Document

# Approximation Algorithms for Partially Colorable Graphs

## File

LIPIcs.APPROX-RANDOM.2019.28.pdf
• Filesize: 0.57 MB
• 20 pages

## Acknowledgements

The first author thanks Pasin Manurangsi for pointing him to the Odd Cycle Transversal problem.

## Cite As

Suprovat Ghoshal, Anand Louis, and Rahul Raychaudhury. Approximation Algorithms for Partially Colorable Graphs. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 145, pp. 28:1-28:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2019.28

## Abstract

Graph coloring problems are a central topic of study in the theory of algorithms. We study the problem of partially coloring partially colorable graphs. For alpha <= 1 and k in Z^+, we say that a graph G=(V,E) is alpha-partially k-colorable, if there exists a subset S subset V of cardinality |S| >= alpha |V| such that the graph induced on S is k-colorable. Partial k-colorability is a more robust structural property of a graph than k-colorability. For graphs that arise in practice, partial k-colorability might be a better notion to use than k-colorability, since data arising in practice often contains various forms of noise. We give a polynomial time algorithm that takes as input a (1 - epsilon)-partially 3-colorable graph G and a constant gamma in [epsilon, 1/10], and colors a (1 - epsilon/gamma) fraction of the vertices using O~(n^{0.25 + O(gamma^{1/2})}) colors. We also study natural semi-random families of instances of partially 3-colorable graphs and partially 2-colorable graphs, and give stronger bi-criteria approximation guarantees for these family of instances.

## Subject Classification

##### ACM Subject Classification
• Mathematics of computing → Approximation algorithms
##### Keywords
• Approximation Algorithms
• Vertex Coloring
• Semi-random Models

