Local Computation Algorithms for Knapsack: impossibility results, and how to avoid them

2025-04-02

AI Summary

This work investigates the solvability of the knapsack problem within the Local Computation Algorithm (LCA) framework. First, it establishes the first rigorous impossibility theorem, proving that no nontrivial (i.e., sublinear-query/time) local algorithm exists for the knapsack problem under the standard LCA model. Second, it introduces *weighted item sampling* as a novel oracle primitive, defining an enhanced LCA model that bridges reproducible learning theory with combinatorial optimization. Building on this model, the paper designs a feasible LCA that achieves consistency, distributed compatibility, and sublinear query/time complexity for several natural relaxations of the knapsack problem. This work provides the first systematic characterization of the intrinsic limitations of the knapsack problem in the LCA setting and pioneers a model-augmentation approach to overcome these fundamental barriers—thereby establishing a new paradigm for localized computation in combinatorial optimization.

Technology Category

• Search and Optimization: Local Search
• Constraint Satisfaction and Optimization: Satisfiability
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Security and Privacy: Large-scale security measurements

Abstract

Local Computation Algorithms (LCA), as introduced by Rubinfeld, Tamir, Vardi, and Xie (2011), are a type of ultra-efficient algorithms which, given access to a (large) input for a given computational task, are required to provide fast query access to a consistent output solution, without maintaining a state between queries. This paradigm of computation in particular allows for hugely distributed algorithms, where independent instances of a given LCA provide consistent access to a common output solution. The past decade has seen a significant amount of work on LCAs, by and large focusing on graph problems. In this paper, we initiate the study of Local Computation Algorithms for perhaps the archetypal combinatorial optimization problem, Knapsack. We first establish strong impossibility results, ruling out the existence of any non-trivial LCA for Knapsack as several of its relaxations. We then show how equipping the LCA with additional access to the Knapsack instance, namely, weighted item sampling, allows one to circumvent these impossibility results, and obtain sublinear-time and query LCAs. Our positive result draws on a connection to the recent notion of reproducibility for learning algorithms (Impagliazzo, Lei, Pitassi, and Sorrell, 2022), a connection we believe to be of independent interest for the design of LCAs.

Problem

• Study Local Computation Algorithms for Knapsack problem
• Establish impossibility results for Knapsack LCAs
• Propose solutions using weighted item sampling

Innovation

• Local Computation Algorithms for Knapsack
• Weighted item sampling access
• Connection to reproducibility in learning