Improving package recommendations through query relaxation

Recommendation systems aim to identify items that are likely to be of interest to users. In many cases, users are interested in package recommendations as collections of items. For example, a dietitian may wish to derive a dietary plan as a collection of recipes that is nutritionally balanced, and a travel agent may want to produce a vacation package as a coordinated collection of travel and hotel reservations. Recent work has explored extending recommendation systems to support packages of items. These systems need to solve complex combinatorial problems, enforcing various properties and constraints defined on sets of items. Introducing constraints on packages makes recommendation queries harder to evaluate, but also harder to express: Queries that are under-specified produce too many answers, whereas queries that are over-specified frequently miss interesting solutions. In this paper, we study query relaxation techniques that target package recommendation systems. Our work offers three key insights: First, even when the original query result is not empty, relaxing constraints can produce preferable solutions. Second, a solution due to relaxation can only be preferred if it improves some property specified by the query. Third, relaxation should not treat all constraints as equals: some constraints are more important to the users than others. Our contributions are threefold: (a) we define the problem of deriving package recommendations through query relaxation, (b) we design and experimentally evaluate heuristics that relax query constraints to derive interesting packages, and (c) we present a crowd study that evaluates the sensitivity of real users to different kinds of constraints and demonstrates that query relaxation is a powerful tool in diversifying package recommendations.

💡 Research Summary

Package recommendation systems aim to suggest collections of items rather than single items, which introduces combinatorial complexity and a rich set of constraints. The authors identify three primary constraint categories: (i) base constraints that apply to each individual item (e.g., “each meal must be gluten‑free”), (ii) global constraints that apply to the whole package (e.g., “total calories ≤ 1500”), and (iii) cardinality constraints that restrict the number of items in the package. In addition to these constraints, a query includes an objective function that must be maximized (e.g., total protein) or minimized (e.g., total travel cost).

The paper’s central thesis is that relaxing a subset of the original constraints—i.e., removing them entirely—can yield packages that are preferable to the top‑k results that strictly satisfy all constraints, even when the original query already returns non‑empty results. Proposition 1 formally guarantees that any relaxation will not worsen the objective value: for maximization, the relaxed package’s objective is at least as high as the original; for minimization, it is at most as low. Consequently, the authors define the “optimal relaxation” as the relaxation that maximizes improvement in the objective while minimizing the total deviation from the original constraints. Improvement (I) is measured as the percentage change in the objective, and deviation (E) is captured by the mean absolute percentage error (MAPE) across all violated constraints.

Because the space of possible relaxations grows exponentially with the number of constraints, exhaustive search is computationally infeasible. The authors therefore propose two heuristic families. The first, “Exhaustive‑I,” exhaustively evaluates all relaxations that remove a fixed percentage of constraints but optimizes only I, ignoring E. The second, “Exhaustive‑IE,” optimizes the ratio (1 + I)/(1 + E), thereby balancing objective improvement against constraint violation. Experiments on a real‑world recipe dataset (≈8 000 recipes) with ten randomly generated queries (3–10 constraints each) show that modest relaxation (10–20 % of constraints) yields rapid gains in the objective while keeping error low; further relaxation yields diminishing returns.

To achieve interactive response times, the authors design greedy algorithms. “Greedy‑I” iteratively removes the single constraint that yields the largest immediate improvement in the objective; “Greedy‑IE” removes the constraint that maximizes the improvement‑to‑error ratio. Both greedy approaches achieve nearly the same improvement and error as their exhaustive counterparts for low relaxation levels, but run orders of magnitude faster. An additional “bidirectional greedy” variant starts from the full constraint set when the target relaxation is below 50 % and from the empty set when it is above 50 %, further reducing runtime.

The practical impact of relaxation is validated through a crowdsourced user study focused on meal‑plan recommendations. Participants evaluated packages under a set of nutritional and preparation‑time constraints. Results indicate that 76 % of users preferred at least one relaxed package, compared with 71 % who preferred a non‑relaxed top‑k package. Moreover, users displayed varying sensitivity to different constraint types: they were more willing to relax nutritional upper bounds than cost constraints, suggesting that constraint importance is user‑dependent.

Overall, the paper contributes three key insights: (1) query relaxation can improve the objective even when the original query yields results; (2) a small, well‑chosen subset of constraints suffices to obtain substantial gains, and greedy heuristics can find such subsets efficiently; (3) real users accept relaxed recommendations and exhibit heterogeneous preferences regarding which constraints to relax. The authors conclude by outlining future work on fine‑grained relaxations (adjusting constraint values rather than removing them) and personalized relaxation strategies that adapt to individual user sensitivity profiles.