Decision-Aware Approximation of Belief Functions for Evidential Combinatorial Optimization

2026-08-11

AI Summary

This work addresses the limitation of conventional distance-based belief function approximation methods in Dempster–Shafer evidence theory, which fail to guarantee high-quality decision outcomes under combination and optimization. To overcome this, the paper proposes a decision-quality-oriented approximation approach that, for the first time, formulates decision regret as the optimization objective. It introduces a decision-aware focal element merging strategy and establishes a single-point regret bound grounded in the true optimal solution. In scalar settings, exact approximation is achieved via dynamic programming, while in scenarios with unknown costs, an online pruning algorithm is developed to proactively compress focal elements. Empirical results demonstrate that, compared to traditional representation-aware methods, the proposed approach substantially reduces decision reversal frequency in tasks such as shortest-path planning and is applicable to both linear and nonlinear decision criteria.

Abstract

Reducing the number of focal elements of a mass function is classically driven by an intrinsic distance, such as Jaccard or Jousselme, that keeps the approximation close to the original as a body of evidence. We consider instead the case where the mass function feeds a linear combinatorial optimisation problem with evidential costs. What should then be preserved is not the closeness of the two mass functions, but the quality of the decision they induce. We introduce a decision-aware approximation that targets the regret of the decision: one decides with the cheaper approximation and is evaluated under the true mass function. On a minimal shortest path, the distance-optimal approximation flips the decision while a decision-aware merge preserves it, and this occurs on a non-negligible fraction of random instances. We prove a one-point bound that localises the regret at the true optimum, turn it into an exact dynamic program for the scalar case, and extend it to an online version that prunes focal elements before the final cost is known. In experiments the decision-aware compressor flips the decision less often than representation-aware compression, for both the linear criterion and a non-linear proxy read-out.

Problem

• belief functions
• combinatorial optimization
• decision-aware approximation
• evidential reasoning
• focal elements

Innovation

• decision-aware approximation
• evidential combinatorial optimization
• belief functions
• regret minimization
• focal element pruning