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# Approximate Hypergraph Coloring under Low-discrepancy and Related Promises

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## Cite As

Vijay V. S. P. Bhattiprolu, Venkatesan Guruswami, and Euiwoong Lee. Approximate Hypergraph Coloring under Low-discrepancy and Related Promises. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 40, pp. 152-174, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)
https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2015.152

## Abstract

A hypergraph is said to be X-colorable if its vertices can be colored with X colors so that no hyperedge is monochromatic. 2-colorability is a fundamental property (called Property B) of hypergraphs and is extensively studied in combinatorics. Algorithmically, however, given a 2-colorable k-uniform hypergraph, it is NP-hard to find a 2-coloring miscoloring fewer than a fraction 2^(-k+1) of hyperedges (which is trivially achieved by a random 2-coloring), and the best algorithms to color the hypergraph properly require about n^(1-1/k) colors, approaching the trivial bound of n as k increases. In this work, we study the complexity of approximate hypergraph coloring, for both the maximization (finding a 2-coloring with fewest miscolored edges) and minimization (finding a proper coloring using fewest number of colors) versions, when the input hypergraph is promised to have the following stronger properties than 2-colorability: (A) Low-discrepancy: If the hypergraph has a 2-coloring of discrepancy l << sqrt(k), we give an algorithm to color the hypergraph with about n^(O(l^2/k)) colors. However, for the maximization version, we prove NP-hardness of finding a 2-coloring miscoloring a smaller than 2^(-O(k)) (resp. k^(-O(k))) fraction of the hyperedges when l = O(log k) (resp. l=2). Assuming the Unique Games conjecture, we improve the latter hardness factor to 2^(-O(k)) for almost discrepancy-1 hypergraphs. (B) Rainbow colorability: If the hypergraph has a (k-l)-coloring such that each hyperedge is polychromatic with all these colors (this is stronger than a (l+1)-discrepancy 2-coloring), we give a 2-coloring algorithm that miscolors at most k^(-Omega(k)) of the hyperedges when l << sqrt(k), and complement this with a matching Unique Games hardness result showing that when l = sqrt(k), it is hard to even beat the 2^(-k+1) bound achieved by a random coloring. (C) Strong Colorability: We obtain similar (stronger) Min- and Max-2-Coloring algorithmic results in the case of (k+l)-strong colorability.
##### Keywords
• Hypergraph Coloring
• Discrepancy
• Rainbow Coloring
• Stong Coloring
• Algorithms
• Semidefinite Programming
• Hardness of Approximation

