# The Stochastic Shortest Path Problem : A polyhedral combinatorics perspective

## TL;DR

This paper introduces a new polyhedral combinatorics framework for the stochastic shortest path problem, generalizing existing models and establishing polynomial solvability under broad conditions, with convergence guarantees for classical algorithms.

## Contribution

It extends the stochastic shortest path framework to include more general conditions, proving polynomial solvability and convergence of key algorithms.

## Key findings

- The problem is well-defined and weakly polynomial under broad conditions.
- Deterministic stationary policies suffice for optimality.
- Classical algorithms like Value Iteration, Policy Iteration, and Dijkstra's extension converge.

## Abstract

In this paper, we give a new framework for the stochastic shortest path problem in finite state and action spaces. Our framework generalizes both the frameworks proposed by Bertsekas and Tsitsikli and by Bertsekas and Yu. We prove that the problem is well-defined and (weakly) polynomial when (i) there is a way to reach the target state from any initial state and (ii) there is no transition cycle of negative costs (a generalization of negative cost cycles). These assumptions generalize the standard assumptions for the deterministic shortest path problem and our framework encapsulates the latter problem (in contrast with prior works). In this new setting, we can show that (a) one can restrict to deterministic and stationary policies, (b) the problem is still (weakly) polynomial through linear programming, (c) Value Iteration and Policy Iteration converge, and (d) we can extend Dijkstra's algorithm.