Reducing Matroid Optimization to Basis Search

2024-08-07

arXiv.org

AI Summary

This paper investigates parallel optimization of binary matroids under the independence oracle model, aiming to break the classical greedy algorithm’s Ω(n) lower bound on adaptive rounds without relying on graph-structural assumptions. The method introduces a novel optimality criterion grounded in local optimality of dual matroid circuits, transforming global optimal basis search into a parallelizable basis verification task. Integrating the KUW88 parallel basis-search framework with structural properties of binary matroids, we design the first adaptive algorithm achieving sublinear round complexity—O(n⁰·⁹⁹). Key contributions include: (i) establishing a new optimality characterization via dual circuits; (ii) achieving, for the first time, adaptive complexity strictly better than greedy for non-graphic matroids; and (iii) substantially reducing adaptivity overhead in parallel matroid optimization.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Reasoning under Uncertainty: Stochastic Optimization
• Machine Learning: Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

Matroids provide one of the most elegant structures for algorithm design. This is best identified by the Edmonds-Rado theorem relating the success of the simple greedy algorithm to the anatomy of the optimal basis of a matroid [Edm71; Rad57]. As a response, much energy has been devoted to understanding a matroid's computational properties. Yet, less is understood where parallel algorithms are concerned. In response, we initiate the study of parallel matroid optimization in the adaptive complexity model [BS18]. First, we reexamine Bor {u}vka's classical minimum weight spanning tree algorithm [Bor26b; Bor26a] in the abstract language of matroid theory, and identify a new certificate of optimality for the basis of any matroid as a result. In particular, a basis is optimal if and only if it contains the points of minimum weight in every circuit of the dual matroid. Hence, we can witnesses whether any specific point belongs to the optimal basis via a test for local optimality in a circuit of the dual matroid, thereby revealing a general design paradigm towards parallel matroid optimization. To instantiate this paradigm, we use the special structure of a binary matroid to identify an optimization scheme with low adaptivity. Here, our key technical step is reducing optimization to the simpler task of basis search in the binary matroid, using only logarithmic overhead of adaptive rounds of queries to independence oracles. Consequentially, we compose our reduction with the parallel basis search method of [KUW88] to obtain an algorithm for finding the optimal basis of a binary matroid terminating in sublinearly many adaptive rounds of queries to an independence oracle. To the authors' knowledge, this is the first algorithm for matroid optimization to outperform the greedy algorithm in terms of adaptive complexity in the independence query model without assuming the matroid is encoded by a graph.

Problem

• Parallel Optimization
• Matroid
• Adaptive Query

Innovation

• Matroid Optimization
• Boruvka's Algorithm Adaptation
• Binary Matroid Adaptive Queries