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## References

1. Geir Agnarsson and Magnús M Halldórsson. Strong colorings of hypergraphs. In Approximation and Online Algorithms, pages 253-266. Springer, 2005.
2. Noga Alon. Personal communication, 2014.
3. Noga Alon, Pierre Kelsen, Sanjeev Mahajan, and Ramesh Hariharan. Approximate hypergraph coloring. Nordic Journal of Computing, 3(4):425-439, 1996.
4. Kazuhiko Aomoto et al. Analytic structure of schläfli function. Nagoya Math. J, 68:1-16, 1977.
5. Per Austrin, Venkatesan Guruswami, and Johan Håstad. (2 + ε)-SAT is NP-hard. In Foundations of Computer Science (FOCS), 2014 IEEE 55th Annual Symposium on. IEEE, 2014.
6. Per Austrin and Johan Håstad. On the usefulness of predicates. ACM Trans. Comput. Theory, 5(1):1:1-1:24, May 2013.
7. Per Austrin and Elchanan Mossel. Approximation resistant predicates from pairwise independence. computational complexity, 18(2):249-271, 2009.
8. Nikhil Bansal. Constructive algorithms for discrepancy minimization. In Foundations of Computer Science (FOCS), 2014 IEEE 51st Annual Symposium on, FOCS'10, pages 3-10. IEEE, 2010.
9. Nikhil Bansal and Subhash Khot. Inapproximability of hypergraph vertex cover and applications to scheduling problems. In Proceedings of the 37th International Colloquium on Automata, Languages and Programming, ICALP'10, pages 250-261, 2010.
10. V. Bhattiprolu, V. Guruswami, and E. Lee. Approximate Hypergraph Coloring under Low-discrepancy and Related Promises. Arxiv, 2015. arXiv:1506.06444.
11. Béla Bollobás, David Pritchard, Thomas Rothvoß, and Alex Scott. Cover-decomposition and polychromatic numbers. SIAM Journal on Discrete Mathematics, 27(1):240-256, 2013.
12. Károly Böröczky Jr and Martin Henk. Random projections of regular polytopes. Archiv der Mathematik, 73(6):465-473, 1999.
13. Hui Chen and Alan M Frieze. Coloring bipartite hypergraphs. In Proceedings of the 5th international conference on Integer Programming and Combinatorial Optimization, IPCO'96, pages 345-358, 1996.
14. Irit Dinur and Venkatesan Guruswami. PCPs via low-degree long code and hardness for constrained hypergraph coloring. In Foundations of Computer Science (FOCS), 2014 IEEE 54th Annual Symposium on, FOCS 13, pages 340-349, 2013.
15. Irit Dinur, Oded Regev, and Clifford D. Smyth. The hardness of 3-Uniform hypergraph coloring. Combinatorica, 25(1):519-535, 2005.
16. Paul Erdos and László Lovász. Problems and results on 3-chromatic hypergraphs and some related questions. Infinite and finite sets, 10(2):609-627, 1975.
17. Venkatesan Guruswami, Johan Håstad, Prahladh Harsha, Srikanth Srinivasan, and Girish Varma. Super-polylogarithmic hypergraph coloring hardness via low-degree long codes. In Proceedings of the 46th annual ACM Symposium on Theory of Computing, STOC'14, 2014.
18. Venkatesan Guruswami, Johan Håstad, and Madhu Sudan. Hardness of approximate hypergraph coloring. SIAM Journal on Computing, 31(6):1663-1686, 2002.
19. Venkatesan Guruswami and Euiwoong Lee. Strong inapproximability results on balanced rainbow-colorable hypergraphs. In Proceedings of the 26th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA'15, pages 822-836, 2015.
20. Venkatesan Guruswami and Euiwoong Lee. Towards a characterization of approximation resistance for symmetric CSPs. To Appear, The 18th. International Workshop on Approximation Algorithms for Combinatorial Optimization Problems (APPROX), 2015.
21. Johan Håstad. Some optimal inapproximability results. Journal of the ACM, 48(4):798-859, July 2001.
22. Sangxia Huang. 2^(log N)^1/4 - o(1) hardness for hypergraph coloring. Electronic Colloquium on Computational Complexity (ECCC), 15-062, 2015.
23. David Karger, Rajeev Motwani, and Madhu Sudan. Approximate graph coloring by semidefinite programming. Journal of the ACM (JACM), 45(2):246-265, 1998.
24. Subhash Khot, Guy Kindler, Elchanan Mossel, and Ryan O'Donnell. Optimal inapproximability results for Max-Cut and other 2-variable CSPs? SIAM Journal on Computing, 37(1):319-357, 2007.
25. Subhash Khot and Rishi Saket. Hardness of coloring 2-colorable 12-uniform hypergraphs with exp(log^Ω (1) n) colors. In Foundations of Computer Science (FOCS), 2014 IEEE 55th Annual Symposium on, pages 206-215. IEEE, 2014.
26. Hellmuth Kneser. Der simplexinhalt in der nichteuklidischen geometrie. Deutsche Math, 1:337-340, 1936.
27. László Lovász. On the shannon capacity of a graph. Information Theory, IEEE Transactions on, 25(1):1-7, 1979.
28. Shachar Lovett and Raghu Meka. Constructive discrepancy minimization by walking on the edges. In Foundations of Computer Science (FOCS), 2014 IEEE 53rd Annual Symposium on, FOCS 12, pages 61-67, 2012.
29. Colin McDiarmid. A random recolouring method for graphs and hypergraphs. Combinatorics, Probability and Computing, 2(03):363-365, 1993.
30. Jun Murakami and Masakazu Yano. On the volume of a hyperbolic and spherical tetrahedron. Communications in analysis and geometry, 13(2):379, 2005.
31. CA Rogers. An asymptotic expansion for certain schläfli functions. Journal of the London Mathematical Society, 1(1):78-80, 1961.
32. Claude Ambrose Rogers. Packing and covering. Number 54 in Cambridge Tracts in Mathematics and Mathematical Physics. Cambridge University Press, 1964.
33. Thomas Rothvoß. Constructive discrepancy minimization for convex sets. In Foundations of Computer Science (FOCS), 2014 IEEE 55th Annual Symposium on, pages 140-145. IEEE, 2014.
34. S. Sachdeva and R. Saket. Optimal inapproximability for scheduling problems via structural hardness for hypergraph vertex cover. In Proceedings of the 28th annual IEEE Conference on Computational Complexity, CCC'13, pages 219-229, 2013.
35. Ludwig Schläfli. On the multiple integral ∫ … ∫ dx dy... dz, whose limits are p1 = a1x+ b1y+… + h1z > 0, p2 > 0,..., pn > 0, and x2+ y2+… + z2 < 1. Quart. J. Math, 2(1858):269-300, 1858.
36. Joel Spencer. Six standard deviations suffice. Transactions of the American Mathematical Society, 289(2):679-706, 1985.
37. Cenny Wenner. Parity is positively useless. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques: The 17th. International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, pages 433-448. Schloss Dagstuhl, 2014.
38. Avi Wigderson. Improving the performance guarantee for approximate graph coloring. Journal of the ACM (JACM), 30(4):729-735, 1983.
39. Uri Zwick. Approximation algorithms for constraint satisfaction problems involving at most three variables per constraint. In Proceedings of the 9th Annual ACM-SIAM Symposium on Discrete Algorithms, volume 98 of SODA'98, pages 201-210, 1998.