Back from my trip with a simple problem to explain, but a hard reduction to do.
The problem: NxN checkers. This is problem GP10 in the appendix.
The description: Given a position on an NxN checkerboard, does black have a forced win? It turns out the reduction will also work if we restrict the board to only having kings on the board (and so no “un-kinged” pieces)
Example: The “NxN” requirement is there since on a standard 8×8 checkerboard, there is a finite set of moves, and so theoretically you could solve the problem in O(1) time (for a really large constant factor, of course). The starting configuration adds extra rows and columns of pieces to the board, still leaving two blank rows in between the two pieces.
So, let’s do an example on a 4×4 board. The starting configuration is this:
(the dashes are empty spaces, * is Black, O is White)
Here is a configuration of pieces that will lead to a black win:
If it’s Black’s turn, they should move the piece in the second row up to either location on the first row (recall that all pieces are kings). Then White’s only move is to go to the space Black just vacated, where it will be jumped, giving Black the win.
The paper by Fraenkel, Garey, Johnson, Schaefer, and Yesha contains a pretty detailed description of the reduction, which contains lots of complicated structures. I’ll just give the general idea here.
The reduction is going to be from Geography, which is still NP-Complete even if the graph is bipartite and planar. They create several structures to help them build their instance of the checkers game.
The first is what the call a phalanx– an open rectangle of (say) White kings that surround the (say) Black pieces. The idea is that since there is no way for the Black pieces to jump anything in the rectangle, then White can “shrink” the phalanx towards Black, running them out of room to maneuver. Here is a picture of a small phalanx on a 6×6 board:
..notice that whatever Black does, they will be captured on their next turn. This remains true no matter how many Black pieces are trapped inside the phalanx, and no matter how much open space is inside the phalanx (White can use their moves to shrink it over time).
The key to the reduction is to build a set of interlocking “potential” phalanxes- situations where a Black king may be able to escape the phalanx. If it can, Black can jump White’s pieces and win, but if it can’t, the phalanx will engulf Black and they will lose. The geography instance is placed in the center of these potential phalanxes in such a way that a Black king can “escape” the Geography instance if and only if Black can win the geography game. The reason why the Geography graph had to be planar was so that we could directly represent the vertices in the graph as positions on the checkerboard. The reason why the Geography graph had to be bipartite was so that edges going from the first vertex set to the second could be all Black pieces, but the edges going from the second set to the first could be all White pieces.
The game starts with black at the “start vertex” for the geography problem, and jumping a line of White checkers:
When a vertex has more than one possible exit, that leads to more than one possible set of checkers to jump for the other player:
(This is part of figure 10 from the paper. Here, after White jumps down the chain of Black pieces, Black can choose the chain of White pieces to jump through.)
The construction takes advantage of the rule in checkers (which I was not aware of until I was in my twenties!) that if a player can make a jump, they must make a jump. So as long as players can jump checkers along these chains (alternately, as long as they can follow edges in the geography graph), they will. As soon as a player cannot make a jump they will be able to deal with the Black king that can either escape the phalanx structure (and win for Black) or trap it (and win for White).
This is the general idea of the reduction, there are a lot of details that I am glossing over.
Difficulty: 8. This is a bit hard to see and very hard to come up with, and it’s very easy to get lost in the weeds of the details. I do like the way that the “removal” of edges from the Geography problem is modeled by the actual removal of pieces from the checkerboard, though.