LP-branching algorithms based on biased graphs

TL;DR

This paper introduces a new combinatorial framework based on biased graphs for developing efficient LP-based fixed-parameter tractable algorithms for a wide range of graph optimization problems, simplifying applicability checks.

Contribution

The authors present a novel LP-branching approach leveraging biased graphs, broadening the scope of problems solvable with fixed-parameter algorithms and easing applicability verification.

Findings

Framework captures many existing VCSP-based problems

Includes new generalizations like Group Feedback Vertex Set

Provides an $O( ext{log OPT})$-approximation for all problems

Abstract

We give a combinatorial condition for the existence of efficient, LP-based FPT algorithms for a broad class of graph-theoretical optimisation problems. Our condition is based on the notion of biased graphs known from matroid theory. Specifically, we show that given a biased graph $\Psi=(G,\mathcal{B})$, where $\mathcal{B}$ is a class of balanced cycles in $G$, the problem of finding a set $X$ of at most $k$ vertices in $G$ which intersects every unbalanced cycle in $G$ admits an FPT algorithm using an LP-branching approach, similar to those previously seen for VCSP problems (Wahlstr\"om, SODA 2014). This framework captures many of the problems previously solved via the VCSP approach to LP-branching, as well as new generalisations, such as Group Feedback Vertex Set for infinite groups (e.g., for graphs whose edges are labelled by matrices). A major advantage compared to previous work is that it is immediate to check the applicability of the result for a given problem, whereas testing applicability of the VCSP approach for a specific VCSP requires determining the existence of an embedding language with certain algebraically defined properties, which is not known to be decidable in general. Additionally, we study the approximation question, and show that every problem of this category admits an $O(\log \text{OPT})$-approximation.