FO and MSO approach to Some Graph Problems: Approximation and Poly time Results

TL;DR

This paper introduces a logical framework using FO and MSO logic to develop approximation algorithms and linear-time solutions for various graph modification, coloring, and domination problems, especially on special graph classes.

Contribution

It provides a unified logical approach to approximate and solve graph problems efficiently, extending known results to new problems and graph classes.

Findings

Constant factor polynomial-time approximation algorithms for node and edge deletion problems.

Linear-time algorithms for several graph problems on bounded tree-width graphs.

Unified logical framework for a broad class of graph problems.

Abstract

The focus of this paper is two fold. Firstly, we present a logical approach to graph modification problems such as minimum node deletion, edge deletion, edge augmentation problems by expressing them as an expression in first order (FO) logic. As a consequence, it follows that these problems have constant factor polynomial-time approximation algorithms. In particular, node deletion/edge deletion on a graph $G$ whose resultant is cograph, split, threshold, comparable, interval and permutation are $O(1)$ approximable. Secondly, we present a monadic second order (MSO) logic to minimum graph modification problems, minimum dominating set problem and minimum coloring problem and their variants. As a consequence, it follows that these problems have linear-time algorithms on bounded tree-width graphs. In particular, we show the existance of linear-time algorithms on bounded tree-width graphs for star coloring, cd-coloring, rainbow coloring, equitable coloring, total dominating set, connected dominating set. In a nut shell, this paper presents a unified framework and an algorithmic scheme through logical expressions for some graph problems through FO and MSO.