# Integer programming on the junction tree polytope for influence diagrams

## TL;DR

This paper introduces an integer programming approach on the junction tree polytope for solving influence diagrams, providing a mixed integer linear formulation and valid inequalities that improve computational efficiency and solution optimality.

## Contribution

It develops a novel mixed integer linear programming formulation for influence diagrams using junction trees, enhancing solution efficiency and theoretical understanding.

## Key findings

- Linear relaxation often yields optimal solutions for certain instances.
- The approach improves computational efficiency over existing methods.
- Valid inequalities strengthen the formulation and solution process.

## Abstract

Influence Diagrams (ID) are a flexible tool to represent discrete stochastic optimization problems, including Markov Decision Process (MDP) and Partially Observable MDP as standard examples. More precisely, given random variables considered as vertices of an acyclic digraph, a probabilistic graphical model defines a joint distribution via the conditional distributions of vertices given their parents. In ID, the random variables are represented by a probabilistic graphical model whose vertices are partitioned into three types : chance, decision and utility vertices. The user chooses the distribution of the decision vertices conditionally to their parents in order to maximize the expected utility. Leveraging the notion of rooted junction tree, we present a mixed integer linear formulation for solving an ID, as well as valid inequalities, which lead to a computationally efficient algorithm. We also show that the linear relaxation yields an optimal integer solution for instances that can be solved by the "single policy update", the default algorithm for addressing IDs.