Merge branch 'memory-estimation'

Author: GiggleLiu

docs/src/index.md

@@ -5,7 +5,7 @@ CurrentModule = GraphTensorNetworks
 # GraphTensorNetworks
 
 This package uses generic tensor network to compute properties of combinatorial problems defined on graph.
-The properties includes the size of the maximum set size, the number of sets of a given size and the enumeration of configurations of a given set size.
+The properties includes the size of the maximum/minimum set size, the number of sets of a given size and the enumeration of configurations of a given set size.
 
 ## Background knowledge
 

examples/IndependentSet.jl

@@ -24,15 +24,15 @@ show_graph(graph; locs=locations)
 
 # ## Tensor network representation
 # Let ``G=(V,E)`` be the target graph that we want to solve.
-# 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.
+# We 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.
 # For each label ``s_i``, we defined a parametrized rank-one vertex tensor ``W(x_i)`` as
 # ```math
 # W(x_i) = \left(\begin{matrix}
 #     1 \\
-#     x_i
-# \end{matrix}\right).
+#     x_i^{w_i}
+# \end{matrix}\right),
 # ```
-# 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``.
+# where ``W(x_i)_0=1`` is the first element associated with ``s_i=0`` and ``W(x_i)_1=x_i^{w_i}`` is the second element associated with ``s_i=1``, and `w_i` is the weight of vertex ``i``.
 # Similarly, on each edge ``(u, v)``, we define a matrix ``B`` indexed by ``s_u`` and ``s_v`` as
 # ```math
 # B = \left(\begin{matrix}

examples/MaxCut.jl

@@ -30,11 +30,13 @@ show_graph(graph; locs=locations)
 # Then the maximum cutting problem can be encoded to tensor networks by mapping an edge ``(i,j)\in E`` to an edge matrix labelled by ``s_is_j``
 # ```math
 # B(x_{\langle i, j\rangle}) = \left(\begin{matrix}
-#     1 & x_{\langle i, j\rangle}\\
-#     x_{\langle i, j\rangle} & 1
+#     1 & x_{\langle i, j\rangle}^{w_{\langle i,j \rangle}}\\
+#     x_{\langle i, j\rangle}^{w_{\langle i,j \rangle}} & 1
 # \end{matrix}\right),
 # ```
-# where variable ``x_{\langle i, j\rangle}`` represents a cut on edge ``(i, j)`` or a domain wall of an Ising spin glass.
+# If and only if there is a cut on edge ``(i, j)``,
+# this tensor contributes a factor ``x_{\langle i, j\rangle}^{w_{\langle i,j \rangle}}``,
+# where ``w_{\langle i,j\rangle}`` is the weight of this edge.
 # Similar to other problems, we can define a polynomial about edges variables by setting ``x_{\langle i, j\rangle} = x``,
 # where its k-th coefficient is two times the number of configurations of cut size k.
 # We define the cutting problem as

examples/MaximalIS.jl

@@ -29,12 +29,13 @@ show_graph(graph; locs=locations)
 # We defined the restriction on its neighbourhood ``N(v)``:
 # ```math
 # T(x_v)_{s_1,s_2,\ldots,s_{|N(v)|},s_v} = \begin{cases}
-#     s_vx_v & s_1=s_2=\ldots=s_{|N(v)|}=0,\\
+#     s_vx_v^{w_v} & s_1=s_2=\ldots=s_{|N(v)|}=0,\\
 #     1-s_v& \text{otherwise}.
 # \end{cases}
 # ```
-# Intuitively, it means if all the neighbourhood vertices are not in ``I_{m}``, i.e., ``s_1=s_2=\ldots=s_{|N(v)|}=0``, then ``v`` should be in ``I_{m}`` and contribute a factor ``x_{v}``,
-# otherwise, if any of the neighbourhood vertices is in ``I_{m}``, then ``v`` cannot be in ``I_{m}``.
+# The first case corresponds to all the neighbourhood vertices of ``v`` are not in ``I_{m}``, then ``v`` must be in ``I_{m}`` and contribute a factor ``x_{v}^{w_v}``,
+# where ``w_v`` is the weight of vertex ``v``.
+# Otherwise, if any of the neighbouring vertices of ``v`` is in ``I_{m}``, ``v`` must not be in ``I_{m}`` by the independence requirement.
 # We can use [`MaximalIS`](@ref) to construct the tensor network for solving the maximal independent set problem as
 problem = MaximalIS(graph; optimizer=TreeSA());