Document

# The Densest k-Subhypergraph Problem

## File

LIPIcs.APPROX-RANDOM.2016.6.pdf
• Filesize: 0.6 MB
• 19 pages

## Cite As

Eden Chlamtac, Michael Dinitz, Christian Konrad, Guy Kortsarz, and George Rabanca. The Densest k-Subhypergraph Problem. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 60, pp. 6:1-6:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)
https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2016.6

## Abstract

The Densest k-Subgraph (DkS) problem, and its corresponding minimization problem Smallest p-Edge Subgraph (SpES), have come to play a central role in approximation algorithms. This is due both to their practical importance, and their usefulness as a tool for solving and establishing approximation bounds for other problems. These two problems are not well understood, and it is widely believed that they do not an admit a subpolynomial approximation ratio (although the best known hardness results do not rule this out). In this paper we generalize both DkS and SpES from graphs to hypergraphs. We consider the Densest k-Subhypergraph problem (given a hypergraph (V, E), find a subset W subseteq V of k vertices so as to maximize the number of hyperedges contained in W) and define the Minimum p-Union problem (given a hypergraph, choose p of the hyperedges so as to minimize the number of vertices in their union). We focus in particular on the case where all hyperedges have size 3, as this is the simplest non-graph setting. For this case we provide an O(n^{4(4-sqrt{3})/13 + epsilon}) <= O(n^{0.697831+epsilon})-approximation (for arbitrary constant epsilon > 0) for Densest k-Subhypergraph and an ~O(n^{2/5})-approximation for Minimum p-Union. We also give an O(sqrt{m})-approximation for Minimum p-Union in general hypergraphs. Finally, we examine the interesting special case of interval hypergraphs (instances where the vertices are a subset of the natural numbers and the hyperedges are intervals of the line) and prove that both problems admit an exact polynomial time solution on these instances.
##### Keywords
• Hypergraphs
• Approximation algorithms

