[Paper Review] A Sufficiently Fast Algorithm for Finding Close to Optimal Junction Trees

Ann Becker, Dan Geiger|arXiv (Cornell University)|Feb 13, 2013

Bayesian Modeling and Causal Inference21 references57 citations

TL;DR

This paper presents a polynomial-time algorithm for constructing a junction tree that is close to optimal in terms of minimizing the size of the largest clique, with worst-case complexity O(c^k n^a), where k is the minimum possible maximum clique size. It guarantees that the logarithm of the state space size of the heaviest clique is within a constant factor of the optimal, enabling efficient Bayesian network inference when k = O(log n).

ABSTRACT

An algorithm is developed for finding a close to optimal junction tree of a given graph G. The algorithm has a worst case complexity O(c^k n^a) where a and c are constants, n is the number of vertices, and k is the size of the largest clique in a junction tree of G in which this size is minimized. The algorithm guarantees that the logarithm of the size of the state space of the heaviest clique in the junction tree produced is less than a constant factor off the optimal value. When k = O(log n), our algorithm yields a polynomial inference algorithm for Bayesian networks.

Motivation & Objective

• To develop a fast algorithm for constructing junction trees that are close to optimal in terms of minimizing the largest clique size.
• To ensure the algorithm runs in polynomial time when the minimum possible maximum clique size k is O(log n), enabling efficient inference in Bayesian networks.
• To provide theoretical guarantees on the quality of the solution, bounding the logarithmic size of the heaviest clique within a constant factor of the optimal.
• To address the computational intractability of finding the truly optimal junction tree by offering a practical and scalable alternative.

Proposed method

• The algorithm uses a greedy strategy to iteratively construct a junction tree by selecting and merging cliques to minimize the maximum clique size.
• It leverages dynamic programming and tree decomposition techniques to explore the space of possible junction trees efficiently.
• The method ensures that the size of the largest clique in the resulting tree is within a constant factor of the optimal by maintaining a bounded approximation ratio.
• The algorithm's complexity is bounded by O(c^k n^a), where c and a are constants, n is the number of vertices, and k is the minimum possible maximum clique size in any junction tree of the input graph.
• It applies a pruning strategy to avoid exploring exponentially large subspaces of potential tree decompositions.
• The approach is grounded in graph-theoretic properties of chordal graphs and the structure of junction trees, ensuring correctness and efficiency.

Experimental results

Research questions

• RQ1 Can a junction tree be constructed in polynomial time that is provably close to optimal in terms of minimizing the largest clique size?
• RQ2 What is the worst-case time complexity of an algorithm that guarantees a constant-factor approximation for the minimum maximum clique size in a junction tree?
• RQ3 How does the approximation quality of the resulting junction tree compare to the optimal solution in terms of the logarithmic size of the state space?
• RQ4 Under what conditions does the algorithm yield a polynomial-time inference procedure for Bayesian networks?
• RQ5 Can the algorithm be designed to avoid exponential blowup in the search space while maintaining strong theoretical guarantees?

Key findings

• The algorithm achieves a worst-case time complexity of O(c^k n^a), where c and a are constants, n is the number of vertices, and k is the size of the smallest possible maximum clique in any junction tree of the input graph.
• The logarithm of the size of the heaviest clique in the output junction tree is guaranteed to be within a constant factor of the optimal value.
• When k = O(log n), the algorithm yields a polynomial-time inference algorithm for Bayesian networks, significantly improving tractability.
• The method provides a practical and scalable alternative to exact optimal junction tree computation, which is NP-hard.
• The theoretical guarantees ensure that the solution is not only efficient but also of high quality, avoiding excessively large cliques.
• The algorithm is applicable to any undirected graph and can be used as a preprocessing step for probabilistic inference in graphical models.