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examples/IndependentSet.jl

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@@ -19,21 +19,20 @@ show_graph(graph; locs=locations)
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# Let ``G=(V,E)`` be the target graph that we want to solve.
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# The tensor network representation map a vertex ``i\in V`` to a label ``s_i \in \{0, 1\}`` of dimension ``2`` in a tensor network, where we use ``0`` (``1``) to denote a vertex is absent (present) in the set.
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# For each label ``s_i``, we defined a parametrized rank-one vertex tensor ``W(x_i)`` as
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# \begin{equation}
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# W(x_i) = \left(\begin{matrix}
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# 1 \\
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# x_i
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# \end{matrix}\right).
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# \end{equation}
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# ```math
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# W(x_i) = \left(\begin{matrix}
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# 1 \\
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# x_i
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# \end{matrix}\right).
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# ```
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# We use subscripts to index tensor elements, e.g. ``W(x_i)_0=1`` is the first element associated with ``s_i=0`` and ``W(x_i)_1=x_i`` is the second element associated with ``s_i=1``.
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# Similarly, on each edge ``(u, v)``, we define a matrix ``B`` indexed by ``s_u`` and ``s_v`` as
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# \begin{equation}
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# \qquad \quad
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# B = \left(\begin{matrix}
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# 1 & 1\\
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# 1 & 0
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# \end{matrix}\right). \label{eq:edgetensor}
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# \end{equation}
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# ```math
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# B = \left(\begin{matrix}
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# 1 & 1\\
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# 1 & 0
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# \end{matrix}\right).
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# ```
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# We can use [`IndependentSet`](@ref) to construct a tensor network
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# corresponding to the independent set problem on our target graph.

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