Deterministic polynomial kernel accelerates odd cycle transversal problem
A Deterministic Polynomial Kernel for Odd Cycle Transversal
Data Structures and Algorithms
Summary
The paper deals with a puzzle in computer science called the Odd Cycle Transversal problem, which involves finding and removing certain odd loops in a network. The authors developed a method to simplify this problem into a smaller version that’s easier to solve and does it deterministically, without relying on chance. They achieved this by using a mathematical structure called gammoids and advanced ways to represent them efficiently. This makes solving or approximating the problem more practical in polynomial time.
What this means in practice
- •For network designers: Simplify large network problems involving odd cycles into smaller, manageable versions for efficient analysis and correction.
- •For software engineers: Create faster deterministic algorithms for checking and fixing odd cycle issues in graph-based systems and data structures.
A theory result. No direct application yet.
Authors
Tomohiro Koana, Soh Kumabe
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.