1}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 8. z_2 + z_6 + z_8 \geqslant 1}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 9. z_3 \geqslant 1}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 10. z_5 \geqslant 1}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 11. z_3 + z_7 \geqslant 1}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 12. z_2 + z_5 + z_8 \geqslant 1}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 13. z_3 \geqslant 1}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 14. z_4 + z_5 + z_6 \geqslant 1}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 15. z_4 + z_6 \geqslant 1}

From constraint {5,9,13}, we can obtain Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z_3 = 1} . Thus we no longer need constraint 2 and 11 as they are satisfied when Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z_3 = 1} . With Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z_3 = 1} determined, the constraints left are: