Average-Case Local Computation Algorithms

2024-02-29

arXiv.org

AI Summary

This paper investigates performance optimization of Local Computation Algorithms (LCAs) on random graphs under average-case analysis. Addressing the Erdős–Rényi and preferential attachment models, it pioneers the extension of the LCA framework to the average-case setting, designing sublinear-query local access schemes for combinatorial structures—including *k*-spanners and maximum independent sets. The main contributions are threefold: (1) It demonstrates that structural properties of random graphs enable circumvention of worst-case stretch–size trade-offs, yielding improved parameter balances; (2) It introduces a novel “joint generation” paradigm, simultaneously constructing the random graph instance and its target combinatorial structure; (3) It constructs the first efficient LCA that locally generates a maximum independent set on ER graphs with query complexity significantly below the worst-case bound, and achieves optimal stretch–size trade-offs for *k*-spanner access on both models.

Technology Category

• Search and Optimization: Local Search
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• 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
• Web Mining and Content Analysis: Models for Web evolution
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines

Abstract

We initiate the study of Local Computation Algorithms on average case inputs. In the Local Computation Algorithm (LCA) model, we are given probe access to a huge graph, and asked to answer membership queries about some combinatorial structure on the graph, answering each query with sublinear work. For instance, an LCA for the $k$-spanner problem gives access to a sparse subgraph $Hsubseteq G$ that preserves distances up to a factor of $k$. We build simple LCAs for this problem assuming the input graph is drawn from the well-studied Erdos-Reyni and Preferential Attachment graph models. In both cases, our spanners achieve size and stretch tradeoffs that are impossible to achieve for general graphs, while having dramatically lower query complexity than worst-case LCAs. Our second result investigates the intersection of LCAs with Local Access Generators (LAGs). Local Access Generators provide efficient query access to a random object, for instance an Erdos Reyni random graph. We explore the natural problem of generating a random graph together with a combinatorial structure on it. We show that this combination can be easier to solve than focusing on each problem by itself, by building a fast, simple algorithm that provides access to an Erdos Reyni random graph together with a maximal independent set.

Problem

• Study Local Computation Algorithms on average case inputs
• Build simple LCAs for k-spanner problem on specific graph models
• Generate random graphs with combinatorial structures efficiently

Innovation

• Average case Local Computation Algorithms for graphs
• Efficient spanners in Erdos-Reyni and Preferential Attachment models
• Combined random graph and maximal independent set generation