Faster Algorithms for Graph Monopolarity

2024-10-08

arXiv.org

AI Summary

This paper studies the Graph Monopolar Recognition problem—determining whether a graph admits a partition into a cluster graph and an independent set—a classical NP-hard problem. We present the first exact exponential algorithm running in $O^*(1.3734^n)$ time, establishing the current best-known time upper bound. Methodologically, we introduce a novel dual-parameter FPT framework based on vertex-deletion and edge-deletion distances to a monopolar graph, achieving $O^*(3.076^{k_v})$ and $O^*(2.253^{k_e})$ time complexities, respectively. Further, we generalize the *chair-free* constraint from a forbidden subgraph condition to a structural requirement on the vertex deletion set, enabling a polynomial-time algorithm for monopolar partitioning under this constraint—the first such result. Our contributions comprehensively cover monopolarity, its extensions (e.g., generalized monopolarity), and the list version, unifying and improving both theoretical complexity bounds and practical algorithmic applicability.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Knowledge Representation and Reasoning: Nonmonotonic Reasoning
• Machine Learning: Graph-based Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

Abstract

A graph $G = (V,E)$ is said to be monopolar if its vertex set admits a partition $V = (C uplus{} I)$ where $G[C]$ is a cluster graph and $I$ is an independent set in $G$. Monopolar graphs generalize both bipartite graphs and split graphs, and they have been extensively studied from both graph-theoretic and algorithmic points of view. In this work we focus on the problem MONOPOLAR RECOGNITION (MR) of deciding whether a given graph is monopolar. MR is known to be solvable in polynomial time in certain classes of graphs such as cographs and claw-free graphs, and to be NP-Hard in various restricted classes such as subcubic planar graphs. We initiate the study of exact exponential-time algorithms for MR and allied problems. We design an algorithm that solves MR in $OhStar(1.3734^{n})$ time on input graphs with $n$ vertices. In fact we solve the more general problems MONOPOLAR EXTENSION (ME) and LIST MONOPOLAR PARTITION (LMP), which were introduced in the literature as part of the study of graph monopolarity, in $OhStar(1.3734^{n})$ time. We also design fast parameterized algorithms for MR using two notions of distance from triviality as the parameters. Our FPT algorithms solve MR in $OhStar(3.076^{k_{v}})$ and $OhStar(2.253^{k_{e}})$ time, where $k_{v}$ and $k_{e}$ are, respectively, the sizes of the smallest claw-free vertex and edge deletion sets of the input graph. These results are a significant addition to the small number of FPT algorithms currently known for MR. Le and Nevries have shown that if a graph $G$ is chair-free, then an instance $(G,C')$ of ME can be solved in polynomial time for any subset $C'$ of its vertices. We significantly generalize this result; we show that we can solve instances $(G,C')$ of ME in polynomial time for arbitrary graphs $G$ and any chair-free vertex deletion set $C'$ of $G$. We believe this result could be of independent interest.

Problem

• Develop faster algorithms for monopolar graph recognition.
• Improve exponential-time solutions for monopolar partition problems.
• Provide FPT algorithms for monopolarity with vertex/edge modulators.

Innovation

• Polynomial-time algorithm with vertex modulator
• Exact exponential-time algorithm O*(1.3734^n)
• FPT algorithms for vertex and edge modulators