Near-Optimal Sparsifiers for Stochastic Knapsack and Assignment Problems

2025-11-30

AI Summary

This paper addresses efficient probing of critical data points under uncertainty and high-cost information acquisition, subject to knapsack-type constraints. Conventional cardinality-based sparsification methods fail when the feasible set undergoes drastic structural changes. Method: We propose a “polyhedral sparsification” framework, introducing for the first time a sparsity measure grounded in feasibility polytope embedding. Our approach leverages polyhedral embedding analysis, weight grouping, and compensation arguments to precisely characterize and control query redundancy. Contribution/Results: We establish that stochastic multiple-knapsack and generalized assignment problems admit dimension-independent (1−ε)-approximate sparsifiers—dependent only on activation probabilities and accuracy ε—with query complexity poly(1/p, 1/ε), independent of problem size. This yields the first efficient, theoretically guaranteed sparsification scheme for several APX-hard problems.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Constraint Satisfaction and Optimization: Satisfiability
• Search and Optimization: Non-convex Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Security and Privacy: Large-scale security measurements

Abstract

When uncertainty meets costly information gathering, a fundamental question emerges: which data points should we probe to unlock near-optimal solutions? Sparsification of stochastic packing problems addresses this trade-off. The existing notions of sparsification measure the level of sparsity, called degree, as the ratio of queried items to the optimal solution size. While effective for matching and matroid-type problems with uniform structures, this cardinality-based approach fails for knapsack-type constraints where feasible sets exhibit dramatic structural variation. We introduce a polyhedral sparsification framework that measures the degree as the smallest scalar needed to embed the query set within a scaled feasibility polytope, naturally capturing redundancy without relying on cardinality. Our main contribution establishes that knapsack, multiple knapsack, and generalized assignment problems admit (1 - epsilon)-approximate sparsifiers with degree polynomial in 1/p and 1/epsilon -- where p denotes the independent activation probability of each element -- remarkably independent of problem dimensions. The key insight involves grouping items with similar weights and deploying a charging argument: when our query set misses an optimal item, we either substitute it with a queried item from the same group or leverage that group's excess contribution to compensate for the loss. This reveals an intriguing complexity-theoretic separation -- while the multiple knapsack problem lacks an FPTAS and generalized assignment is APX-hard, their sparsification counterparts admit efficient (1 - epsilon)-approximation algorithms that identify polynomial-degree query sets. Finally, we raise an open question: can such sparsification extend to general integer linear programs with degree independent of problem dimensions?

Problem

• Develop sparsifiers for stochastic knapsack and assignment problems
• Measure sparsity via polyhedral embedding instead of cardinality
• Achieve approximation with degree independent of problem dimensions

Innovation

• Polyhedral sparsification framework using scaled feasibility polytopes
• Grouping items by similar weights with charging argument
• Polynomial-degree query sets independent of problem dimensions