polish weighted

Author: GiggleLiu

examples/PaintShop.jl

@@ -3,7 +3,7 @@
 # !!! note
 #     It is highly recommended to read the [Independent set problem](@ref) chapter before reading this one.
 
-# ## Problme Definition
+# ## Problem Definition
 # The [binary paint shop problem](http://m-hikari.com/ams/ams-2012/ams-93-96-2012/popovAMS93-96-2012-2.pdf) is defined as follows:
 # we are given a ``2m`` length sequence containing ``m`` cars, where each car appears twice.
 # Each car need to be colored red in one occurrence, and blue in the other.

examples/saveload.jl

@@ -28,7 +28,7 @@ save_sumproduct(filename, all_independent_sets_tree)
 loaded_sets_tree = load_sumproduct(filename)
 
 # ## Loading solutions to python
-# The following python script loads and unpacks the solutions as an array from a `:binary` formatted file.
+# The following python script loads and unpacks the solutions as a numpy array from a `:binary` format file.
 # ```python
 # import numpy as np
 #

examples/weighted.jl

@@ -5,10 +5,12 @@ using GenericTensorNetworks, Graphs
 
 graph = Graphs.smallgraph(:petersen)
 
-# The following code computes the weighted MIS problem.
+# The following code constructs a weighted MIS problem instance.
 problem = IndependentSet(graph; weights=collect(1:10));
 
-# `weights` is a key word argument that can be a vector for weighted graphs or a `NoWeight()` object for unweighted graphs. The maximum independent set can be found as follows.
+# Here, the `weights` keyword argument can be a vector for weighted graphs or `NoWeight()` for unweighted graphs.
+# Most solution space properties work for unweighted graphs also work for the weighted graphs.
+# For example, the maximum independent set can be found as follows.
 
 max_config_weighted = solve(problem, SingleConfigMax())[]
 
@@ -20,14 +22,17 @@ locations = [[rot15(0.0, 1.0, i) for i=0:4]..., [rot15(0.0, 0.6, i) for i=0:4]..
 show_graph(graph; locs=locations, vertex_colors=
           [iszero(max_config_weighted.c.data[i]) ? "white" : "red" for i=1:nv(graph)])
 
-# For weighted MIS problem, a property that many people care about is the "energy spectrum", or the largest weights.
-# We just feed a positional argument in the [`SizeMax`](@ref) constructor as the number of largest weights.
+# The only solution space property that can not be defined for general real-weighted (not including integer-weighted) graphs is the [`GraphPolynomial`](@ref).
+
+# For the weighted MIS problem, a useful solution space property is the "energy spectrum", i.e. the largest several configurations and their weights.
+# We can use the solution space property is [`SizeMax`](@ref)`(10)` to compute the largest 10 weights.
 spectrum = solve(problem, SizeMax(10))[]
 
-# The return value has type [`ExtendedTropical`](@ref), which contains one field `orders`. The `orders` is a vector of [`Tropical`](@ref) numbers.
+# The return value has type [`ExtendedTropical`](@ref), which contains one field `orders`.
 spectrum.orders
 
-# We can get weighted independent sets with maximum 5 sizes.
+# We can see the `order` is a vector of [`Tropical`](@ref) numbers.
+# Similarly, we can get weighted independent sets with maximum 5 sizes as follows.
 max5_configs = solve(problem, SingleConfigMax(5))[]
 
 # The return value also has type [`ExtendedTropical`](@ref), but this time the element type of `orders` has been changed to [`CountingTropical`](@ref)`{Float64,`[`ConfigSampler`](@ref)`}`.