Abstract: The classical stochastic shortest path (SSP) problems asks to find a policy for traversing a weighted stochastic graph until reaching a distinguished goal state that minimizes the expected accumulated weight. SSP problems have numerous applications in, e.g., operations research, artificial intelligence, robotics and verification of probabilistic systems. The underlying graph model is a finite-state Markov decision process (MDP) with integer weights for its state-action pairs. Prominent results are the existence of optimal memoryless deterministic policies together with linear programming techniques and value and policy iteration to compute such policies and their values. These results rely on the assumption that the minimum under all proper policies that reach the goal state almost surely exists. Early work on the SSP problems goes back to the 1960s-1990s and makes additional assumptions. Complete algorithms that only require the existence of proper policies combine these techniques with a pre-analysis of end components, an elegant graph-theoretic concept for MDPs that has been introduced by de Alfaro in the late 1990s. The talk will start with a summary of these results. The second part of the talk presents more recent results for variants of the classical SSP. The conditional and partial SSP drop the assumption that the goal state must be reached almost surely and ask to minimize the expected accumulated weight under the condition that the goal will be reached (conditional SSP) resp. assign value 0 to all paths that do not reach the goal state (partial SSP). Other variants take into account aspects of risk-awareness, e.g., by studying the conditional value-at-risk or the variance-penalized expected accumulated weight. While the classical SSP problem is solvable in polynomial time, such non-classical SSP problems are computationally much harder. For the general case, the decidability status of such non-classical SSP problems is unknown, but they have been shown to be at least as hard as the Skolem problem (and even as the positivity problem). However, for non-positive weights, the conditional, partial and variance-penalized SSP problem are solvable in exponential time with a PSPACE lower bounds for acyclic MDPs
Speaker bio:
Christel Baier is a full professor and head of the chair for Algebraic and Logic Foundations of Computer Science at the Faculty of Computer Science of the Technische Universität Dresden since 2006. Since 2022 she holds an honorary doctorate (Dr. rer. nat. h.c.) from RWTH Aachen. From the University of Mannheim she received her Diploma in Mathematics in 1990, her Ph.D. in Computer Science in 1994 and her Habilitation in 1999. She was an associate professor for Theoretical Computer Science at the University of Bonn from 1999 to 2006. She was member of the DFG (German Research Foundation) review board for computer science from 2012 to 2019, and is a member of the DFG senate commission on Research Training Groups since 2020. Since 2011 she is a member of Academia Europaea. Her research focuses on modeling, specification and verification of reactive systems, quantitative analysis of stochastic systems, probabilistic model checking, temporal and modal logics and automata theory.
