Constraint Programming Solver

The approach

The implementation follows the ideas described in Chapter 12, "Concepts, Techniques, and Models of Computer Programming" by Peter Van Roy and Seif Haridi.

An overview of CP implementation in Mozart/Oz.

Status

Proof of concept. Not suitable for use in production. Significant API changes and core implementation rewrites are expected.

Intro

Implemented constraints

• equal, not_equal, less_or_equal
• all_different
• sum
• element, element2d
• circuit

Features

• views (linear combinations of variables in constraints)
• solving constraint satisfaction (CSP) and constrained optimization (COP) problems
• parallel search
• pluggable search strategies
• distributed solving

Installation

The package can be installed by adding fixpoint to your list of dependencies in mix.exs:

def deps do
  [
    {:fixpoint, "~> 0.8.4"}
  ]
end

Usage

Getting started

Let's solve the following constraint satisfaction problem:

Given two sets of values

x = {1,2}, y = {0, 1}

, find all solutions such that $x$ $\neq$ $y$

First step is to create a model that describes the problem we want to solve. The model consists of variables and constraints over the variables. In this example, we have 2 variables $x$ and $y$ and a single constraint $x$ $\neq$ $y$

alias CPSolver.IntVariable
alias CPSolver.Constraint.NotEqual
alias CPSolver.Model
## Variable constructor takes a domain (i.e., set of values), and optional parameters, such as `name`
x = IntVariable.new([1, 2], name: "x")
y = IntVariable.new([0, 1], name: "y")
## Create NotEqual constraint
neq_constraint =  NotEqual.new(x, y)

Now create an instance of CPSolver.Model:

model = Model.new([x, y], [neq_constraint])

Once we have a model, we pass it to CPSolver.solve/1,2.

We can either solve asynchronously:

## Asynchronous solving doesn't block 
{:ok, solver} = CPSolver.solve(model)
Process.sleep(10)
## We can check for solutions and solver state and/or stats,
## for instance:
## There are 3 solutions: {x = 1, y = 0}, {x = 2,  y = 0}, {x = 2, y = 1} 
iex(46)> CPSolver.solutions(solver)
[[1, 0], [2, 0], [2, 1]]

## Solver reports it has found all solutions    
iex(47)> CPSolver.status(solver)
:all_solutions 

## Some stats
iex(48)> CPSolver.statistics(solver)
%{
  elapsed_time: 2472,
  solution_count: 3,
  active_node_count: 0,
  failure_count: 0,
  node_count: 5
}

, or use a blocking call:

iex(49)> {:ok, results} = CPSolver.solve_sync(model)
{:ok,
 %{
   status: :all_solutions,
   statistics: %{
     elapsed_time: 3910,
     solution_count: 3,
     active_node_count: 0,
     failure_count: 0,
     node_count: 5
   },
   variables: ["x", "y"],
   objective: nil,
   solutions: [[2, 1], [1, 0], [2, 0]]
 }}

API

#################
# Solving       
#################
# 
# Asynchronous solving.
# Takes CPSolver.Model instance and solver options as a Keyword. 
# Creates a solver process that runs asynchronously
# and could be controlled and queried for produced solutions and/or status as it runs.
# The solver process is alive even after the solving is completed.
# It's the responsibility of a caller to dispose of it when no longer needed.
# (by calling CPSolver.dispose/1)
  
{:ok, solver} = CPSolver.solve(model, solver_opts)

# Synchronous solving.
# Takes CPSolver.Model instance and solver options as a Keyword. 
# Starts the solver and gets the results (solutions and/or solver stats) once the solver finishes.
{:ok, solver_results} = CPSolver.solve_sync(model, solver_opts)

, where

• model - specification of the model;
• solver_opts (optional) - solver options.

Model specification

For CSP (constraint satisfaction problem):

model = CPSolver.Model.new(variables, constraints)

, where

• variables is a list of variables up to a concrete implementation.

Currently, the only implementation supported is for variables over integer finite domain.

• constraints is a list of constraints.

  Available constraints

For COP (constraint optimization problem):

model = CPSolver.Model.new(variables, constraints, objective: objective)

The same as for CSP, but with additional :objective option. The objective is constructed by using CPSolver.Objective.minimize/1 and CPSolver.Objective.maximize/1.

Example of COP model

Configuring solver

Available options:

• solution_handler: function()

  A callback that gets called performed every time the solver finds a new solution. The single argument is a list of tuples

  {variable_name, variable_value}

• timeout: integer()

  Time to wait (in milliseconds) for terminating CPSolver.solve_sync/2 call. Defaults to 30_000.