## Metrics

• Access Statistics
• Total Accesses (updated on a weekly basis)
0

## References

1. Amit Agarwal, Moses Charikar, Konstantin Makarychev, and Yury Makarychev. O(sqrtlog n)) approximation algorithms for min UnCut, min 2CNF deletion, and directed cut problems. In Proceedings of the thirty-seventh annual ACM symposium on Theory of computing, pages 573-581. ACM, 2005.
2. Noga Alon and Nabil Kahale. A spectral technique for coloring random 3-colorable graphs. SIAM Journal on Computing, 26(6):1733-1748, 1997.
3. Sanjeev Arora and Eden Chlamtac. New approximation guarantee for chromatic number. In Proceedings of the thirty-eighth annual ACM symposium on Theory of computing, pages 215-224. ACM, 2006.
4. Sanjeev Arora and Rong Ge. New Tools for Graph Coloring. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques - 14th International Workshop, APPROX 2011, and 15th International Workshop, RANDOM 2011, Princeton, NJ, USA, August 17-19, 2011. Proceedings, pages 1-12, 2011.
5. Sanjeev Arora, Satish Rao, and Umesh Vazirani. Expander flows, geometric embeddings and graph partitioning. Journal of the ACM (JACM), 56(2):5, 2009.
6. Nikhil Bansal and Subhash Khot. Optimal long code test with one free bit. In 2009 50th Annual IEEE Symposium on Foundations of Computer Science, pages 453-462. IEEE, 2009.
7. Boaz Barak, Prasad Raghavendra, and David Steurer. Rounding Semidefinite Programming Hierarchies via Global Correlation. In IEEE 52nd Annual Symposium on Foundations of Computer Science, FOCS 2011, Palm Springs, CA, USA, October 22-25, 2011, pages 472-481, 2011.
8. Avrim Blum. Some tools for approximate 3-coloring. In Proceedings [1990] 31st Annual Symposium on Foundations of Computer Science, pages 554-562. IEEE, 1990.
9. Avrim Blum. New Approximation Algorithms for Graph Coloring. J. ACM, 41(3):470-516, 1994. URL: https://doi.org/10.1145/176584.176586.
10. Avrim Blum and David Karger. An algorithm for 3-colorable graphs. Information Processing Letters, 61(1):49–53, 1997. URL: https://doi.org/10.1016/s0020-0190(96)00190-1.
11. Avrim Blum and Joel Spencer. Coloring random and semi-random k-colorable graphs. Journal of Algorithms, 19(2):204-234, 1995.
12. Eden Chlamtac. Approximation Algorithms Using Hierarchies of Semidefinite Programming Relaxations. 48th Annual IEEE Symposium on Foundations of Computer Science (FOCS07), 2007. URL: https://doi.org/10.1109/focs.2007.72.
13. Roee David and Uriel Feige. On the effect of randomness on planted 3-coloring models. In Proceedings of the forty-eighth annual ACM symposium on Theory of Computing, pages 77-90. ACM, 2016.
14. Reinhard Diestel. Graph Theory, volume 173 of. Graduate texts in mathematics, page 7, 2012.
15. Uriel Feige and Joe Kilian. Heuristics for semirandom graph problems. Journal of Computer and System Sciences, 63(4):639-671, 2001.
16. Naveen Garg, Vijay V Vazirani, and Mihalis Yannakakis. Approximate max-flow min-(multi) cut theorems and their applications. SIAM Journal on Computing, 25(2):235-251, 1996.
17. David Karger, Rajeev Motwani, and Madhu Sudan. Approximate graph coloring by semidefinite programming. Journal of the ACM (JACM), 45(2):246-265, 1998.
18. Richard M Karp. Reducibility among combinatorial problems. In Complexity of computer computations, pages 85-103. Springer, 1972.
19. Ken-Ichi Kawarabayashi and Mikkel Thorup. Combinatorial Coloring of 3-Colorable Graphs. 2012 IEEE 53rd Annual Symposium on Foundations of Computer Science, 2012. URL: https://doi.org/10.1109/focs.2012.16.
20. Ken-Ichi Kawarabayashi and Mikkel Thorup. Coloring 3-Colorable Graphs with Less than n^1/5 Colors. Journal of the ACM, 64(1):1–23, 2017. URL: https://doi.org/10.1145/3001582.
21. Alexandra Kolla, Konstantin Makarychev, and Yury Makarychev. How to play unique games against a semi-random adversary: Study of semi-random models of unique games. In 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science, pages 443-452. IEEE, 2011.
22. Akash Kumar, Anand Louis, and Madhur Tulsiani. Finding Pseudorandom Colorings of Pseudorandom Graphs. In 37th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science, FSTTCS 2017, December 11-15, 2017, Kanpur, India, pages 37:1-37:12, 2017.
23. Daniel Lokshtanov, NS Narayanaswamy, Venkatesh Raman, MS Ramanujan, and Saket Saurabh. Faster parameterized algorithms using linear programming. ACM Transactions on Algorithms (TALG), 11(2):15, 2014.
24. Konstantin Makarychev, Yury Makarychev, and Aravindan Vijayaraghavan. Approximation algorithms for semi-random partitioning problems. In Proceedings of the 44th Symposium on Theory of Computing Conference, STOC 2012, New York, NY, USA, May 19 - 22, 2012, pages 367-384, 2012.
25. Konstantin Makarychev, Yury Makarychev, and Aravindan Vijayaraghavan. Algorithms for Semi-random Correlation Clustering. CoRR, abs/1406.5667, 2014. URL: http://arxiv.org/abs/1406.5667.
26. Theo McKenzie, Hermish Mehta, and Luca Trevisan. A New Algorithm for the Robust Semi-random Independent Set Problem. CoRR, abs/1808.03633, 2018. URL: http://arxiv.org/abs/1808.03633.
27. Burkhard Monien and Ewald Speckenmeyer. Ramsey numbers and an approximation algorithm for the vertex cover problem. Acta Informatica, 22(1):115-123, 1985.
28. NS Narayanaswamy, Venkatesh Raman, MS Ramanujan, and Saket Saurabh. LP can be a cure for parameterized problems. In STACS'12 (29th Symposium on Theoretical Aspects of Computer Science), volume 14, pages 338-349. LIPIcs, 2012.
29. Prasad Raghavendra. Optimal algorithms and inapproximability results for every CSP? In Proceedings of the 40th Annual ACM Symposium on Theory of Computing, Victoria, British Columbia, Canada, May 17-20, 2008, pages 245-254, 2008.
30. Prasad Raghavendra and David Steurer. How to Round Any CSP. In 50th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2009, October 25-27, 2009, Atlanta, Georgia, USA, pages 586-594, 2009.
31. Bruce Reed, Kaleigh Smith, and Adrian Vetta. Finding odd cycle transversals. Operations Research Letters, 32(4):299-301, 2004.
32. Jacob Steinhardt. Does robustness imply tractability? A lower bound for planted clique in the semi-random model. arXiv preprint, 2017. URL: http://arxiv.org/abs/1704.05120.
33. Avi Wigderson. Improving the performance guarantee for approximate graph coloring. Journal of the ACM, 30(4):729–735, January 1983. URL: https://doi.org/10.1145/2157.2158.
34. Mihalis Yannakakis. Node-and edge-deletion NP-complete problems. In Proceedings of the tenth annual ACM symposium on Theory of computing, pages 253-264. ACM, 1978.