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# Inapproximability of Maximum Edge Biclique, Maximum Balanced Biclique and Minimum k-Cut from the Small Set Expansion Hypothesis

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## Cite As

Pasin Manurangsi. Inapproximability of Maximum Edge Biclique, Maximum Balanced Biclique and Minimum k-Cut from the Small Set Expansion Hypothesis. In 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 80, pp. 79:1-79:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
https://doi.org/10.4230/LIPIcs.ICALP.2017.79

## Abstract

The Small Set Expansion Hypothesis (SSEH) is a conjecture which roughly states that it is NP-hard to distinguish between a graph with a small set of vertices whose expansion is almost zero and one in which all small sets of vertices have expansion almost one. In this work, we prove conditional inapproximability results for the following graph problems based on this hypothesis: - Maximum Edge Biclique (MEB): given a bipartite graph G, find a complete bipartite subgraph of G with maximum number of edges. We show that, assuming SSEH and that NP != BPP, no polynomial time algorithm gives n^{1 - epsilon}-approximation for MEB for every constant epsilon > 0. - Maximum Balanced Biclique (MBB): given a bipartite graph G, find a balanced complete bipartite subgraph of G with maximum number of vertices. Similar to MEB, we prove n^{1 - epsilon} ratio inapproximability for MBB for every epsilon > 0, assuming SSEH and that NP != BPP. - Minimum k-Cut: given a weighted graph G, find a set of edges with minimum total weight whose removal splits the graph into k components. We prove that this problem is NP-hard to approximate to within (2 - epsilon) factor of the optimum for every epsilon > 0, assuming SSEH. The ratios in our results are essentially tight since trivial algorithms give n-approximation to both MEB and MBB and 2-approximation algorithms are known for Minimum k-Cut [Saran and Vazirani, SIAM J. Comput., 1995]. Our first two results are proved by combining a technique developed by Raghavendra, Steurer and Tulsiani [Raghavendra et al., CCC, 2012] to avoid locality of gadget reductions with a generalization of Bansal and Khot's long code test [Bansal and Khot, FOCS, 2009] whereas our last result is shown via an elementary reduction.
##### Keywords
• Hardness of Approximation
• Small Set Expansion Hypothesis

