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# Editing Graphs to Satisfy Diversity Requirements

• Huda Chuangpishit
• Manuel Lafond
• Lata Narayanan
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11346)

## Abstract

Let G be a graph where every vertex has a colour and has specified diversity constraints, that is, a minimum number of neighbours of every colour. Every vertex also has a max-degree constraint: an upper bound on the total number of neighbours. In the Min-Edit-Cost problem, we wish to transform G using edge additions and/or deletions into a graph $$G'$$ where every vertex satisfies all diversity as well as max-degree constraints. We show an $$O(n^5 \log n)$$ algorithm for the Min-Edit-Cost problem, and an $$O(n^3 \log n \log \log n)$$ algorithm for the bipartite case. Given a specified number of edge operations, the Max-Satisfied-Nodes problem is to find the maximum number of vertices whose diversity constraints can be satisfied while ensuring that all max-degree constraints are satisfied. We show that the Max-Satisfied-Nodes problem is W[1]-hard, in parameter $$r+ \ell$$, where r is the number of edge operations and $$\ell$$ is the number of vertices to be satisfied. We also show that it is inapproximable to within a factor of $$n^{1/2-\epsilon }$$. For certain relaxations of the max-degree constraints, we are able to show constant-factor approximation algorithms for the problem.

## Notes

### Acknowledgement

We thank Jaroslav Opatrny for useful discussions.

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## Copyright information

© Springer Nature Switzerland AG 2018

## Authors and Affiliations

• Huda Chuangpishit
• 1
• Manuel Lafond
• 2
Email author
• Lata Narayanan
• 3
1. 1.Department of MathematicsRyerson UniversityTorontoCanada
2. 2.Department of Computer ScienceUniversité de SherbrookeSherbrookeCanada
3. 3.Department of Computer Science and Software EngineeringConcordia UniversityMontréalCanada