Solving 0-1 Integer Programs with Unknown Knapsack Constraints Using Membership Oracles
Rosario Messana, Rui Chen, Andrea Lodi, Alberto Ceselli
TL;DR
The paper addresses solving $0$-$1$ integer programs with unknown knapsack constraints accessed via membership oracles, where a budget on oracle calls governs the search. It introduces Interactive Sampling-Enhanced Optimization (ISEO), a framework that iteratively learns surrogate linear constraints through separation, selects informative sub-solutions for oracle queries via sampling, and solves a surrogate IP while maintaining upper and lower bounds to monitor progress. The authors compare separation methods (SVM-based vs. SEP) and sampling strategies (simple margin SIM vs. Closest Cutting Plane CUT) on KNAP, CSPP, and GAP instances, showing that the combination SEP+CUT often yields the best final endings with smaller gaps and competitive run times, albeit at higher per-iteration cost. The work demonstrates that learning-driven, interactive exploration of unknown constraints can yield strong primal solutions under limited oracle feedback, with practical relevance to settings like edge computing orchestration where constraint parameters are unknown a priori. Theoretical results establish a worst-case hardness bound, while empirical findings highlight the value of the proposed hybrids in achieving better solutions within realistic budgets.
Abstract
We consider solving a combinatorial optimization problem with unknown knapsack constraints using a membership oracle for each unknown constraint such that, given a solution, the oracle determines whether the constraint is satisfied or not with absolute certainty. The goal of the decision maker is to find the best possible solution subject to a budget on the number of oracle calls. Inspired by active learning for binary classification based on Support Vector Machines (SVMs), we devise a framework to solve the problem by learning and exploiting surrogate linear constraints. The framework includes training linear separators on the labeled points and selecting new points to be labeled, which is achieved by applying a sampling strategy and solving a 0-1 integer linear program. Following the active learning literature, a natural choice would be SVM as a linear classifier and the information-based sampling strategy known as simple margin, for each unknown constraint. We improve on both sides: we propose an alternative sampling strategy based on mixed-integer quadratic programming and a linear separation method inspired by an algorithm for convex optimization in the oracle model. We conduct experiments on classical problems and variants inspired by realistic applications to show how different linear separation methods and sampling strategies influence the quality of the results in terms of several metrics including objective value, dual bound and running time.