Introduction

dwavebinarycsp is a library to construct a binary quadratic model from a constraint satisfaction problem (CSP) with small constraints over binary variables (represented as either binary values {0, 1} or spin values {-1, 1}).

Constraint Satisfaction Problems

Constraint satisfaction problems require that all a problem’s variables be assigned values, out of a finite domain, that result in the satisfying of all constraints.

The map-coloring CSP, for example, is to assign a color to each region of a map such that any two regions sharing a border have different colors.

image

Coloring a map of Canada with four colors.

The constraints for the map-coloring problem can be expressed as follows:

  • Each region is assigned one color only, of \(C\) possible colors.
  • The color assigned to one region cannot be assigned to adjacent regions.

Binary Constraint Satisfaction Problems

Solving such problems as the map-coloring CSP on a sampler such as the D-Wave system necessitates that the mathematical formulation use binary variables because the solution is implemented physically with qubits, and so must translate to spins \(s_i\in\{-1,+1\}\) or equivalent binary values \(x_i\in \{0,1\}\). This means that in formulating the problem by stating it mathematically, you might use unary encoding to represent the \(C\) colors: each region is represented by \(C\) variables, one for each possible color, which is set to value \(1\) if selected, while the remaining \(C-1\) variables are \(0\).

Another example is logical circuits. Logic gates such as AND, OR, NOT, XOR etc can be viewed as binary CSPs: the mathematically expressed relationships between binary inputs and outputs must meet certain validity conditions. For inputs \(x_1,x_2\) and output \(y\) of an AND gate, for example, the constraint to satisfy, \(y=x_1x_2\), can be expressed as a set of valid configurations: (0, 0, 0), (0, 1, 0), (1, 0, 0), (1, 1, 1), where the variable order is \((x_1, x_2, y)\).

Boolean AND Operation
\(x_1,x_2\) \(y\)
\(0,0\) \(0\)
\(0,1\) \(0\)
\(1,0\) \(0\)
\(1,1\) \(1\)

Binary Quadratic Models

D-Wave systems solve problems that can be mapped onto an Ising model or a quadratic unconstrained binary optimization (QUBO) problem. These can be seen as subsets of a binary quadratic model (BQM).

For example, the Boolean operations of logical gates represented as CSPs can also be represented by a particular type of BQM called a penalty model: penalty functions penalize invalid states; that is, invalid sets of input and output values representing gates have higher penalty values than valid sets.

For example, the AND gate’s constraint \(y=x_1x_2\) can be represented as penalty function

\[x_1 x_2 - 2(x_1+x_2)y +3y,\]

which penalizes invalid configurations while no penalty is applied to valid configurations.

In Table Boolean AND Operation as a Penalty, columns \(Out_{valid}\) and \(Out_{invalid}\) represent, together with the \(in\) column for each row, valid and invalid configurations of an AND gate, with columns \(P_{valid}\) and \(P_{invalid}\) the respective penalty values.

Boolean AND Operation as a Penalty.
in \(\mathbf{out_{valid}}\) \(\mathbf{out_{invalid}}\) \(\mathbf{P_{valid}}\) \(\mathbf{P_{invalid}}\)
\(0,0\) \(0\) \(1\) \(0\) \(3\)
\(0,1\) \(0\) \(1\) \(0\) \(1\)
\(1,0\) \(0\) \(1\) \(0\) \(1\)
\(1,1\) \(1\) \(0\) \(0\) \(1\)

For example, the state \(in=0,0; out_{valid}=0\) of the first row is represented by the penalty function with \(x_1=x_2=0\) and \(z = 0 = x_1 \wedge x_2\). For this valid configuration, the value of \(P_{valid}\) is

\[ \begin{align}\begin{aligned}x_1 x_2 - 2(x_1+x_2)z +3z &= 0 \times 0 -2 \times (0+0) \times 0 + 3 \times 0\\&= 0,\end{aligned}\end{align} \]

not penalizing the valid configuration. In contrast, the state \(in=0,0; out_{invalid}=1\) of the same row is represented by the penalty function with \(x_1=x_2=0\) and \(z = 1 \ne x_1 \wedge x_2\). For this invalid configuration, the value of \(P_{invalid}\) is

\[ \begin{align}\begin{aligned}x_1 x_2 - 2(x_1+x_2)z +3z &= 0 \times 0 -2 \times (0+0) \times 1 + 3 \times 1\\&= 3,\end{aligned}\end{align} \]

adding a penalty of \(3\) to the incorrect configuration.

The samples representing low energy states returned from a sampler such as the D-Wave system correspond to valid configurations, and therefore correctly represent the AND gate.

