Mathematics > Optimization and Control
[Submitted on 12 Aug 2021 (v1), last revised 7 Dec 2021 (this version, v2)]
Title:An SDP-Based Approach for Computing the Stability Number of a Graph
View PDFAbstract:Finding the stability number of a graph, i.e., the maximum number of vertices of which no two are adjacent, is a well known NP-hard combinatorial optimization problem. Since this problem has several applications in real life, there is need to find efficient algorithms to solve this problem. Recently, Gaar and Rendl enhanced semidefinite programming approaches to tighten the upper bound given by the Lovász theta function. This is done by carefully selecting some so-called exact subgraph constraints (ESC) and adding them to the semidefinite program of computing the Lovász theta function.
First, we provide two new relaxations that allow to compute the bounds faster without substantial loss of the quality of the bounds. One of these two relaxations is based on including violated facets of the polytope representing the ESCs, the other one adds separating hyperplanes for that polytope.
Furthermore, we implement a branch and bound (B&B) algorithm using these tightened relaxations in our bounding routine. We compare the efficiency of our B&B algorithm using the different upper bounds. It turns out that already the bounds of Gaar and Rendl drastically reduce the number of nodes to be explored in the B&B tree as compared to the Lovász theta bound. However, this comes with a high computational cost. Our new relaxations improve the run time of the overall B&B algorithm, while keeping the number of nodes in the B&B tree small.
Submission history
From: Elisabeth Gaar [view email][v1] Thu, 12 Aug 2021 13:01:20 UTC (40 KB)
[v2] Tue, 7 Dec 2021 20:38:45 UTC (43 KB)
References & Citations
Bibliographic and Citation Tools
Bibliographic Explorer (What is the Explorer?)
Connected Papers (What is Connected Papers?)
Litmaps (What is Litmaps?)
scite Smart Citations (What are Smart Citations?)
Code, Data and Media Associated with this Article
alphaXiv (What is alphaXiv?)
CatalyzeX Code Finder for Papers (What is CatalyzeX?)
DagsHub (What is DagsHub?)
Gotit.pub (What is GotitPub?)
Hugging Face (What is Huggingface?)
Papers with Code (What is Papers with Code?)
ScienceCast (What is ScienceCast?)
Demos
Recommenders and Search Tools
Influence Flower (What are Influence Flowers?)
CORE Recommender (What is CORE?)
arXivLabs: experimental projects with community collaborators
arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.
Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.
Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.