Question

I’m implementing a combinatorial optimization algorithm in Haskell:

Given an initial candidate solution, repeat until stopping criteria are met:

  1. Determine possible moves
  2. Evaluate possible moves
  3. Choose a move
  4. Make move, record new candidate solution, update search state

I could write functions for steps 1-4 and chain them together inside a recursive function to handle looping and passing state from one iteration to the next, but I have a vague idea that monads apply.

What’s the best way to express this kind of procedure in Haskell?

Answer 1

The best way to express this sort of iterative procedure in Haskell is as an infinite list of each successive result. Piecing together your four steps yields a notion of a function from a solution to a different (better) solution; all you need to do is apply this infinitely many times. The user of your function can then use any list function to get the answer: solve s0 !! numIterations, or find stoppingCondition $ solve s0, or whatever you want.

In order to get here, let’s write out the types for each of these functions.

1. moves :: Solution -> [Move]
   Given a possible solution, figure out the possible changes you can make.
2. value :: Solution -> Move -> Double
   Given a solution and a move, evaluate it and record that value as some real number.
3. choose :: Solution -> [Move] -> Move
   Given a solution and a list of moves, pick the best one.
4. apply :: Solution -> Move -> Solution
   Given a move, apply it to an existing solution to get a new one.

You want to write a function with a type something like solve :: Solution -> (Solution -> Bool) -> Solution which takes an initial solution and a stopping condition to execute your algorithm.

Instead, let’s make this an infinite list; this means that you’ll just remove the predicate and have Solution -> [Solution].

import Data.Ord
import Data.List

-- moves, value, and apply are domain-specific
choose :: Solution -> [Move] -> Move
choose s ms = maximumBy (comparing $ value s) ms

solve :: Solution -> [Solution]
solve = iterate $ \s -> apply s . choose s $ moves s

Here, the key is iterate :: (a -> a) -> a -> [a], which repeatedly applies a function to a value and gives you the results—exactly the description of your algorithm.

However, the way I’d really write this would be the following:

import Data.Ord
import Data.List

solve :: Ord o => (s -> [m]) -> (s -> m -> o) -> (s -> m -> s) -> s -> [s]
solve moves value apply = iterate step
  where step   s = apply s . choose s $ moves s
        choose s = maximumBy (comparing $ value s)

The advantage of this is that you can reuse this same generic structure for any problem domain. All you need to do is to provide the moves, value, and apply functions! And depending on my mood, I might rewrite that as this:

import Control.Applicative
import Data.Ord
import Data.List

solve :: Ord o => (s -> [m]) -> (s -> m -> o) -> (s -> m -> s) -> s -> [s]
solve moves value apply = iterate step
  where step   = (.) <$> apply <*> choose <*> moves
        choose = maximumBy . comparing . value

Here, we use applicative notation to say that we’re effectively just doing (.) apply choose moves (which is just apply . choose $ moves) in a context where each of those functions is implicitly passed a parameter s (the reader applicative). If we really wanted to tersify things, we could write

import Control.Applicative
import Data.Ord
import Data.List

solve :: Ord o => (s -> [m]) -> (s -> m -> o) -> (s -> m -> s) -> s -> [s]
solve moves value apply =
  iterate $ (.) <$> apply <*> maximumBy . comparing . value <*> moves

Any of these snippets will do exactly what you need. (Proviso: there are no effects/monads in any of your functions, so randomness is out. You make this monadic easily, though.)

---

Just for kicks, though, let’s think about the State monad. This represents a computation with some sort of environment, so that State s a is isomorphic to s -> (a,s)—something which can see the state and potentially update it. Here, all the Solution ->s on the left of your function signatures would disappear, as would the -> Solutions on the right. That would leave you with

1. moves :: State Solution [Move]
2. value :: Move -> State Solution Double
3. choose :: [Move] -> State Solution Move
4. apply :: Move -> State Solution ()

This means that you would have some monadic action step:

import Control.Applicative
import Control.Monad.State
import Data.Ord
import Data.List

choose :: [Move] -> State Solution Move
choose = let val m = do v <- value m
                        return (m,v)
         in fst . maximumBy (comparing snd) <$> mapM val ms

step :: State Solution ()
step = apply =<< choose =<< moves

You could make this more point-free, or make it polymorphic just as above, but I won’t do that here. The point is that once you have step, you can generate answers with runState . last $ replicateM_ numIterations step, or given a whileM function, runState $ whileM (stoppingCondition :: State Solution Bool) step. Again, the user can decide how to stop it. Your moves and value functions would probably query the state with get :: State s s; apply would probably use modify :: (s -> s) -> State s () to tweak the state without needing to pull it back out. You can see the similarity with the structure from above in these types; and in fact, you can see that structure in the definition of step, as well. Each one says “string together apply, choose/value, and moves“, which is the definition of your algorithm.

---

The take-home message from both of these is that you want to avoid explicit loops/recursion, as you so rightly realized. If you think about this algorithm imperatively, then the State monad seems like a natural structure, as it hides exactly those imperative features you were thinking of. However, it has downsides: for instance, everything has become monadic, and—worst of all—functions other than apply are able to change the saved solution. If you instead imagine this algorithm as producing a new result each time, you get the notion of step :: Solution -> Solution, and from there you can use iterate to get a well-behaved infinite list.