Hamiltonian path in grid graph

Question

Here is my situation. I have a grid-type graph with obstacles. Every move (horizontally, vertically or diagonally with a range of 1) has a cost of exactly 1 (the graph is not weighted) provided that neither the source or the destination is an obstacle.

I need to, provided a starting node (a square), find any path that visits every single node that is reachable, without visiting any node twice. If there is no such path, then any path that would visit as many nodes as possible. In other words, I'm looking for an algorithm to find a Hamiltonian path in such a graph.

I am not looking for the shortest path.

The most important factor is the time the algorithm takes to complete (complexity).

I have searched Google and this site, and I only find variants of the travelling salesman, or algorithms for shortest path visiting all nodes, which won't work in my situation. My algorithm needs to find a path of any length visiting as many nodes as possible while never visiting a node twice. The path can end anywhere.

Does anyone have any tips? I've tried to do this with a variant of A*, but I couldn't make it work. I've succesfully done it by brute-forcing, but as the graph can be rather large, I need something a faster than O(n!).

It would be even better if the algorithm was non-deterministic, which means that if there are multiple possible solutions, we wouldn't always get the same one with the same graph and starting location. I have no idea how to do this without slowing it down though, and speed is more important.

Answer 1

There isn't going to be a fast algorithm for this problem.

The class of graphs you're dealing with is, in some of the literature, called grid graphs. That is, they're finite induced subgraphs of the infinite graph with vertices $\mathbb{Z}\times\mathbb{Z}$ with an edge from $(x,y)$ to $(x',y')$ if, and only if, $|x-x'|+|y-y'|=1$.

Itai, Papadimitriou and Szwarcfiter (Hamiltonian Paths in Grid Graphs, SIAM Journal on Computing, 11(4):676–686, 1982) showed that it's NP-complete to determine whether a grid graph has a Hamiltonian path. Your problem is NP-hard (under Cook reductions, and possibly also under the more usual Karp reductions): if you can determine the length of the longest path from a vertex, just do that for each vertex in turn and see if any of the longest paths is Hamiltonian.

Answer 2

Since an exact algorithm is most likely off the table, I'll propose the following heuristic: Start at any vertex (though a vertex with no vertices above it would be best), and always move to the first neighbour in the following list that exists and has not yet been visited: UPLEFT, LEFT, DOWNLEFT, DOWN, DOWNRIGHT, RIGHT, UPRIGHT, UP. This should cause you to quickly travel to the top-left part of the graph, and thereafter alternatingly "snake" down and then up adjacent columns, "bending" around any obstacles but always hugging as close to the left border of the graph as possible, and thus gradually covering the graph from left to right.

There are probably many ways for this algorithm to get stuck -- that is, to make a move that leads to the subgraph induced by the unvisited vertices becoming disconnected, meaning that at most one of the resulting components can be visited. One way to ameliorate this somewhat is to first test whether the next valid move will lead to such a disconnection by checking whether there is a path through unvisited vertices from the vertex we would get to at the move after that to the next valid move from current vertex -- this can be done with BFS (or DFS, but BFS will terminate faster in the common case where a short path does exist). If no such path would then exist, and if a heuristic estimating the length of the path reachable from the first move is small (for example: the size of the connected component of unvisited vertices you would find yourself in), try the next move.