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## References

1. Christoph Ambühl, Monaldo Mastrolilli, and Ola Svensson. Inapproximability results for maximum edge biclique, minimum linear arrangement, and sparsest cut. SIAM J. Comput., 40(2):567-596, April 2011.
2. Sanjeev Arora, Carsten Lund, Rajeev Motwani, Madhu Sudan, and Mario Szegedy. Proof verification and the hardness of approximation problems. J. ACM, 45(3):501-555, May 1998.
3. Sanjeev Arora and Shmuel Safra. Probabilistic checking of proofs: A new characterization of NP. J. ACM, 45(1):70-122, January 1998.
4. Nikhil Bansal and Subhash Khot. Optimal long code test with one free bit. In IEEE FOCS, pages 453-462, 2009.
5. Piotr Berman and Georg Schnitger. On the complexity of approximating the independent set problem. Inf. Comput., 96(1):77-94, 1992.
6. Amey Bhangale, Rajiv Gandhi, Mohammad Taghi Hajiaghayi, Rohit Khandekar, and Guy Kortsarz. Bicovering: Covering edges with two small subsets of vertices. In ICALP, pages 6:1-6:12, 2016.
7. Avrim Blum. Algorithms for approximate graph coloring. Technical report, Massachusetts Institute of Technology, Cambridge, MA, USA, 1991.
8. Irit Dinur. Mildly exponential reduction from gap 3SAT to polynomial-gap label-cover. ECCC, 23:128, 2016.
9. Irit Dinur and David Steurer. Analytical approach to parallel repetition. In ACM STOC, pages 624-633, 2014.
10. Uriel Feige. Relations between average case complexity and approximation complexity. In ACM STOC, pages 534-543, 2002.
11. Uriel Feige and Shimon Kogan. Hardness of approximation of the balanced complete bipartite subgraph problem. Technical report, Weizmann Institute of Science, Israel, 2004.
12. Michael R. Garey and David S. Johnson. Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman &Co., New York, NY, USA, 1979.
13. Olivier Goldschmidt and Dorit S Hochbaum. A polynomial algorithm for the k-cut problem for fixed k. Mathematics of operations research, 19(1):24-37, 1994.
14. Johan Håstad. Clique is hard to approximate within n^1-ε. In IEEE FOCS, pages 627-636, 1996.
15. Johan Håstad. Some optimal inapproximability results. J. ACM, 48(4):798-859, July 2001.
16. Russell Impagliazzo, Ramamohan Paturi, and Francis Zane. Which problems have strongly exponential complexity? J. Comput. Syst. Sci., 63(4):512-530, December 2001.
17. David S. Johnson. The NP-completeness column: An ongoing guide. J. Algorithms, 8(5):438-448, September 1987.
18. Subhash Khot. Improved inaproximability results for MaxClique, chromatic number and approximate graph coloring. In IEEE FOCS, pages 600-609, 2001.
19. Subhash Khot. On the power of unique 2-prover 1-round games. In ACM STOC, pages 767-775, 2002.
20. Subhash Khot. Ruling out ptas for graph min-bisection, dense k-subgraph, and bipartite clique. SIAM J. Comput., 36(4), 2006.
21. Subhash Khot, Guy Kindler, Elchanan Mossel, and Ryan O'Donnell. Optimal inapproximability results for MAX-CUT and other 2-variable CSPs? SIAM J. Comput., 37(1):319-357, 2007.
22. Subhash Khot and Ashok Kumar Ponnuswami. Better inapproximability results for MaxClique, chromatic number and Min-3Lin-Deletion. In ICALP, pages 226-237, 2006.
23. Subhash Khot and Oded Regev. Vertex cover might be hard to approximate to within 2 - ε. J. Comput. Syst. Sci., 74(3):335-349, 2008.
24. Jon Kleinberg, Christos Papadimitriou, and Prabhakar Raghavan. Segmentation problems. J. ACM, 51(2):263-280, March 2004.
25. Pasin Manurangsi. Almost-polynomial ratio ETH-hardness of approximating densest k-subgraph. In ACM STOC, 2017. To appear.
26. Pasin Manurangsi and Prasad Raghavendra. A birthday repetition theorem and complexity of approximating dense CSPs. In ICALP, 2017. To appear.
27. Dana Moshkovitz. The projection games conjecture and the NP-hardness of ln n-approximating set-cover. Theory of Computing, 11:221-235, 2015.
28. Elchanan Mossel, Ryan O'Donnell, and Krzysztof Oleszkiewicz. Noise stability of functions with low influences: Invariance and optimality. Ann. Math., pages 295-341, 2010.
29. Joseph (Seffi) Naor and Yuval Rabani. Tree packing and approximating k-cuts. In ACM-SIAM SODA, pages 26-27, 2001.
30. Ryan O'Donnell. Analysis of Boolean Functions. Cambridge University Press, 2014.
31. René Peeters. The maximum edge biclique problem is NP-complete. Discrete Appl. Math., 131(3):651-654, September 2003.
32. Prasad Raghavendra and David Steurer. Graph expansion and the unique games conjecture. In ACM STOC, pages 755-764, 2010.
33. Prasad Raghavendra, David Steurer, and Madhur Tulsiani. Reductions between expansion problems. In IEEE CCC, pages 64-73, 2012.
34. R. Ravi and Amitabh Sinha. Approximating k-cuts via network strength. In ACM-SIAM SODA, pages 621-622, 2002.
35. Huzur Saran and Vijay V. Vazirani. Finding k cuts within twice the optimal. SIAM J. Comput., 24(1):101-108, February 1995.
36. Ola Svensson. Hardness of vertex deletion and project scheduling. Theory of Computing, 9(24):759-781, 2013.
37. Mingyu Xiao, Leizhen Cai, and Andrew Chi-Chih Yao. Tight approximation ratio of a general greedy splitting algorithm for the minimum k-way cut problem. Algorithmica, 59(4):510-520, 2011.
38. Liang Zhao, Hiroshi Nagamochi, and Toshihide Ibaraki. Approximating the minimum k-way cut in a graph via minimum 3-way cuts. J. Comb. Optim., 5(4):397-410, 2001.
39. David Zuckerman. Linear degree extractors and the inapproximability of max clique and chromatic number. Theory of Computing, 3(6):103-128, 2007.
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