Weighted Min-Cut: Sequential, Cut-Query and Streaming Algorithms
Sagnik Mukhopadhyay, Danupon Nanongkai

TL;DR
This paper introduces new efficient algorithms for the 2-respecting min-cut problem in weighted graphs, improving upon previous bounds and providing solutions suitable for sequential, query-based, and streaming models.
Contribution
The paper presents a novel approach to the 2-respecting min-cut problem that simplifies implementation and improves efficiency across multiple computational models.
Findings
Sequential algorithm with improved time complexity over Karger's bounds.
Cut query algorithm requiring O(n) queries for weighted graphs.
Streaming algorithm using O(n) space and O( ext{log} n) passes.
Abstract
Consider the following 2-respecting min-cut problem. Given a weighted graph and its spanning tree , find the minimum cut among the cuts that contain at most two edges in . This problem is an important subroutine in Karger's celebrated randomized near-linear-time min-cut algorithm [STOC'96]. We present a new approach for this problem which can be easily implemented in many settings, leading to the following randomized min-cut algorithms for weighted graphs. * An -time sequential algorithm: This improves Karger's and bounds when the input graph is not extremely sparse or dense. Improvements over Karger's bounds were previously known only under a rather strong assumption that the input graph is simple [Henzinger et al. SODA'17; Ghaffari et al. SODA'20]. For…
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