Computational issues in time-inconsistent planning
  Time-inconsistency refers to a paradox in decision making where agents exhibit inconsistent behaviors over time. Examples are procrastination where agents tends to costly postpone easy tasks, and abandonments where agents start a plan and quit in the middle. These behaviors are undesirable in the sense that agents make clearly suboptimal decisions over optimal ones. To capture such behaviors and more importantly, to quantify inefficiency caused by such behaviors, [Kleinberg & Oren 2014] propose a graph model which is essentially same as the standard planning model except for the cost structure. Using this model, they initiate the study of several interesting problems: 1) cost ratio: the worst ratio between the actual cost of the agent and the optimal cost, over all graph instances; 2) motivating subgraph: how to motivate the agent to reach the goal by deleting nodes and edges; 3) Intermediate rewards: how to motivate agents to reach the goal by placing intermediate rewards. Kleinberg and Oren give partial answers to these questions, but the main problems are still open. In fact, they raise these problems explicitly as open problems in their paper. In this paper, we give answers to all three open problems in [Kleinberg & Oren 2014]. First, we show a tight upper bound of cost ratio for graphs without Akerlof's structure, thus confirm the conjecture by Kleinberg and Oren that Akerlof's structure is indeed the worst case for cost ratio. Second, we prove that finding a motivating subgraph is NP-hard, showing that it is generally inefficient to motivate agents by deleting nodes and edges in the graph. Last but not least, we show that computing a strategy to place minimum amount of total reward is also NP-hard. Therefore, it is computational inefficient to motivate agents by placing intermediate rewards. The techniques we use to prove these results are nontrivial and of independent interests. 
Pingzhong Tang, Yifeng Teng, Zihe Wang, Shenke Xiao, Yichong Xu
cs.GT cs.CC cs.DS cs.SI