Skip to content

9.6 An ILP model for the Subset Sum Problem

Part 2 · Formulations and their properties · Chapter 9 · Chapter 9: Knapsack problems · Lecture notes, volume 2 (PDF) · Slides · Exercises: Knapsack problems · App: LP relaxation and upper bounds for the 0/1 KP

The Subset Sum Problem and its reference instance (Example 2.2) are defined in Chapter 2 «Knapsack problems», Section 2.3.

  • We introduce the following \(n\) binary variables:

    \[\begin{equation*} \label{BIN_VAR_SSP} x_j = \left\{ \begin{array}{ll} 1 & \textrm{if item $j$ is selected in the subset}\\[1.5ex] 0 & \textrm{otherwise} \end{array} \qquad \forall j \in \mathscr{I}. \right. \tag{9.20} \end{equation*}\]

Using these variables, an Integer Linear Programming (ILP) model for the Subset Sum Problem (SSP) reads as follows:

\[\begin{align*} && \max \;\;{\sum_{j \in \mathscr{I}} w_j \, x_j} \label{obj_SSP} \tag{9.21a}\\[1ex] && \text{subject to} \quad {\sum_{j \in \mathscr{I}} w_j \, x_j} &\leq c, \label{cons1_SSP} \tag{9.21b}\\[1ex] &&x_j &\in \{0,1\}, & \forall j \in \mathscr{I}. \label{cons2_SSP} \tag{9.21c} \end{align*}\]

This ILP model is referred to as the \(\mathrm{ILP}_{\mathrm{SSP}}\) model.

  • Description of the objective function and constraints of the \(\mathrm{ILP}_{\mathrm{SSP}}\) model:

    • the linear objective function \(\eqref{obj_SSP}\) maximizes the total weight of the items selected in the subset;

    • constraint \(\eqref{cons1_SSP}\) ensures that the total weight of the selected items does not exceed the target value (one linear constraint);

    • finally, constraints \(\eqref{cons2_SSP}\) define the variables of the ILP model.

  • The \(\mathrm{ILP}_{\mathrm{SSP}}\) model is the \(\mathrm{ILP}_{\mathrm{KP}}\) model with \(p_j = w_j\), for every \(j \in \mathscr{I}\): the answer to the SSP is yes if and only if \(z(\mathrm{ILP}_{\mathrm{SSP}}) = c\).

Example 9.10: \(\mathrm{ILP}_{\mathrm{SSP}}\) model

We consider a target value \(c = 9\) and a set \(\mathscr{I} = \{1,2,3,4,5,6\}\) of \(6\) items (\(n = 6\)), with the following weights:

\[ \begin{array}{l|rrrrrrr} \hline & {j = 1} & {j = 2} & {j = 3} & {j = 4} & {j = 5} & {j = 6} \\ \hline {w_j} & {2} & {3} & {6} & {7} & {5} & {9} \\ \hline \end{array} \]

This is the SSP instance of Example 2.2. The \(\mathrm{ILP}_{\mathrm{SSP}}\) model for this SSP instance is:

\[ \begin{array}{rrrrrrrrrrrrr} {\max} & 2 \, {x_1} & + 3 \, {x_2} & + 6 \, {x_3} & + 7 \, {x_4} & + 5 \, {x_5} & + 9 \, {x_6} \\[0.5ex] \text{subject to} & 2 \, {x_1} & + 3 \, {x_2} & + 6 \, {x_3} & + 7 \, {x_4} & + 5 \, {x_5} & + 9 \, {x_6} & {\le} & 9 \\[0.5ex] & {x_1}, & {x_2}, & {x_3}, & {x_4}, & {x_5}, & {x_6} & { \in} & { \{0,1\}} \end{array} \]

An optimal solution \(~{\tilde{\vet x}}(\mathrm{ILP}_{\mathrm{SSP}})\) is:

\[ \tilde x_1=1,~~\tilde x_2=0,~~\tilde x_3=0,~~\tilde x_4=1,~~\tilde x_5=0,~~\tilde x_6=0 \]

The optimal value \(z(\mathrm{ILP}_{\mathrm{SSP}})\) is \(9 = c\), so the answer to the SSP is yes. The associated subset is \(\tilde{S} =\{1,4\}\). The optimal solution is not unique: the subsets \(\{2,3\}\) and \(\{6\}\) also reach the target value.

Next: 9.7 An ILP model for the Bounded Knapsack Problem