## Abstract

We study the following basic problem called Bi-Covering. Given a graph G(V,E), find two (not necessarily disjoint) sets A ⊆ V and B ⊆ V such that A ∪ B = V and that every edge e belongs to either the graph induced by A or to the graph induced by B. The goal is to minimize max{|A|, |B|}. This is the most simple case of the Channel Allocation problem [Gandhi et. al, Networks, 2006]. A solution that outputs V, 0 gives ratio at most 2. We show that under the similar Strong Unique Game Conjecture by [Bansal-Khot, FOCS, 2009] there is no 2 - ϵ ratio algorithm for the problem, for any constant ϵ > 0. Given a bipartite graph, Max-bi-clique is a problem of finding largest k × k complete bipartite sub graph. For Max-bi-clique problem, a constant factor hardness was known under random 3-SAT hypothesis of Feige [Feige, STOC, 2002] and also under the assumption that NP ⊈ ∩ϵ>0 BPTIME(2^{nϵ}) [Khot, SIAM J. on Comp., 2011]. It was an open problem in [Ambühl et. al., SIAM J. on Comp., 2011] to prove inapproximability of Max-bi-clique assuming weaker conjecture. Our result implies similar hardness result assuming the Strong Unique Games Conjecture. On the algorithmic side, we also give better than 2 approximation for Bi-Covering on numerous special graph classes. In particular, we get 1.876 approximation for Chordal graphs, exact algorithm for Interval Graphs, 1+o(1) for Minor Free Graph, 2-4δ/3 for graphs with minimum degree δn, 2/(1+δ^{2}/8) for δ-vertex expander, 8/5 for Split Graphs, 2-(6/5) · 1/d for graphs with minimum constant degree d etc. Our algorithmic results are quite non-trivial. In achieving these results, we use various known structural results about the graphs, combined with the techniques that we develop tailored to getting better than 2 approximation.

Original language | English |
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Title of host publication | 43rd International Colloquium on Automata, Languages, and Programming, ICALP 2016 |

Editors | Yuval Rabani, Ioannis Chatzigiannakis, Davide Sangiorgi, Michael Mitzenmacher |

Publisher | Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing |

ISBN (Electronic) | 9783959770132 |

DOIs | |

State | Published - 1 Aug 2016 |

Externally published | Yes |

Event | 43rd International Colloquium on Automata, Languages, and Programming, ICALP 2016 - Rome, Italy Duration: 12 Jul 2016 → 15 Jul 2016 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
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Volume | 55 |

ISSN (Print) | 1868-8969 |

### Conference

Conference | 43rd International Colloquium on Automata, Languages, and Programming, ICALP 2016 |
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Country/Territory | Italy |

City | Rome |

Period | 12/07/16 → 15/07/16 |

### Bibliographical note

Funding Information:Research supported in part by NSF grant CCF-1253886. Supported in part by NSF grant number 1218620. Supported in part by NSF CAREER award CCF-1053605, NSF BIGDATA grant IIS-1546108, NSF AF:Medium grant CCF-1161365, DARPA GRAPHS/AFOSR grant FA9550-12-1-0423, and another DARPA SIMPLEX grant. Supported in part by NSF grant number 1218620 and by NSF grant 1540547.

Copyright:

Copyright 2017 Elsevier B.V., All rights reserved.

## Keywords

- Bi-covering
- Max Bi-clique
- Unique Games