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I have a matrix of zeros and ones, i.e., $\mathbf{X}=[x_{ij}]$ with $x_{ij}\in\{0,1\}$ for all $i=1,2,\ldots,m$ and $j=1,2,\ldots,n$. Associated with each row $i$ of the matrix $\mathbf{X}$ a set of …

Let us say we have $m$ bins and $n$ balls. Every bin $i$ has capacity $c_i$ which is the number of balls that can be put into bin $i$. We have $c_i\geq1$ for all $i$. For each bin $i$, there is a coll …

The knapsack problem is NP-hard and can be formulated as: $$\begin{align}&\text{maximize } \sum_{i=1}^n v_i x_i,\tag{P1}\\& \text{subject to } \sum_{i=1}^n w_i x_i \leq W,\\&\text{and } x_i \in \{0,1\ …

There are $n$ jobs where each job $i$ has an arrival time $r_i$, a deadline $d_i$ and a cost $c_i$. The problem is to find a scheduling time $t_i$ (where $r_i\leq t_i\leq d_i$) for each job $i$ in ord …

Given a set of elements $U=\{1,2,\ldots,n\}$ and a collection of $m$ sets $\{S_1,S_2,\ldots,S_m\}$ whose union equals $U$. Each element $e$ of a set $S_i$ has a weight $w_i(e)$. The weight $w(S_i)$ of …

I have a set of $n$ objects $\{1,2,\ldots,n\}$ where object $i$ has weight $w(i)$ and we have a capacity $W$. I would like to pick a subset $S=\{a_1,\ldots,a_m\}\subseteq \{1,2,\ldots,n\}$ of the obje …

I have a single job of unit length, a set of $n$ slots, and a budget of $B$ units. If the job is scheduled at slot $t$, then it will consume $c(t)$ units of the budget $B$. If the job is not scheduled …

We have $m$ bins and $n$ balls. Each bin $i=1,2,\ldots,m$ can contain at most two balls (not any two balls but two balls from some specific set), see 3. Each ball $j=1,2,\ldots,n$ can be put into b …

Problem: In the generalized assignment problem with unit-value items, there are $m$ bins of capacity $C$ each. There are $n$ items where each item $i$ has weight $w_{ij}$ with bin $j$. The objective i …

Given two sets $S_1$ and $S_2$ of $n$ elements each. Each set $S_1$ (resp. $S_2$) has a revenue $R_1$ (resp. $R_2$). Each element $i$ of $S_1$ (resp. $S_2$) has a gain $g_{i1}$ (resp. $g_{i2}$). From …

The multiple choice knapsack problem (MCKP) can be defined as follows: MCKP is known to be NP-hard in general. I have a special case of MCKP for which $N_i=\{1,2,\cdots,|N_i|\}$, for all $1\leqs …

Given a set of jobs $J$ and a set of machines $M$, where the link between machine $i\in M$ and job $j\in J$ has a positive weight $w_{ij}$. The problem is to select a perfect matching between the jobs …