Algorithms and Conditional Lower Bounds for Planning Problems

TL;DR

This paper studies planning problems across graphs, MDPs, and games, providing efficient algorithms and conditional lower bounds that reveal complexity differences and model-specific challenges.

Contribution

It introduces new algorithms and lower bounds that distinguish the complexity of planning problems across different models and objectives.

Findings

Linear-time algorithm for coverage in graphs

Quadratic lower bound for MDPs and games on graphs

Sub-quadratic algorithm for sequential reachability in MDPs

Abstract

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.