Example: Solving a Map-Coloring CSP

This example finds a solution to the map-coloring problem for a map of Canada using four colors. Canada’s 13 provinces are denoted by postal codes:

Canadian Provinces’ Postal Codes
Code Province Code Province
AB Alberta BC British Columbia
MB Manitoba NB New Brunswick
NL Newfoundland and Labrador NS Nova Scotia
NT Northwest Territories NU Nunavut
ON Ontario PE Prince Edward Island
QC Quebec SK Saskatchewan
YT Yukon    

The workflow for solution is as follows:

  1. Formulate the problem as a graph, with provinces represented as nodes and shared borders as edges, using 4 binary variables (one per color) for each province.
  2. Create a binary constraint satisfaction problem and add all the needed constraints.
  3. Convert to a binary quadratic model.
  4. Sample.
  5. Plot a valid solution, if such was found.

The following sample code creates a graph of the map with provinces as nodes and shared borders between provinces as edges (e.g., “(‘AB’, ‘BC’)” is an edge representing the shared border between British Columbia and Alberta). It creates a binary constraint satisfaction problem based on two types of constraints:

  • csp.add_constraint(one_color_configurations, variables) represents the constraint that each node (province) select a single color, as represented by valid configurations one_color_configurations = {(0, 0, 0, 1), (0, 0, 1, 0), (0, 1, 0, 0), (1, 0, 0, 0)}
  • csp.add_constraint(not_both_1, variables) represents the constraint that two nodes (provinces) with a shared edge (border) not both select the same color.
import dwavebinarycsp
from dwave.system.samplers import DWaveSampler
from dwave.system.composites import EmbeddingComposite
import networkx as nx
import matplotlib.pyplot as plt

# Represent the map as the nodes and edges of a graph
provinces = ['AB', 'BC', 'MB', 'NB', 'NL', 'NS', 'NT', 'NU', 'ON', 'PE', 'QC', 'SK', 'YT']
neighbors = [('AB', 'BC'), ('AB', 'NT'), ('AB', 'SK'), ('BC', 'NT'), ('BC', 'YT'), ('MB', 'NU'),
             ('MB', 'ON'), ('MB', 'SK'), ('NB', 'NS'), ('NB', 'QC'), ('NL', 'QC'), ('NT', 'NU'),
             ('NT', 'SK'), ('NT', 'YT'), ('ON', 'QC')]

# Function for the constraint that two nodes with a shared edge not both select one color
def not_both_1(v, u):
    return not (v and u)

# Function that plots a returned sample
def plot_map(sample):
    G = nx.Graph()
    G.add_nodes_from(provinces)
    G.add_edges_from(neighbors)
    # Translate from binary to integer color representation
    color_map = {}
    for province in provinces:
          for i in range(colors):
            if sample[province+str(i)]:
                color_map[province] = i
    # Plot the sample with color-coded nodes
    node_colors = [color_map.get(node) for node in G.nodes()]
    nx.draw_circular(G, with_labels=True, node_color=node_colors, node_size=3000, cmap=plt.cm.rainbow)
    plt.show()

# Valid configurations for the constraint that each node select a single color
one_color_configurations = {(0, 0, 0, 1), (0, 0, 1, 0), (0, 1, 0, 0), (1, 0, 0, 0)}
colors = len(one_color_configurations)

# Create a binary constraint satisfaction problem
csp = dwavebinarycsp.ConstraintSatisfactionProblem(dwavebinarycsp.BINARY)

# Add constraint that each node (province) select a single color
for province in provinces:
    variables = [province+str(i) for i in range(colors)]
    csp.add_constraint(one_color_configurations, variables)

# Add constraint that each pair of nodes with a shared edge not both select one color
for neighbor in neighbors:
    v, u = neighbor
          for i in range(colors):
        variables = [v+str(i), u+str(i)]
                    csp.add_constraint(not_both_1, variables)

# Convert the binary constraint satisfaction problem to a binary quadratic model
bqm = dwavebinarycsp.stitch(csp)

# Set up a solver using the local system’s default D-Wave Cloud Client configuration file
# and sample 50 times
sampler = EmbeddingComposite(DWaveSampler())         # doctest: +SKIP
response = sampler.sample(bqm, num_reads=50)         # doctest: +SKIP

# Plot the lowest-energy sample if it meets the constraints
sample = next(response.samples())      # doctest: +SKIP
if not csp.check(sample):              # doctest: +SKIP
    print("Failed to color map")
else:
    plot_map(sample)

The plot shows a solution returned by the D-Wave solver. No provinces sharing a border have the same color.

image

Solution for a map of Canada with four colors. The graph comprises 13 nodes representing provinces connected by edges representing shared borders. No two nodes connected by an edge share a color.

Terminology

model
A collection of variables with associated linear and quadratic biases.
sampler
A process that samples from low energy states of a problem’s objective function. A binary quadratic model (BQM) sampler samples from low energy states in models such as those defined by an Ising equation or a Quadratic Unconstrained Binary Optimization (QUBO) problem and returns an iterable of samples, in order of increasing energy. A dimod sampler provides ‘sample_qubo’ and ‘sample_ising’ methods as well as the generic BQM sampler method.