Cheating on the N Queens benchmark

Many Solver distributions include an N Queens example, in which n queens need to be placed on a n*n sized chessboard, with no attack opportunities. So when you’re looking for the fastest Solver, it’s tempting to use the N Queens example as a benchmark to compare those solvers. That’s a tragic mistake, because the N Queens problem is solvable in polynomial time, which means there’s a way to cheat.

That being said, OptaPlanner solves the 1 000 000 queens problem in less than 3 seconds :) Here’s a log to prove it (with time spent in milliseconds):

INFO  Opened: data/nqueens/unsolved/10000queens.xml INFO  Solving ended: time spent (23), best score (0), ...INFO  Opened: data/nqueens/unsolved/100000queens.xml INFO  Solving ended: time spent (159), best score (0), ...INFO  Opened: data/nqueens/unsolved/1000000queens.xml INFO  Solving ended: time spent (2981), best score (0), ...

# How to cheat on the N Queens problem

The N Queens problem is not NP-complete, nor NP-hard. That is math speak for stating that there’s a perfect algorithm to solve this problem: the Explicits Solutions algorithm. Implemented with a CustomPhaseCommand in OptaPlanner it looks like this:

public class CheatingNQueensPhaseCommand implements CustomPhaseCommand {

    public void changeWorkingSolution(ScoreDirector scoreDirector) {
        NQueens nQueens = (NQueens) scoreDirector.getWorkingSolution();
        int n = nQueens.getN();
        List<Queen> queenList = nQueens.getQueenList();
        List<Row> rowList = nQueens.getRowList();

        if (n % 2 == 1) {
            Queen a = queenList.get(n - 1);
            scoreDirector.beforeVariableChanged(a, "row");
            a.setRow(rowList.get(n - 1));
            scoreDirector.afterVariableChanged(a, "row");
            n--;
        }
        int halfN = n / 2;
        if (n % 6 != 2) {
            for (int i = 0; i < halfN; i++) {
                Queen a = queenList.get(i);
                scoreDirector.beforeVariableChanged(a, "row");
                a.setRow(rowList.get((2 * i) + 1));
                scoreDirector.afterVariableChanged(a, "row");

                Queen b = queenList.get(halfN + i);
                scoreDirector.beforeVariableChanged(b, "row");
                b.setRow(rowList.get(2 * i));
                scoreDirector.afterVariableChanged(b, "row");
            }
        } else {
            for (int i = 0; i < halfN; i++) {
                Queen a = queenList.get(i);
                scoreDirector.beforeVariableChanged(a, "row");
                a.setRow(rowList.get((halfN + (2 * i) - 1) % n));
                scoreDirector.afterVariableChanged(a, "row");

                Queen b = queenList.get(n - i - 1);
                scoreDirector.beforeVariableChanged(b, "row");
                b.setRow(rowList.get(n - 1 - ((halfN + (2 * i) - 1) % n)));
                scoreDirector.afterVariableChanged(b, "row");
            }
        }

    }

}

Now, one could argue that this implementation doesn’t use any of OptaPlanner’s algorithms (such as the Construction Heuristics or Local Search). But it’s straightforward to mimic this approach in a Construction Heuristic (or even a Local Search). So, in a benchmark, any Solver which simulates that approach the most, is guaranteed to win when scaling out.

# Why doesn’t that work for other planning problems?

This algorithm is perfect for N Queens, so why don’t we use a perfect algorithm on other planning problems? Well, simply because there are none!

Most planning problems, such as vehicle routing, employee rostering, cloud optimization, bin packing, …​are proven to be NP-complete (or NP-hard). This means that these problems are in essence the same: a perfect algorithm for one, would work for all of them. But no human has ever found such an algorithm (and most experts believe no such algorithm exists).

Note: There are a few notable exceptions of planning problems that are not NP-complete, nor NP-hard. For example, finding the shortest distance between 2 points can be solved in polynomial time with A*-Search. But their scope is narrow: finding the shortest distance to visit n points (TSP), on the other hand, is not solvable in polynomial time.

Because N Queens differs intrinsically from real planning problems, it is a terrible use case to benchmark.

# Conclusion

Benchmarks on the N Queens problem are meaningless. Instead, benchmark implementations of a realistic competition. A realistic competition is an official, independent competition:

1. that clearly defines a real-word use case
2. with real-world constraints
3. with multiple, real-world datasets
4. that expects reproducible results within a specific time limit on specific hardware
5. that has had serious participation from the academic and/or enterprise Operations Research community