\begin{algorithm}
 \caption{}\label{alg2}
\begin{algorithmic}[1]
\State Let $\mathcal{J}=\{I_1,I_2,\ldots,I_m\}$ be the set of subintervals formed the chore $[0,1]$.
\State  Solve the following linear program:

\begin{align}\label{eq1}
\min \quad
& \sum_{i,j =1}^{n} \sum_{k=1}^m x_{j,I_k} V_{i,j}(I_k)
\end{align}
s.t.
\begin{align}
\sum_{i=1}^n x_{i,I_k}&
= 1 && \forall k\in [m] \label{eq:2} \\
x_{i,k} &\geq 0&& \forall i\in N, \forall k\in [m] \label{eq:3}\\
\sum_{k=1}^m \sum_{j=1}^n x_{j,I_k}V_{i,j}(I_k)&\leq \frac{1}{n} && \forall i\in N \label{eq:4}
\end{align}
 \begin{equation} \label{eq:5}
\begin{split}
\sum_{k=1}^m x_{i,I_k}V_{i,i}(I_k)+x_{j,I_k} V_{i,j}(I_k) & \leq\sum_{k=1}^m x_{j,I_k}V_{i,i}(I_k)+x_{i,I_k} V_{i,j}(I_k)\\ & \forall i,j \in N
\end{split}
\end{equation}
\State Return an allocation which for all $i\in N$ and $I_k\in \mathcal{J}$ allocates an $x_{i,k}$ fraction of subinterval $I_k$ to agent $i$.
\end{algorithmic}
\end{algorithm}

\begin{algorithm}
\caption{}\begin{algorithmic}
[1]
\State Let $\mathcal{J}=\{I_1,I_2,\ldots,I_m\}$ be the set of subintervals formed the chore $[0,1]$.
\State Solve the following linear program:
\begin{align*}
\min \quad
& \sum_{i,j =1}^{n} \sum_{k=1}^m x_{j,I_k} V_{i,j}(I_k)
\end{align*}
s.t.
\begin{align*}
\sum_{i=1}^n x_{i,I_k}&
= 1 && \forall k\in [m] \\
x_{i,k} &\geq 0&& \forall i\in N, \forall k\in [m] \\
\sum_{k=1}^m \sum_{j=1}^n x_{j,I_k}V_{i,j}(I_k)&\leq \frac{1}{n} && \forall i\in N \end{align*}
\begin{equation*}
\begin{split}
\sum_{k=1}^m x_{i,I_k}V_{i,i}(I_k)+x_{j,I_k} V_{i,j}(I_k) & \leq\sum_{k=1}^m x_{j,I_k}V_{i,i}(I_k)+x_{i,I_k} V_{i,j}(I_k)\\ & \forall i,j \in N
\end{split}
\end{equation*}
\State Return an allocation which for all $i\in N$ and $I_k\in \mathcal{J}$ allocates an $x_{i,k}$ fraction of subinterval $I_k$ to agent $i$.
\end{algorithmic}
\end{algorithm}
{
    "title": "externalities in chore division",
    "id": "2303.12446",
    "abstract": "the chore division problem simulates the fair division of a heterogeneous, undesirable resource among several agents. in the fair division of chores, each agent only gets the disutility from its own piece. agents may, however, also be concerned with the pieces given to other agents; these externalities naturally appear in fair division situations. we first demonstrate the generalization of the classical concepts of proportionality and envy-freeness while extending the classical model by taking externalities into account.",
    "categories": "cs.gt cs.ai cs.ma",
    "doi": "",
    "created": "2023-03-22",
    "updated": "2024-02-24",
    "authors": [
        "mohammad azharuddin sanpui"
    ],
    "affiliation": [],
    "url": "https://arxiv.org/abs/2303.12446"
}
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