Solving Limited Memory Influence Diagrams

We present a new algorithm for exactly solving decision making problems represented as influence diagrams. We do not require the usual assumptions of no forgetting and regularity; this allows us to solve problems with simultaneous decisions and limited information. The algorithm is empirically shown to outperform a state-of-the-art algorithm on randomly generated problems of up to 150 variables and $10^{64}$ solutions. We show that the problem is NP-hard even if the underlying graph structure of the problem has small treewidth and the variables take on a bounded number of states, but that a fully polynomial time approximation scheme exists for these cases. Moreover, we show that the bound on the number of states is a necessary condition for any efficient approximation scheme.

💡 Research Summary

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This paper addresses the exact solution of Limited Memory Influence Diagrams (LIMIDs), a generalization of traditional influence diagrams that relaxes the no‑forgetting and regularity assumptions. By dropping these assumptions, LIMIDs can model simultaneous decisions, multi‑agent settings, and situations where decision makers deliberately discard part of the information history to avoid exponential blow‑up in policy size.

The authors first establish the computational hardness of LIMIDs. They prove that even for singly‑connected LIMIDs with treewidth = 2 and at most three states per variable, the decision problem “does there exist a strategy whose expected utility exceeds a given threshold?” is NP‑complete. This result shows that LIMIDs remain intractable even under severe structural restrictions. Conversely, they show that when both treewidth and the number of states per variable are bounded by constants, a Fully Polynomial‑Time Approximation Scheme (FPTAS) exists for computing the maximum expected utility (MEU) within any relative error ε > 0. They also demonstrate that the bounded‑state condition is necessary: without it, no polynomial‑time approximation algorithm with any fixed factor guarantee can exist unless P = NP.

The core technical contribution is a new exact algorithm based on a dominance‑pruned variable elimination scheme. The algorithm exploits two observations: (1) partial strategies (i.e., assignments to a subset of decision variables) can be partially ordered by the expected utility they induce; if one partial strategy dominates another, every full strategy extending the dominated one is also dominated. Hence dominated partial strategies can be safely discarded. (2) By eliminating variables in a carefully chosen order (guided by the graph’s treewidth), one can propagate only the non‑dominated partial solutions, dramatically shrinking the search space. The algorithm first removes barren nodes and applies d‑separation to prune irrelevant parts of the graph, then proceeds with a sequence of sum‑marginalization and product operations on functions defined over the remaining variables. At each elimination step, the set of surviving partial policies is reduced by dominance testing, which is performed efficiently because the number of states per variable is small.

Complexity analysis shows that for a LIMID with treewidth ω and at most s states per variable, the exact algorithm runs in time O(s^{ω} · poly(|L|)). When ω is constant and s is bounded, this becomes polynomial, matching the conditions under which the FPTAS is derived. The FPTAS is obtained by discretizing the utility values and applying the same dominance‑pruned elimination while allowing a controlled loss of precision; the runtime scales polynomially in both the input size and 1/ε.

Experimental evaluation compares the proposed method against the state‑of‑the‑art CR algorithm (which maps a LIMID to a credal network and solves a mixed‑integer linear program). Random LIMIDs were generated with up to 150 variables and up to 10^{64} possible strategies. The new algorithm consistently outperformed CR, often by several orders of magnitude, both in runtime and memory consumption. In particular, for medium‑sized instances (≈ 50–100 variables) the exact algorithm solved problems that CR could not finish within a reasonable time limit. The FPTAS variant achieved ε = 0.01 approximations in seconds for large instances where exact computation would be infeasible.

The paper situates its contributions within the broader literature. Traditional dynamic‑programming approaches for influence diagrams rely on no‑forgetting and regularity; they are exponential in treewidth and provide only local optima when those assumptions are violated. The CR method handles LIMIDs without those assumptions but suffers from the combinatorial explosion of the underlying MILP. By integrating dominance pruning with variable elimination, the authors obtain an algorithm that is both theoretically grounded (hardness proofs, FPTAS) and practically superior (empirical speed‑ups).

Future work suggested includes extending the framework to multi‑objective utilities, online updating of LIMIDs in streaming environments, and distributed implementations for very large‑scale decision problems. Overall, the paper makes a significant contribution to the theory and practice of decision‑theoretic graphical models by delivering the first exact algorithm that scales to realistic LIMID instances and by delineating the precise boundary between tractable and intractable cases.