We study qualitative multi-objective reachability problems for Ordered
Branching Markov Decision Processes (OBMDPs), or equivalently context-free
MDPs, building on prior results for single-target reachability on Branching
Markov Decision Processes (BMDPs).
We provide two separate algorithms for "almost-sure" and "limit-sure"
multi-target reachability for OBMDPs. Specifically, given an OBMDP,
A, given a starting non-terminal, and given a set of target
non-terminals K of size k=∣K∣, our first algorithm decides whether the
supremum probability, of generating a tree that contains every target
non-terminal in set K, is 1. Our second algorithm decides whether there is
a strategy for the player to almost-surely (with probability 1) generate a
tree that contains every target non-terminal in set K.
The two separate algorithms are needed: we show that indeed, in this context,
"almost-sure" = "limit-sure" for multi-target reachability, meaning that
there are OBMDPs for which the player may not have any strategy to achieve
probability exactly 1 of reaching all targets in set K in the same
generated tree, but may have a sequence of strategies that achieve probability
arbitrarily close to 1. Both algorithms run in time 2O(k)⋅∣A∣O(1), where ∣A∣ is the total bit encoding length
of the given OBMDP, A. Hence they run in polynomial time when k
is fixed, and are fixed-parameter tractable with respect to k. Moreover, we
show that even the qualitative almost-sure (and limit-sure) multi-target
reachability decision problem is in general NP-hard, when the size k of the
set K of target non-terminals is not fixed.Comment: 47 page