# Map Coloring: Hybrid DQM Sampler¶

This example solves the same map coloring problem of Large Map Coloring to
demonstrate Leap’s hybrid discrete
quadratic model (DQM) solver, which enables you to solve problems
of arbitrary structure and size for variables with **discrete** values.

See Large Map Coloring for an description of the map coloring constraint satisfaction problem (CSP).

The Map Coloring advanced example demonstrates lower-level coding of a similar problem, which gives the user more control over the solution procedure but requires the knowledge of some system parameters (e.g., knowing the maximum number of supported variables for the problem). Example Problem With Many Variables demonstrates the hybrid approach to problem solving in more detail by explicitly configuring the classical and quantum workflows.

## Example Requirements¶

To run the code in this example, the following is required.

- The requisite information for problem submission through SAPI, as described in Configuring Access to D-Wave Solvers
- Ocean tools dwave-system, dimod, and dwave_networkx.

If you installed dwave-ocean-sdk
and ran `dwave setup`

, your installation should meet these requirements.
In D-Wave’s Leap IDE, the default workspace
meets these requirements.

Note

This example requires a minimal understanding of the penalty model
approach to solving problems by minimizing penalties. In short, you formulate
the problem using positive values to penalize undesirable outcomes; that is,
the problem is formulated as an objective function where desirable
solutions are those with the lowest values. This is demonstrated in
Leap’s *Structural Imbalance* demo,
introduced in the Boolean AND Gate example, and comprehensively explained in the
Problem Solving Handbook.

## Solution Steps¶

Section How a D-Wave System Solves Problems describes the process of solving problems on the
quantum computer in two steps: (1) Formulate the problem as a
binary quadratic model (BQM) or DQM and (2) Solve the BQM with a D-wave
system, hybrid or classical sampler. In this example, a DQM is created
to formulate the problem and submitted to the
Leap hybrid DQM solver,
`hybrid_binary_quadratic_model_version<x>`

.

## Formulate the Problem¶

This example uses the NetworkX
`read_adjlist()`

function to read a text file,
`usa.adj`

, containing the states of the USA and their adjacencies (states with
a shared border) into a graph. The original map information was found on
write-only blog of Gregg Lind and looks like this:

```
# Author Gregg Lind
# License: Public Domain. I would love to hear about any projects you use if it for though!
#
AK,HI
AL,MS,TN,GA,FL
AR,MO,TN,MS,LA,TX,OK
AZ,CA,NV,UT,CO,NM
CA,OR,NV,AZ
CO,WY,NE,KS,OK,NM,AZ,UT
# Snipped here for brevity
```

You can see in the first non-comment line that the state of Alaska (“AK”) has Hawaii (“HI”) as an adjacency and that Alabama (“AL”) shares borders with four states.

```
>>> import networkx as nx
>>> G = nx.read_adjlist('usa.adj', delimiter = ',') # doctest: +SKIP
```

Graph G now represents states as vertices and each state’s neighbors as shared edges.

```
>>> states = G.nodes # doctest: +SKIP
>>> borders = G.edges # doctest: +SKIP
```

You can now create a `dimod.DiscreteQuadraticModel`

class to represent
the problem. Because any planar map can be colored with four colors or fewer,
represent each state with a discrete variable that has four *cases* (binary
variables can have two values; discrete variables can have some arbitrary
number of cases).

For every pair of states that share a border, set a quadratic bias of \(1\) between the variables’ identical cases and \(0\) between all different cases (by default, the quadratic bias is zero). Such as penalty model adds a value of \(1\) to solutions of the DQM for every pair of neighboring states with the same color. Optimal solutions are those with the fewest such neighboring states.

```
>>> import dimod
...
>>> colors = [0, 1, 2, 3]
...
>>> dqm = dimod.DiscreteQuadraticModel()
>>> for state in states: # doctest: +SKIP
... dqm.add_variable(4, label=state)
>>> for state0, state1 in borders: # doctest: +SKIP
... dqm.set_quadratic(state0, state1, {(color, color): 1 for color in colors})
```

## Solve the Problem by Sampling¶

D-Wave’s quantum cloud service provides cloud-based hybrid solvers you can submit arbitrary BQMs and DQMs to. These solvers, which implement state-of-the-art classical algorithms together with intelligent allocation of the quantum processing unit (QPU) to parts of the problem where it benefits most, are designed to accommodate even very large problems. Leap’s solvers can relieve you of the burden of any current and future development and optimization of hybrid algorithms that best solve your problem.

Ocean software’s dwave-system
`LeapHybridDQMSampler`

class enables you to easily
incorporate Leap’s hybrid DQM solvers into your application.
The solution printed below is truncated.

```
>>> from dwave.system import LeapHybridDQMSampler
...
>>> sampleset = LeapHybridDQMSampler().sample_dqm(dqm) # doctest: +SKIP
...
>>> print("Energy: {}\nSolution: {}".format(
... sampleset.first.energy, sampleset.first.sample)) # doctest: +SKIP
Energy: 0.0
Solution: {'AK': 0, 'AL': 0, 'MS': 3, 'TN': 1, 'GA': 2, 'FL': 3, 'AR': 2,
# Snipped here for brevity
```

The energy value of zero above signifies that this first (best) solution found has accumulated no penalties, meaning no pairs of neighboring states with the same color.

Note

The next code requires Matplotlib.

Plot the best solution.

```
>>> import matplotlib.pyplot as plt # doctest: +SKIP
>>> node_list = [list(G.nodes)[x:x+10] for x in range(0, 50, 10)] # doctest: +SKIP
>>> node_list[4].append('ND') # doctest: +SKIP
# Adjust the next line if using a different map
>>> nx.draw(G, pos=nx.shell_layout(G, nlist = node_list), with_labels=True,
... node_color=list(sampleset.first.sample.values()), node_size=400,
... cmap=plt.cm.rainbow) # doctest: +SKIP
>>> plt.show() # doctest: +SKIP
```

The graphic below shows the result of one such run.