A Deterministic Polynomial Kernel for Odd Cycle Transversal

2026-09-24

AI Summary

This study addresses the challenge of derandomizing kernelization for the Odd Cycle Transversal problem in parameterized complexity. Methodologically, it constructs quasi-matroid representations of gammoids and integrates NC algorithms for matching with matrix rank approximation techniques to establish a deterministic polynomial-time framework for computing representative families. The primary contribution is achieving the first deterministic polynomial kernelization for this problem, successfully derandomizing the classical randomized kernelization result of Kratsch and Wahlström. By completely eliminating the uncertainty inherent in existing algorithms, this work resolves a long-standing theoretical open problem in the field.

Abstract

We give a deterministic polynomial kernel for Odd Cycle Transversal, derandomizing the randomized kernel of Kratsch and Wahlström (TALG 2014). Our algorithm uses a deterministic polynomial-time construction of almost multilinear representations of gammoids. Such a representation assigns a block of columns to each element so that, for every subset of elements, the normalized matrix rank approximates its matroid rank to within a prescribed additive error $δ$. The construction builds on recent breakthroughs in NC algorithms for matching. Our kernelization algorithm then computes the required representative families from these representations.

Problem

• Odd Cycle Transversal
• Polynomial Kernel
• Derandomization
• Kernelization

Innovation

• Odd Cycle Transversal
• Deterministic Polynomial Kernel
• Gammoids
• Almost Multilinear Representations
• Representative Families