Layered Graphs: Applications and Algorithms

Bhadrachalam Chitturi,Krithic Puthiyoppil,Srijith Balachander,Sandeep Satheesh

doi:10.3390/a11070093

Bhadrachalam Chitturi, Krithic Puthiyoppil + Show 2 more

Open Access

https://doi.org/10.3390/a11070093

Copy DOI

Abstract

The computation of distances between strings has applications in molecular biology, music theory and pattern recognition. One such measure, called short reversal distance, has applications in evolutionary distance computation. It has been shown that this problem can be reduced to the computation of a maximum independent set on the corresponding graph that is constructed from the given input strings. The constructed graphs primarily fall into a class that we call layered graphs. In a layered graph, each layer refers to a subgraph containing, at most, some k vertices. The inter-layer edges are restricted to the vertices in adjacent layers. We study the MIS, MVC, MDS, MCV and MCD problems on layered graphs where MIS computes the maximum independent set; MVC computes the minimum vertex cover; MDS computes the minimum dominating set; MCV computes the minimum connected vertex cover; and MCD computes the minimum connected dominating set. MIS, MVC and MDS run in polynomial time if k=Θ(log|V|). MCV and MCD run in polynomial time ifk=O((log|V|)α), where α<1. If k=Θ((log|V|)1+ϵ), for ϵ>0, then MIS, MVC and MDS run in quasi-polynomial time. If k=Θ(log|V|), then MCV and MCD run in quasi-polynomial time.

Highlights

A string is a sequence of symbols from an alphabet Σ, in which a symbol can be repeated
If k = Θ((log | V |)1+e ), for e > 0, MIS, MVC and MDS run in quasi-polynomial time
If k = Θ(log | V |), MCV and MCD run in quasi-polynomial time

Summary

Introduction

A string is a sequence of symbols from an alphabet Σ, in which a symbol can be repeated. The computation of the short reversal distance between α and β is reduced to the computation of the cardinality of the maximum independent set on a conflict graph constructed from α and β. The maximum independent set problem on a general graph is known to be NP-complete [6]. Hsiao et al designed an O(n) time algorithm to solve the maximum weight independent set problem on an interval graph with n vertices, given its interval representation with a sorted endpoints list [14]. Garey and Johnson proved that the MCV problem is NP-hard in planar graphs, with a maximum degree of 4 [6]. Garey and Johnson showed that the MDS problems on planar graphs with maximum vertex degree. We define a new class of graphs that we call layered graphs and design algorithms for various graph-theoretic problems

Layered Graph

MVC and MCV

MDS and MCD

Compatible Algorithm

Correctness and Complexity

Conclusions