Inapproximability of Maximum Biclique Problems, Minimum k-Cut and Densest At-Least-k-Subgraph from the Small Set Expansion Hypothesis
The Small Set Expansion Hypothesis is a conjecture which roughly states that it is NP-hard to distinguish between a graph with a small subset of vertices whose (edge) expansion is almost zero and one in which all small subsets of vertices have expansion almost one. In this work, we prove conditional...
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| Format: | Article |
| Language: | English |
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MDPI AG
2018-01-01
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| Series: | Algorithms |
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| Online Access: | http://www.mdpi.com/1999-4893/11/1/10 |
| _version_ | 1819076453046681600 |
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| author | Pasin Manurangsi |
| author_facet | Pasin Manurangsi |
| author_sort | Pasin Manurangsi |
| collection | DOAJ |
| description | The Small Set Expansion Hypothesis is a conjecture which roughly states that it is NP-hard to distinguish between a graph with a small subset of vertices whose (edge) expansion is almost zero and one in which all small subsets of vertices have expansion almost one. In this work, we prove conditional inapproximability results with essentially optimal ratios for the following graph problems based on this hypothesis: Maximum Edge Biclique, Maximum Balanced Biclique, Minimum k-Cut and Densest At-Least-k-Subgraph. Our hardness results for the two biclique problems are proved by combining a technique developed by Raghavendra, Steurer and Tulsiani to avoid locality of gadget reductions with a generalization of Bansal and Khot’s long code test whereas our results for Minimum k-Cut and Densest At-Least-k-Subgraph are shown via elementary reductions. |
| first_indexed | 2024-12-21T18:41:32Z |
| format | Article |
| id | doaj.art-f6c9bd85f04e4b5f8b4eb70f332de106 |
| institution | Directory Open Access Journal |
| issn | 1999-4893 |
| language | English |
| last_indexed | 2024-12-21T18:41:32Z |
| publishDate | 2018-01-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Algorithms |
| spelling | doaj.art-f6c9bd85f04e4b5f8b4eb70f332de1062022-12-21T18:54:00ZengMDPI AGAlgorithms1999-48932018-01-011111010.3390/a11010010a11010010Inapproximability of Maximum Biclique Problems, Minimum k-Cut and Densest At-Least-k-Subgraph from the Small Set Expansion HypothesisPasin Manurangsi0Department of Electrical Engineering and Computer Sciences, University of California, Berkeley, CA 94709, USAThe Small Set Expansion Hypothesis is a conjecture which roughly states that it is NP-hard to distinguish between a graph with a small subset of vertices whose (edge) expansion is almost zero and one in which all small subsets of vertices have expansion almost one. In this work, we prove conditional inapproximability results with essentially optimal ratios for the following graph problems based on this hypothesis: Maximum Edge Biclique, Maximum Balanced Biclique, Minimum k-Cut and Densest At-Least-k-Subgraph. Our hardness results for the two biclique problems are proved by combining a technique developed by Raghavendra, Steurer and Tulsiani to avoid locality of gadget reductions with a generalization of Bansal and Khot’s long code test whereas our results for Minimum k-Cut and Densest At-Least-k-Subgraph are shown via elementary reductions.http://www.mdpi.com/1999-4893/11/1/10hardness of approximationsmall set expansion hypothesismaximum edge bicliquemaximum balanced bicliqueminimum k-cutdensest at-least-k-subgraph |
| spellingShingle | Pasin Manurangsi Inapproximability of Maximum Biclique Problems, Minimum k-Cut and Densest At-Least-k-Subgraph from the Small Set Expansion Hypothesis Algorithms hardness of approximation small set expansion hypothesis maximum edge biclique maximum balanced biclique minimum k-cut densest at-least-k-subgraph |
| title | Inapproximability of Maximum Biclique Problems, Minimum k-Cut and Densest At-Least-k-Subgraph from the Small Set Expansion Hypothesis |
| title_full | Inapproximability of Maximum Biclique Problems, Minimum k-Cut and Densest At-Least-k-Subgraph from the Small Set Expansion Hypothesis |
| title_fullStr | Inapproximability of Maximum Biclique Problems, Minimum k-Cut and Densest At-Least-k-Subgraph from the Small Set Expansion Hypothesis |
| title_full_unstemmed | Inapproximability of Maximum Biclique Problems, Minimum k-Cut and Densest At-Least-k-Subgraph from the Small Set Expansion Hypothesis |
| title_short | Inapproximability of Maximum Biclique Problems, Minimum k-Cut and Densest At-Least-k-Subgraph from the Small Set Expansion Hypothesis |
| title_sort | inapproximability of maximum biclique problems minimum k cut and densest at least k subgraph from the small set expansion hypothesis |
| topic | hardness of approximation small set expansion hypothesis maximum edge biclique maximum balanced biclique minimum k-cut densest at-least-k-subgraph |
| url | http://www.mdpi.com/1999-4893/11/1/10 |
| work_keys_str_mv | AT pasinmanurangsi inapproximabilityofmaximumbicliqueproblemsminimumkcutanddensestatleastksubgraphfromthesmallsetexpansionhypothesis |