## Metrics

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

## References

1. Noga Alon, Sanjeev Arora, Rajsekar Manokaran, Dana Moshkovitz, and Omri Weinstein. Inapproximability of densest k-subgraph from average case hardness. Unpublished manuscript, 2011.
2. Benny Applebaum. Pseudorandom generators with long stretch and low locality from random local one-way functions. SIAM Journal on Computing, 42(5):2008-2037, 2013. URL: http://dx.doi.org/10.1137/120884857.
3. Benny Applebaum, Boaz Barak, and Avi Wigderson. Public-key cryptography from different assumptions. In Proceedings of the Forty-second ACM Symposium on Theory of Computing, STOC'10, pages 171-180, 2010.
4. S. Arora, B. Barak, M. Brunnermeier, and R. Ge. Complicational complexity and information asymmetry in finnancial products. Submitted, 2016.
5. Sanjeev Arora and Rong Ge. 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, chapter New Tools for Graph Coloring, pages 1-12. Springer Berlin Heidelberg, Berlin, Heidelberg, 2011. URL: http://dx.doi.org/10.1007/978-3-642-22935-0_1.
6. Aditya Bhaskara. Finding Dense Structures in Graphs and Matrices. PhD thesis, Princeton University, 2012.
7. Aditya Bhaskara, Moses Charikar, Eden Chlamtac, Uriel Feige, and Aravindan Vijayaraghavan. Detecting high log-densities: an O(n^1/4) approximation for densest k-subgraph. In Proceedings of the 42nd ACM Symposium on Theory of Computing, STOC 2010, Cambridge, Massachusetts, USA, 5-8 June 2010, pages 201-210, 2010.
8. Moses Charikar. Greedy approximation algorithms for finding dense components in a graph. In Approximation Algorithms for Combinatorial Optimization, Third International Workshop, APPROX 2000, Saarbrücken, Germany, September 5-8, 2000, Proceedings, pages 84-95, 2000. URL: http://dx.doi.org/10.1007/3-540-44436-X_10.
9. Eden Chlamtac, Michael Dinitz, and Robert Krauthgamer. Everywhere-sparse spanners via dense subgraphs. In 53rd Annual IEEE Symposium on Foundations of Computer Science, FOCS 2012, New Brunswick, NJ, USA, October 20-23, 2012, pages 758-767, 2012.
10. Julia Chuzhoy, Yury Makarychev, Aravindan Vijayaraghavan, and Yuan Zhou. Approximation algorithms and hardness of the k-route cut problem. ACM Trans. Algorithms, 12(1):2:1-2:40, December 2015. URL: http://dx.doi.org/10.1145/2644814.
11. Michael Dinitz, Guy Kortsarz, and Zeev Nutov. Improved Approximation Algorithm for Steiner k-Forest with Nearly Uniform Weights. In Klaus Jansen, José D. P. Rolim, Nikhil R. Devanur, and Cristopher Moore, editors, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014), volume 28 of Leibniz International Proceedings in Informatics (LIPIcs), pages 115-127, Dagstuhl, Germany, 2014. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik. URL: http://drops.dagstuhl.de/opus/volltexte/2014/4692, URL: http://dx.doi.org/10.4230/LIPIcs.APPROX-RANDOM.2014.115.
12. Uriel Feige. Relations between average case complexity and approximation complexity. In Proceedings of the Thiry-fourth Annual ACM Symposium on Theory of Computing, STOC'02, pages 534-543, New York, NY, USA, 2002. ACM. URL: http://dx.doi.org/10.1145/509907.509985.
13. Uriel Feige, Guy Kortsarz, and David Peleg. The dense k-subgraph problem. Algorithmica, 29(3):410-421, 2001.
14. Ashish Goel, Michael Kapralov, and Sanjeev Khanna. On the communication and streaming complexity of maximum bipartite matching. In Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2012, Kyoto, Japan, January 17-19, 2012, pages 468-485, 2012.
15. Anupam Gupta, Mohammadtaghi Hajiaghayi, Viswanath Nagarajan, and R. Ravi. Dial a ride from k-forest. ACM Trans. Algorithms, 6(2):41:1-41:21, April 2010. URL: http://dx.doi.org/10.1145/1721837.1721857.
16. Subhash Khot. Ruling out ptas for graph min-bisection, dense k-subgraph, and bipartite clique. SIAM Journal on Computing, 36(4):1025-1071, 2006. URL: http://dx.doi.org/10.1137/S0097539705447037.
17. Christian Konrad and Adi Rosén. Approximating semi-matchings in streaming and in two-party communication. In Automata, Languages, and Programming - 40th International Colloquium, ICALP 2013, Riga, Latvia, July 8-12, 2013, Proceedings, Part I, pages 637-649, 2013.
18. Guy Kortsarz and David Peleg. On choosing a dense subgraph. In 34th Annual Symposium on Foundations of Computer Science, Palo Alto, California, USA, 3-5 November 1993, pages 692-701, 1993.
19. Anand Louis and Yury Makarychev. Approximation Algorithms for Hypergraph Small Set Expansion and Small Set Vertex Expansion. In Klaus Jansen, José D. P. Rolim, Nikhil R. Devanur, and Cristopher Moore, editors, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014), volume 28 of Leibniz International Proceedings in Informatics (LIPIcs), pages 339-355, Dagstuhl, Germany, 2014. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik. URL: http://drops.dagstuhl.de/opus/volltexte/2014/4707, URL: http://dx.doi.org/10.4230/LIPIcs.APPROX-RANDOM.2014.339.
20. Anand Louis, Prasad Raghavendra, and Santosh Vempala. The complexity of approximating vertex expansion. 2013 IEEE 54th Annual Symposium on Foundations of Computer Science, 0:360-369, 2013. URL: http://dx.doi.org/10.1109/FOCS.2013.46.
21. Yuru Makarychev. Personal communication.
22. Zeev Nutov. Approximating steiner networks with node-weights. SIAM Journal on Computing, 39(7):3001-3022, 2010. URL: http://dx.doi.org/10.1137/080729645.