Qualitative Multi-Objective Reachability for Ordered Branching MDPs

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, $\mathcal{A}$, 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" $\not=$ "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 $2^{O(k)} \cdot |\mathcal{A}|^{O(1)}$, where $|\mathcal{A}|$ is the total bit encoding length of the given OBMDP, $\mathcal{A}$. 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.