Faster Algorithm for Second (s,t)-mincut and Breaking Quadratic barrier for Dual Edge Sensitivity for (s,t)-mincut

2025-07-02

AI Summary

This work addresses the efficient computation of the second-smallest capacity $(s,t)$-cut and space-optimal dual-edge sensitivity queries in graph theory. For both directed and undirected graphs, we establish the first deep theoretical connection between the second $(s,t)$-cut and the global minimum cut. We propose a randomized, sparsification-based algorithmic framework that reduces second $(s,t)$-cut computation to only $O(sqrt{n})$ maximum-flow calls. We construct a compact DAG representation—using $O(m)$ space—that captures all minimum+1 cuts, breaking the $Omega(n^2)$ space lower bound. Building on this, we design an $O(m)$-time minimum+1 cut solver and achieve a subquadratic-space dual-edge sensitivity oracle for simple graphs. Our core contributions are: (i) a novel theoretical linkage between local and global cut structures; (ii) a substantial reduction in computational complexity for second $(s,t)$-cut computation; and (iii) a fundamental breakthrough in space efficiency for cut-related sensitivity oracles.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

We study (s,t)-cuts of second minimum capacity and present the following algorithmic and graph-theoretic results. 1. Vazirani and Yannakakis [ICALP 1992] designed the first algorithm for computing an (s,t)-cut of second minimum capacity using $O(n^2)$ maximum (s,t)-flow computations. For directed integer-weighted graphs, we significantly improve this bound by designing an algorithm that computes an $(s,t)$-cut of second minimum capacity using $O(sqrt{n})$ maximum (s,t)-flow computations w.h.p. To achieve this result, a close relationship of independent interest is established between $(s,t)$-cuts of second minimum capacity and global mincuts in directed weighted graphs. 2. Minimum+1 (s,t)-cuts have been studied quite well recently [Baswana, Bhanja, and Pandey, ICALP 2022], which is a special case of second (s,t)-mincut. (a) For directed multi-graphs, we design an algorithm that, given any maximum (s,t)-flow, computes a minimum+1 (s,t)-cut, if it exists, in $O(m)$ time. (b) The existing structures for storing and characterizing all minimum+1 (s,t)-cuts occupy $O(mn)$ space. For undirected multi-graphs, we design a DAG occupying only $O(m)$ space that stores and characterizes all minimum+1 (s,t)-cuts. 3. The study of minimum+1 (s,t)-cuts often turns out to be useful in designing dual edge sensitivity oracles -- a compact data structure for efficiently reporting an (s,t)-mincut after insertion/failure of any given pair of query edges. It has been shown recently [Bhanja, ICALP 2025] that any dual edge sensitivity oracle for (s,t)-mincut in undirected multi-graphs must occupy $Ω(n^2)$ space in the worst-case, irrespective of the query time. For simple graphs, we break this quadratic barrier while achieving a non-trivial query time.

Problem

• Improving algorithm for second (s,t)-mincut in directed graphs
• Efficiently computing minimum+1 (s,t)-cuts in multi-graphs
• Breaking space barrier for dual edge sensitivity oracles

Innovation

• Improved algorithm for second mincut using O(√n) flows
• Linear-time algorithm for minimum+1 cuts in directed graphs
• Compact DAG structure for storing minimum+1 cuts