Graph Planning with Expected Finite Horizon

TL;DR

This paper studies graph planning with an adversarial stopping time distribution, showing stationary plans are sufficient and can be computed efficiently, unlike in fixed horizon scenarios.

Contribution

It proves that stationary plans are optimal under adversarial stopping times with expected horizon T and provides a polynomial-time algorithm for their computation.

Findings

Stationary plans are sufficient for optimality under adversarial stopping times.

Computing optimal stationary plans under adversarial conditions is polynomial-time solvable.

Optimal plans for fixed horizon are NP-complete to compute.

Abstract

Graph planning gives rise to fundamental algorithmic questions such as shortest path, traveling salesman problem, etc. A classical problem in discrete planning is to consider a weighted graph and construct a path that maximizes the sum of weights for a given time horizon $T$. However, in many scenarios, the time horizon is not fixed, but the stopping time is chosen according to some distribution such that the expected stopping time is $T$. If the stopping time distribution is not known, then to ensure robustness, the distribution is chosen by an adversary, to represent the worst-case scenario. A stationary plan for every vertex always chooses the same outgoing edge. For fixed horizon or fixed stopping-time distribution, stationary plans are not sufficient for optimality. Quite surprisingly we show that when an adversary chooses the stopping-time distribution with expected stopping time $T$, then stationary plans are sufficient. While computing optimal stationary plans for fixed horizon is NP-complete, we show that computing optimal stationary plans under adversarial stopping-time distribution can be achieved in polynomial time. Consequently, our polynomial-time algorithm for adversarial stopping time also computes an optimal plan among all possible plans.