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Consider the following table for an airline company:
                                                     table
Each flight has an origin-destination airport $i$ and time of departure-arrival airport $i$.

Task: find the minimum number of planes required to perform all flights.

One way of solving this is with minimum network flow:
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Nodes $k=1,...5$ represent a flight, $k^\text{'}$ origin node, $k^\text{''}$ destination node, $s$ source node, $t$ sink node. Edges represent all possible valid combinations of flight paths.

Formulating the problem in the LP form:
$$\text{Min} \sum_{j} x_j$$ $$\text{subject to } x_j \in \{0,1\}$$ where $x$ is planes, $j=1,...,5$ is plane number.

Question1: how to complete the LP form?
Question2: what will the coefficients of matrices $A$ and $b$ be for this network?

I do not know of other methods for solving this task, if there is a simpler one, an explanation would be much appreciated.

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To complete the LP formulation, add an arc from $t$ to $s$ with a unit cost: a minimum cost flow will minimize the number of units of flow that go from $t$ to $s$, in other words the number of airplanes. Let $x_{ij}\in \mathbb{N}$ be the flow through arc $(i,j)$.

You then want to minimize $x_{ts}$, subject to usual flow conservation constraints.

Note. Another way of doing this is by working with the complement of your graph: add an edge between a destination node and an origin node if and only if the same airplane cannot do both flights. Minimizing the number of airplanes is then equivalent to finding the chromatic number $\chi(G)$ of the graph. By construction, this graph is an interval graph, for which greedy algorithms can optimally color the graph. For example, determine a perfect elimination order of the nodes, and sequentially color them in the reverse order will give you a coloring with exactly $\chi(G)$ colors, in linear time.

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  • $\begingroup$ Would finding the chromatic number help me in formulating a LP? i.e. Could I formulate a different/simpler LP knowing that information? $\endgroup$ – user2974951 Feb 18 '16 at 14:37
  • $\begingroup$ Yes you could, but it would not be a good idea: if you formulate an LP, you will have to solve it with integer programming, which could take a long time. $\endgroup$ – Kuifje Feb 18 '16 at 15:33
  • $\begingroup$ Suppose that is not an issue, I have the information about which node has which color, how would i proceed from here? $\endgroup$ – user2974951 Feb 19 '16 at 18:45
  • $\begingroup$ LP for the chromatic number: binary variables $y_i$ if color $i$ is used, $x_{vi}$ if node $v$ takes color $i$. You want to minimize $\sum_i y_i$ subject to $\sum_i x_{vi}=1$ for all $v$, $x_{ui}+x_{vi}\le y_i$ for all edges $(u,v)$. $\endgroup$ – Kuifje Feb 19 '16 at 19:45

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