MaxMin Separation Problems: FPT Algorithms for $st$-Separator and Odd Cycle Transversal

2025-02-11

AI Summary

This paper investigates the parameterized complexity of two classical separation problems—Max-Min $st$-separator and Max-Min Odd Cycle Transversal (OCT)—parameterized by solution size $k$. The former asks whether a graph contains a minimal $st$-separator of size at least $k$; the latter asks whether it contains a minimal odd cycle transversal of size at least $k$. We resolve an open question posed by Hanaka et al. by showing both problems are fixed-parameter tractable (FPT). Our approach leverages a meta-theorem of Lokshtanov et al. to reduce each problem to highly inseparable graphs, then combines minimal solution enumeration, verification, and structured dynamic programming to design $f(k) cdot n^{O(1)}$-time algorithms. This work establishes the first FPT algorithms for these Max-Min separation problems, thereby confirming their parameterized tractability, and precisely delineates the complexity boundary between the basic problems and their extended variants.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Topic discovery and tracking
• Social Networks and Social Media: Social mining and social search on the Web

Abstract

In this paper, we study the parameterized complexity of the MaxMin versions of two fundamental separation problems: Maximum Minimal $st$-Separator and Maximum Minimal Odd Cycle Transversal (OCT), both parameterized by the solution size. In the Maximum Minimal $st$-Separator problem, given a graph $G$, two distinct vertices $s$ and $t$ and a positive integer $k$, the goal is to determine whether there exists a minimal $st$-separator in $G$ of size at least $k$. Similarly, the Maximum Minimal OCT problem seeks to determine if there exists a minimal set of vertices whose deletion results in a bipartite graph, and whose size is at least $k$. We demonstrate that both problems are fixed-parameter tractable parameterized by $k$. Our FPT algorithm for Maximum Minimal $st$-Separator answers the open question by Hanaka, Bodlaender, van der Zanden and Ono (TCS 2019). One unique insight from this work is the following. We use the meta-result of Lokshtanov, Ramanujan, Saurabh and Zehavi (ICALP 2018) that enables us to reduce our problems to highly unbreakable graphs. This is interesting, as an explicit use of the recursive understanding and randomized contractions framework of Chitnis, Cygan, Hajiaghayi, Pilipczuk and Pilipczuk (SICOMP 2016) to reduce to the highly unbreakable graphs setting (which is the result that Lokshtanov et al. tries to abstract out in their meta-theorem) does not seem obvious because certain ``extension'' variants of our problems are W[1]-hard.

Problem

• FPT algorithms for MaxMin separation problems
• Parameterized complexity of st-Separator and OCT
• Reduction to highly unbreakable graphs

Innovation

• Fixed-parameter tractable (FPT) algorithms
• Reduction to highly unbreakable graphs
• Meta-result for problem abstraction