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

TL;DR

The paper establishes that the maximized MinMax separation problems—Maximum Minimal $st$-Separator and Maximum Minimal OCT—are fixed-parameter tractable when parameterized by the solution size $k$. It treats CMSO-definable formulations and leverages the Lokshtanov et al. unbreakable-graphs metatheorem to reduce to $(q,k)$-unbreakable instances, where two problem-specific algorithms are developed: a branching approach for MaxMin $st$-Separator on unbreakable graphs with a certificate-based framework, and a combination of two safe structural lemmas (long induced odd cycles and Sunflower-based short cycles) for MaxMin OCT, complemented by an irrelevance-detection and bounded-instance brute-force step. Together these yield running times $(k-1)^{2q} \, n^{O(1)}$ for the $st$-separator and $2^{(qk)^{O(q)}} \, n^{O(1)}$ for OCT on unbreakable graphs, leading to overall FPT algorithms by the CMSO meta-theorem. The results resolve an open question and provide structural insights, highlighting how highly unbreakable graph structure can be exploited to tackle intrinsically extension-hard MaxMin variants. The work also outlines future directions, including faster FPT bounds and weighted or edge-deletion variants.

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.

Theorems & Definitions (55)

• Lemma 1
• proof
• Lemma 2: HAAS2006360
• proof
• Lemma 3
• proof
• Theorem 1
• Theorem 2
• Proposition 1.1: Theorem 1, Lokshtanov2018ReducingCM
• Theorem 3