We consider planning problems for graphs, Markov Decision Processes (MDPs), and games on graphs in an explicit state space. 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 with k different target sets: (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 a given 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, based on the boolean matrix multiplication (BMM) conjecture and strong exponential time hypothesis (SETH), 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) problem-separation results showing that for MDPs the coverage problem is harder than the sequential target problem.

我们考虑在显式状态空间中针对图，马尔可夫决策过程（MDP）和图上的博弈的计划问题。图表示最基本的计划模型，而MDP表示与自然的相互作用，图上的博弈表示与对抗性环境的相互作用。我们考虑k的两个计划问题不同的目标集：（a）覆盖率问题询问每个目标集是否有计划；（b）顺序目标可达性问题询问是否可以按给定顺序达到目标。对于覆盖率问题，我们提出了一种针对图的线性时间算法，以及针对图上的MDP和博弈的二次条件下界。对于顺序目标问题，我们提出了一种用于图形的线性时间算法，一种用于MDP的亚二次算法以及一种用于图形游戏的二次条件下界。我们基于布尔矩阵乘法（BMM）猜想和强指数时间假设（SETH）的条件下界结果，建立了（i）模型分离结果，该结果表明，对于覆盖率问题，图上的MDP和博弈比图难，对于顺序可及性问题，图上的游戏比MDP和图难。（ii）问题分离结果表明，对于MDP，覆盖问题比顺序目标问题更难。