SAT formula for connected graphs on the grid

Question

In the answer to an earlier question "SAT algorithm for determining if a graph is disjoint" a formula is constructed that is satisfiable iff a given graph is connected.

The formula uses a lot of variables, as basically a breadth first search is performed and the variables code the number of steps needed to reach a vertex from the origin (which is an arbitrary, fixed vertex).

Here I want to specialise this to graphs that are actually embedded on the grid. The vertices are the points of a $m\times n$ grid, and edges are only between the east/north/west/south neighbours in the grid.

Is there a solution that is more variable-efficient, using the restricted topology?

score 0 · Accepted Answer · answered Jul 20 '21 at 00:22

0

I added a new answer, https://cs.stackexchange.com/a/142450/755, to that question. Applied to your situation, my solution uses $O(mn \lg(m+n))$ variables, which is asymptotically better than Yuval's algorithm (which uses $mn(m+n)$ variables).

answered Jul 20 '21 at 00:22

D.W.

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Thanks. Now I had to solve a dilemma: I decided to vote on the "real answer", at the earlier question. – Hendrik Jan Jul 20 '21 at 08:22
But, as I could not award the same bounty for two different answers, a bounty goes here. – Hendrik Jan Dec 03 '21 at 00:32

SAT formula for connected graphs on the grid

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