Algorithms and Conditional Lower Bounds for Planning Problems
This work addresses computational complexity challenges in planning models for researchers in theoretical computer science and AI, establishing model and objective separation results, but it is incremental as it builds on existing planning frameworks.
The paper tackles planning problems in graphs, MDPs, and games on graphs by developing algorithms and conditional lower bounds for coverage and sequential target reachability, showing linear-time algorithms for graphs and quadratic lower bounds for MDPs and games in coverage, and linear-time for graphs, sub-quadratic for MDPs, and quadratic lower bounds for games in sequential reachability.
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; (ii) objective-separation results showing that for MDPs the coverage problem is harder than the sequential target problem.