Qualitative Multi-Objective Reachability for Ordered Branching MDPs
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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, ๐, 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" โ "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)ยท |๐|^O(1), where |๐| is the total bit encoding length of the given OBMDP, ๐. 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.
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