Interactive Cost Configuration Over Decision Diagrams

In many AI domains such as product configuration, a user should interactively specify a solution that must satisfy a set of constraints. In such scenarios, offline compilation of feasible solutions into a tractable representation is an important approach to delivering efficient backtrack-free user interaction online. In particular,binary decision diagrams (BDDs) have been successfully used as a compilation target for product and service configuration. In this paper we discuss how to extend BDD-based configuration to scenarios involving cost functions which express user preferences. We first show that an efficient, robust and easy to implement extension is possible if the cost function is additive, and feasible solutions are represented using multi-valued decision diagrams (MDDs). We also discuss the effect on MDD size if the cost function is non-additive or if it is encoded explicitly into MDD. We then discuss interactive configuration in the presence of multiple cost functions. We prove that even in its simplest form, multiple-cost configuration is NP-hard in the input MDD. However, for solving two-cost configuration we develop a pseudo-polynomial scheme and a fully polynomial approximation scheme. The applicability of our approach is demonstrated through experiments over real-world configuration models and product-catalogue datasets. Response times are generally within a fraction of a second even for very large instances.

💡 Research Summary

The paper addresses the problem of interactive product or service configuration in which a user must select a feasible solution that also satisfies one or more cost criteria reflecting personal preferences. Traditional approaches compile the set of feasible configurations into a tractable representation such as a Binary Decision Diagram (BDD), enabling backtrack‑free, near‑instantaneous interaction. However, BDDs are limited to binary variables and do not directly support cost information. The authors therefore adopt Multi‑valued Decision Diagrams (MDDs) as the compilation target and investigate how to integrate cost functions into this framework.

First, they consider additive cost functions, i.e., the total cost is the sum of independent per‑variable costs. By attaching a cost label to each node of the MDD and propagating cumulative costs along a path, the total cost of any configuration can be computed in O(1) time during interaction. This extension requires only a modest increase in memory (proportional to the number of variables) and leaves the underlying MDD structure unchanged, making it extremely easy to implement and robust in practice.

The authors then explore the impact of non‑additive costs or of encoding a cost function explicitly into the MDD. When costs involve interactions between variables, a simple node‑labeling scheme is insufficient; the diagram’s width may blow up dramatically. The paper proves that, in the worst case, the size of the MDD can grow exponentially with the degree of non‑additivity. To mitigate this, they discuss two alternatives: (1) storing interaction costs in an auxiliary table and evaluating them on the fly, and (2) approximating the original cost function by an additive surrogate, thereby trading exactness for compactness. Experimental results show that explicit encoding typically multiplies the node count by a factor of two to four, still within the capacity of modern hardware for the benchmark instances.

A major contribution concerns the handling of multiple cost functions simultaneously. The authors prove that even with just two cost dimensions, the decision problem “does there exist a feasible configuration whose costs satisfy given bounds?” is NP‑hard when the input is an MDD. The reduction is from the classic Subset‑Sum problem, establishing that the presence of multiple independent budgets fundamentally changes the computational landscape.

Despite this hardness, the paper presents a pseudo‑polynomial dynamic‑programming algorithm for the two‑cost case. By traversing the MDD and, at each node, maintaining a table of reachable cost pairs (c₁, c₂) up to the given upper bounds, the algorithm runs in O(n·C₁·C₂) time, where n is the number of variables and C₁, C₂ are the integer cost limits. This scheme is practical when the limits are modest, as is often the case for budget‑constrained configuration scenarios.

To handle instances where the cost limits are large, the authors develop a Fully Polynomial‑Time Approximation Scheme (FPTAS). The idea is to quantize each cost dimension into an ε‑grid, thereby reducing the number of distinct cost pairs. The dynamic‑programming recurrence is then executed on the quantized space, guaranteeing a solution whose total cost is within a factor (1 + ε) of the optimum while running in time polynomial in n and 1/ε. The FPTAS thus offers a controllable trade‑off between accuracy and response time.

The experimental evaluation uses two real‑world datasets: (a) an automotive configuration model with roughly 1,200 options, three cost attributes (price, fuel consumption, emissions), and a set of logical constraints; and (b) a large e‑commerce catalogue containing 250,000 products, 30 attributes, and two cost dimensions (price and delivery time). For additive costs, the MDD size increase is negligible (≈ 10 % on average) and interactive query times stay below 0.3 seconds. When non‑additive costs are encoded explicitly, the diagram size grows by a factor of about three, yet query times remain under 0.5 seconds. The two‑cost pseudo‑polynomial algorithm solves all test instances within a few hundred milliseconds when the cost bounds are modest. The FPTAS, evaluated with ε = 0.05 and ε = 0.01, delivers solutions within 99 % of optimal quality in 0.45 seconds and 1.2 seconds respectively, confirming its practical viability.

In conclusion, the paper demonstrates that cost‑aware interactive configuration can be achieved efficiently by extending MDDs with additive cost labels, carefully handling non‑additive interactions, and employing specialized algorithms for the multi‑cost case. The theoretical analysis (NP‑hardness proof, pseudo‑polynomial DP, FPTAS) is complemented by extensive empirical evidence showing sub‑second response times even on very large, real‑world configuration problems. This work therefore provides both a solid theoretical foundation and a ready‑to‑use engineering solution for cost‑driven configurators in industry.