# 0/1/all CSPs, Half-Integral $A$-path Packing, and Linear-Time FPT Algorithms

## TL;DR

This paper introduces a fast $O(km)$-time LP solving algorithm for FPT problems, enabling linear-time algorithms with minimal parameter dependency for several complex graph problems.

## Contribution

It presents a novel $O(km)$ algorithm for solving LP relaxations in FPT algorithms, improving efficiency and enabling linear-time solutions for multiple problems.

## Key findings

- Achieved linear-time FPT algorithms for Group Feedback Vertex Set and Non-monochromatic Cycle Transversal.
- Developed an $O(km)$ LP solving algorithm based on 0/1/all constraints and $A$-path packing.
- Provided the first linear-time FPT algorithms for certain graph problems.

## Abstract

A recent trend in the design of FPT algorithms is exploiting the half-integrality of LP relaxations. In other words, starting with a half-integral optimal solution to an LP relaxation, we assign integral values to variables one-by-one by branch and bound. This technique is general and the resulting time complexity has a low dependency on the parameter. However, the time complexity often becomes a large polynomial in the input size because we need to compute half-integral optimal LP solutions. In this paper, we address this issue by providing an $O(km)$-time algorithm for solving the LPs arising from various FPT problems, where $k$ is the optimal value and $m$ is the number of edges/constraints. Our algorithm is based on interesting connections among 0/1/all constraints, which has been studied in the field of constraints satisfaction, $A$-path packing, which has been studied in the field of combinatorial optimization, and the LPs used in FPT algorithms. With the aid of this algorithm, we obtain improved FPT algorithms for various problems, including Group Feedback Vertex Set, Subset Feedback Vertex Set, Node Multiway Cut, Node Unique Label Cover, and Non-monochromatic Cycle Transversal. The obtained running time for each of these problems is linear in the input size and has the current smallest dependency on the parameter. In particular, these algorithms are the first linear-time FPT algorithms for problems including Group Feedback Vertex Set and Non-monochromatic Cycle Transversal.