• stop_on: term() | condition_fun()

  Condition for stopping the solving. Currently, only {:max_solutions, max_solutions} condition is available. Defaults to nil.

• search: {variable_choice(), value_choice()}

  Search strategy.

• space_threads: integer()

  Defines the number of processes for parallel search. Defaults to 8.

• distributed: boolean() | [Node.t()]

  If true, all connected nodes will participate in distributed solving. Alternatively, one can specify the sublist of connected nodes. Defaults to false.

Distributed solving

Fixpoint allows to solve an instance of CSP/COP problem using multiple cluster nodes.

Note: Fixpoint will not configure the cluster nodes! It's assumed that each node has the cluster membership and the fixpoint dependency is installed on it. The solving starts on a 'leader' node, and then the work is distributed across participating nodes. The 'leader' node coordinates the process of solving through shared solver state.

Let's collect all solutions for 8-Queens problem using distributed solving.

For demonstration purposes, we will spawn peer nodes like so:

iex --name leader --cookie solver -S mix

### Let's spawn 2 worker nodes...
worker_nodes = Enum.map(["node1", "node2"], fn node -> 
  {:ok, _pid, node_name} = :peer.start(%{name: node, longnames: true, args: ['-setcookie', 'solver']})
  :erpc.call(node_name, :code, :add_paths, [:code.get_path()])
  node_name
end)

Then we'll pass spawned worker nodes to the solver:

## To convince ourselves that the solving runs on worker nodes, we'll use a solution handler:
solution_handler = fn solution -> IO.puts("#{inspect Enum.map(solution, fn {_name, solution} -> solution end)} <- #{inspect Node.self()}") end 
{:ok, _solver} = CPSolver.solve(CPSolver.Examples.Queens.model(8), 
  distributed: worker_nodes, 
  solution_handler: solution_handler)

Search

Fixpoint allows to specify strategies for searching for feasible and/or optimal solutions.

This is controlled by :search option, which is a tuple {variable_choice, value_choice}.

Generally, variable_choice is either an implementation of variable selector, or an identificator of out-of-box implementation that fronts such an implementation.

Likewise, value_choice is either an implementation of value partition, or an identificator of out-of-box implementation.

Available standard search strategies:

• For variable_choice:

  • :first_fail : choose the unfixed variable with smallest domain size
  • :input_order : choose the first unfixed variable in the order defined by the model

• For value_choice

  • :indomain_min, :indomain_max, :indomain_random : choose minimal, maximal and random value from the variable domain, respectively

Default search strategy is {:first_fail, :indomain_min}

The choice of search strategy may significantly affect the performance of solving,

as the following example shows:

Let's use some out-of-box strategies for solving an instance of Knapsack problem,

alias CPSolver.Examples.Knapsack
## First, use the default strategy
{:ok, results} = CPSolver.solve_sync(Knapsack.tourist_knapsack_model())
results.statistics

%{
  elapsed_time: 689543,
  solution_count: 114,
  active_node_count: 0,
  failure_count: 1614,
  node_count: 3455
}

## Now, use the :indomain_max for the value choice. 
## Decision variables for items have {0,1} domain, where 1 means that the item will be packed.
## Hence, :indomain_max tells the solver to try to include the items first
## before excluding them.
##
{:ok, results} = CPSolver.solve_sync(Knapsack.tourist_knapsack_model(), search: {:first_fail, :indomain_max})

iex(main@zephyr.local)21> results.statistics
%{
  elapsed_time: 301501,
  solution_count: 14,
  active_node_count: 0,
  failure_count: 693,
  node_count: 1413
}

The solution time for :indomain_max is more than twice less compared to the default value choice strategy

Examples

Reindeer Ordering

Shows how to put together a model that solves a simple riddle.

N-Queens

Classical N-Queens problem

Sudoku

No explanation needed :-)

SEND+MORE=MONEY

Cryptoarithmetics problem - a riddle that involves arithmetics.

Knapsack

Constraint Optimization Problem - packing items so they fit the knapsack and maximize the total value. Think Indiana Jones trying to fill his backpack with treasures in the best way possible :-)

Quadratic Assignment

Constraint Optimization Problem - assign facilities to locations so the cost of moving goods between facilities is minimized.

Travelling Salesman problem

https://en.wikipedia.org/wiki/Travelling_salesman_problem

xkcd comic

Two combinatorial problems from https://xkcd.com/287/