We consider planning problems for graphs, Markov decision processes (MDPs), and games on graphs. While graphs represent the most basic planning model, MDPs represent interaction with nature and games on graphs represent interaction with an adversarial environment.We consider two planning problems where there are k different target sets, and the problems are as follows: (a) the coverage problem asks whether there is a plan for each individual target set, and (b) the sequential target reachability problem asks whether the targets can be reached in sequence. For the coverage problem, we present a linear-time algorithm for graphs, and quadratic conditional lower bound for MDPs and games on graphs.For the sequential target problem, we present a linear-time algorithm for graphs, a sub-quadratic algorithm for MDPs, and a quadratic conditional lower bound for games on graphs.Our results with conditional lower bounds establish (i) model-separation results showing that for the coverage problem MDPs and games on graphs are harder than graphs and for the sequential reachability problem games on graphs are harder than MDPs and graphs;and (ii) objective-separation results showing that for MDPs the coverage problem is harder than the sequential target problem.

Partially observable Markov decision processes (POMDPs) are widely used in probabilistic planning problems in which an agent interacts with an environment using noisy and imprecise sensors. We study a setting in which the sensors are only partially defined and the goal is to synthesize “weakest” additional sensors, such that in the resulting POMDP, there is a small-memory policy for the agent that almost-surely (with probability 1) satisfies a reachability objective. We show that the problem is NP-complete, and present a symbolic algorithm by encoding the problem into SAT instances. We illustrate trade-offs between the amount of memory of the policy and the number of additional sensors on a simple example. We have implemented our approach and consider three classical POMDP examples from the literature, and show that in all the examples the number of sensors can be significantly decreased (as compared to the existing solutions in the literature) without increasing the complexity of the policies.

We develop a topological approach to stratification learning. Given point cloud data drawn from a stratified space, our objective is to infer which points belong to the same strata. First we define a multi-scale notion of a stratified space, giving a stratification for each radius level. We then use methods derived from kernel and cokernel persistent homology to cluster the data points into different strata, and we prove a result which guarantees the correctness of our clustering, given certain topological conditions. We later give bounds on the minimum number of sample points required to infer, with probability, which points belong to the same strata. Finally, we give an explicit algorithm for the clustering and apply it to some simulated data.