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3.1
Graph Minimizer
The graph minimizer solves an energy minimization problem where we have n
nodes \(N_i\), with each node having several possible solutions.
Every solution has a self energy \(E_{self}\) and pairwise energies in between nodes
are possible. The goal is to select exactly one solution per node to obtain
a set \(X=[x_1, x_2, ..., x_n]\) that minimizes:
\[F(X)=\displaystyle\sum_iE_{self}(N_i[x_i]) +\displaystyle \sum_i \displaystyle \sum_{j>i}E_{pair}(N_i[x_i], N_j[x_j])\]

class
promod3.core. GraphMinimizer

AddNode (self_energies)
Adds a node to the graph.
Parameters:  self_energies (list of float ) – Directly controls the number of possible solutions in that
node and assigns the corresponding self energies 
Returns:  The idx of the added node 
Return type:  int 

AddEdge (node_idx_one, node_idx_two, pairwise_energies)
Adds an edge between the specified nodes.
Parameters: 
 node_idx_one (
int ) – Index of the first node
 node_idx_two (
int ) – Index of the second node
 pairwise_energies (
list of list of float ) – The pairwise energies that contribute to the
overall energy function. There must be a list for
every possible solution in node one. All of those
lists must have exactly the length of possible
solutions in node two.

Returns:  The idx of the added edge

Return type:  int

Raises:  RuntimeError if node_idx_one or node_idx_two
specify non existent nodes or when pairwise_energies is
inconsistent with the number of solutions in the specified nodes.


ApplyDEE (node_idx[, e_cut=0.0])
Applies dead end elimination on one particular node and potentially
deactivates certain solutions. The goldstein criterion is described in
[goldstein1994].
Parameters: 
 node_idx (
int ) – Node to apply dead end elimination
 e_cut (
float ) – If set to
0.0, the default goldstein criterion is applied =>
a solution is removed if it’s dominated by all other
solutions in the same node. If you increase this value,
a solution must be dominated by at least this e_cut.

Returns:  bool whether any solution has been deactivated.


ApplyEdgeDecomposition (edge_idx, epsilon)
Applies edge decomposition on one particular edge and potentially
deactivates it. The exact decomposition procedure is described in
[krivov2009].
Parameters: 
 edge_idx (
int ) – Edge to decompose.
 epsilon (
float ) – The energy threshold to perform edge decomposition.

Returns:  bool whether the edge has been decomposed and
deactivated


Prune (epsilon[, e_cut=0.0, consider_all_nodes=False])
Performs edge decomposition followed by dead end elimination in an
iterative manner until no changes can be observed anymore given
epsilon.
Parameters: 
 epsilon (
float ) – The energy threshold to perform edge decomposition.
 e_cut (
float ) – Parameter to control dead end elimination.
 consider_all_nodes (
bool ) – Flag, wether the dead end elimination should be
applied to all nodes, or only those who are
connected with an edge removed by edge
decomposition. This is useful if already a Prune
operation has been performed => only those nodes
connected to a removed edge have the chance for
successful dead end elimination.


Reset ()
Resets the graph by undoing any pruning operation (DEE and edge decomposition)

TreeSolve ([max_complexity=inf, initial_epsilon=0.02])
The method solves a graph using a minimal width tree decomposition
approach in an iterative manner. In every iteration, the algorithm performs
a pruning step (DEE / Edge Decomposition) and subsequently tries to solve
each connected component in the graph separately.
If the number of possible enumerations in the tree constructetd from a
particular connected component is is larger max_complexity,
this component is solved in a later iteration. The algorithm iterates until
all connected components are solved and steadily increases the epsilon value
resulting in a more and more agressive edge decomposition.
Algorithm further descsribed in [krivov2009].
Parameters: 
 max_complexity (
int ) – Max number of possible permutations, that have to
be enumerated in the resulting tree after tree
decomposition.
 initial_epsilon (
float ) – The initial energy threshold to perform edge
decomposition.

Returns:  A tuple with the first element being a list of indices
representing the single solutions minimizing
the overall energy function.
The second element is the according energy value.


AStarSolve (e_thresh[, max_n=100, max_visited_nodes=100000000])
The method solves a graph using the A* algorithm. Besides creating the
minimal energy solution, the function produces a maximum of max_n
solutions sorted by energy. It aborts as soon as it sees the first solution
with an energy difference of e_thresh to the optimal solution or hits
max_n. If you’re only interested in the optimal solution you should use
the TreeSolve function since it’s much faster and uses less memory.
There is no automatic pruning of the graph using DEE or edge decomposition,
so you have to do it by yourself, otherwise you’ll have a looooooong
runtime or even hit the max_visited_nodes parameter that caps the memory
usage.
To have a valid solution you have to take care that you set the e_cut
parameter in the pruning function to e_tresh.
Algorithm is described in [leach1998].
Parameters: 
 e_thresh (
float ) – Maximal energy difference of a solution to the
optimal solution to be considered.
 max_n (
int ) – The maximum number of solutions that will be generated.
 max_visited_nodes (
int ) – Caps the memory usage of the algorithm. Besides
The memory used for pairwise energies and self
energies, the algorithm uses about
max_visited_nodes * 20 bytes.

Returns:  A tuple with the first element being a list of
solutions. The second element is a list
of energies for the according solutions.


MCSolve ([n=100, mc_steps=100000, start_temperature=1000.0, change_frequency=1000, cooling_factor=0.9, seed=0])
Does a total of n Monte Carlo runs to find low energy solutions
of the graph. Each run starts with a random
configuration. At each of the mc_steps steps, a random node and
a random solution at that location is selected and an energy difference
of that random selection relative to the current configuration is
estimated. If the difference in energy is negative, the step is
accepted. If not, the step is accepted with a probability given by
the temperature dependent Metropolis criterion
\(exp^{\left(\frac{e_{diff}}{T}\right)}\).
The temperature for every run starts with start_temperature
and is multiplied every change_frequency steps with cooling_factor
to achieve a simulated annealing effect.
There is no automatic pruning of the graph using DEE or edge decomposition,
you have to do that by yourself.
Parameters: 
 n (
int ) – Number of Monte Carlo runs
 mc_steps (
int ) – Number of Monte Carlo steps per run
 start_temperature (
float ) – Start temperature for the temperature dependent
Metropolis criterion
 change_frequency (
int ) – Number of steps the temperature stays the same
 cooling_factor (
float ) – Factor to multiply temperature each
change_frequency steps
 seed (
float ) – Seed for random number generator

Returns:  A tuple with the first element being a list of
solutions. The second element is a list
of energies for the according solutions.


NaiveSolve ()
Don’t even think of using this... This function only exists for debug
purposes and does the full enumeration of the solution space.
It might become faster with the appropriate
techniques.
Returns:  A tuple with the first element being a list of indices
representing the single solutions minimizing
the overall energy function.
The second element is the according energy value. 

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