## Rummikub Solver

From: andrew cooke <andrew@...>

Date: Thu, 12 Jan 2023 16:48:36 -0300

I have been working on a solver for Rummikub, but I haven't actually
made much progress.  This email is to note down some of my ideas and
related issues.

The underlying insight is that this is a graph colouring problem.  The
nodes of the graph are the known tiles.  Edges connect any two tiles
that can appear "next to each other" in a group.  The colouring
identifies each group (a successful colouring meets the appropriate
rules).

There may be an efficiency gain in using the previous colouring as a
starting point when searching for a new one.

This is fine as far as it goes.  It becomes more problematic when you
include jokers.

A naive first approach is to include the jokers as additional nodes.
The problem is that these are connected to *every* other node.  This
complicates hugely the search process when trying t find a colouring.

In practice, when you play, you tend to separate things into two
stages: first, liberating a joker and second, using that joker.

Using this approach the joker tiles become "normal" tiles in the graph
(ie take their assumed values) when searching for liberation.  Then,
in the next step, thereare many possible graphs (one for each possible
value of the joker).

I currently don't see how to handle this explosion in the number of
graphs.

So I'm stuck. It seems like the colouring view doesn't reduce the
combinatorial complexity sufficiently.  Or doesn't allow for the
complexity to be reduced using a "natural" compromise.

Andrew

### Better Approach

From: andrew cooke <andrew@...>

Date: Sat, 21 Jan 2023 19:56:42 -0300

In the note above I modeled the game as a graph with colourings.  But
things kinda fell apart because I needed multiple graphs.

After thinking some more, it may be better to ditch colourings
altogether and simply search for graphs.  These graphs have
connections only where there are "played" relationships between the
tiles.

Now, the problem with this is that there are an awful lot of possible
graphs.  How do you generate them efficiently (where efficiency
includes memory, speed, and accuracy (generating no or few
"impossible" graphs))?

It seems like there may be a sweet-spot implementation that uses
association tables where each row is in fact a list.  Each entry in
the list describes a locally-consistent set of neigbours (so, for
example, a neigbour that belongs to a run and another that belongs to
a group would not appear in the list together).

To first aproximation the graphs we want to generate are then all the
different combinations possible by selecting one list entry from each
node.

But that ignores remote (contrast w local) consistency - if a node A
has selected node B as part of a run then node B must select A as part
of a run too (for example).

So one can imagine a search process in which selecting a list entry
for one node constraints the available list entries in neghbouring
nodes (in fact, constraints can cascade across multiple connections,
which complicaes things somewhat).  I think this would be most
efficient as a "push" - then the entry is selected all neighbours are
corrected immediately.

Thinking in more detail about the implementation, each list entry
might have a tag that indicates whether it is a run or group locally.
That tag could then have a third value - unused - when filtered by
remote constraints (and maybe a fourth indicating that the tile is
unused).

A further advantage of this approach is that jokers are more naturally
handled (their "assumed value" can be stored in the list entry?).

Andrew