Are there graphs for which A* cannot be admissible?

Question

Wikipedia states that "On infinite graphs with a finite branching factor and edge costs that are bounded away from zero $ (d(x,y)>\varepsilon >0$ for some fixed $\varepsilon$), A* is guaranteed to terminate only if there exists a solution."

Does this mean that, if I have such a graph, then there is no admissible A* which exists for this graph? When one says that A* is not admissible, this means that its heuristic is not admissible, right?

Furthermore, is it correct to say that a heuristic is only admissible with respect to a graph - not generally admissible?

For example, if I have a infinite graph which has a finite branching factor, and the cost of each edge is half of the cost of the edge preceding it (something like this: $goal\leftarrow_{_{c=2}} start \rightarrow_{_{c=1}}q1 \rightarrow_{_{c=1/2}} q_2 \rightarrow ...$), the heuristic and therefore the A* is necessarily inadmissible as there exists no fixed $\epsilon>0$ that is less than the cost of any edge?

To generalize, the $epsilon$ constraint is to make sure that there is no infinite path who's total cost converges, thereby assuring termination?

Any clarification is appreciated. Thanks!

Answer 1

The heuristic function $h:V \longrightarrow \mathbb{R}_{\geq 0}$ is *an input to the $A^*$ algorithm. If the function is admissible then $A^*$ algorithm gives you the solution; however as you have noted for infinite graphs the edge weights must have a positive lower bound.

The point of the heuristic function is to find the shortest path in the least amount of time, i.e. lower the computational complexity; because you are making informed decisions based on a heuristic (hence the name).

Recall the heuristic function is admissible if $h(v)$ is always less than (or equal to) the true path cost to the goal node.

There is always an admissable $h$ namely $h(v)= 0$ for all $v$. In this extreme case it becomes Dijkstra's algorithm.

Returning to the example you gave if you plug in the input $A^*(G,h)$ where $G$ is a description of $G$ and $h = 0$ then $A^*$ won't halt (geometric series $r = 2 \implies \sum_i r^i = \frac{1-r^{n+1}}{1-r} \leq 2$ for all finite partial sums). But lets see if we can bypass that; lets try: \begin{equation} h(v) = \begin{cases} 3 & \text{if } v = q_1 \\ 0 & \text{else}\end{cases} \end{equation} as you can see $h$ is admissible (because $d(q_1 , \text{goal}) = 3$ on the nose) and $A^*$ will pick "goal" as its first node (because its two choices are $f(q_1) = 4$ or $f(\text{goal}) = 2$).