## Abstract

In the parameterized problem MaxLin2-AA[k], we are given a system with variables x

The problem Max-r-Lin2-AA[k, r] is the same as MaxLin2-AA[k] with two differences: each equation has at most r variables and r is the second parameter. We prove a theorem on Max-r-Lin2-AA[k, r] which implies that Max-r-Lin2-AA[k, r] has a kernel with at most (2k − 1)r variables, improving a number of results including one by Kim and Williams (2010). The theorem also implies a lower bound on the maximum of a function f : {−1, 1}

_{1}, . . . , x_{n}consisting of equations of the form ∏_{ i∈I}xi = b, where xi , b ∈ {−1, 1} and I ⊆ [n], each equation has a positive integral weight, and we are to decide whether it is possible to simultaneously satisfy equations of total weight at least W/2 + k, where W is the total weight of all equations and k is the parameter (if k = 0, the possibility is assured). We show that MaxLin2-AA[k] has a kernel with at most O(k 2 log k) variables and can be solved in time 2^{O(k log k)}(nm)^{O(1)}. This solves an open problem of Mahajan et al. (2006).The problem Max-r-Lin2-AA[k, r] is the same as MaxLin2-AA[k] with two differences: each equation has at most r variables and r is the second parameter. We prove a theorem on Max-r-Lin2-AA[k, r] which implies that Max-r-Lin2-AA[k, r] has a kernel with at most (2k − 1)r variables, improving a number of results including one by Kim and Williams (2010). The theorem also implies a lower bound on the maximum of a function f : {−1, 1}

^{n}→ R whose Fourier expansion (which is a multilinear polynomial) is of degree r. We show applicability of the lower bound by giving a new proof of the Edwards-Erdős bound (each connected graph on n vertices and m edges has a bipartite subgraph with at least m/2+ (n−1)/4 edges) and obtaining a generalization.Original language | English |
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Title of host publication | Proceedings of IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science |

Editors | Supratik Chakraborty, Amit Kumar |

Place of Publication | Germany |

Publisher | Dagstuhl Publishing |

Pages | 229-240 |

Number of pages | 12 |

Volume | 13 |

ISBN (Print) | 978-3-939897-34-7 |

DOIs | |

Publication status | Published - 2011 |

Event | International Conference on Foundations of Software Technology and Theoretical Computer Science (FST&TCS 2011 31st) - India, India Duration: 12 Dec 2011 → 14 Dec 2011 Conference number: 2011 (31st) |

### Conference

Conference | International Conference on Foundations of Software Technology and Theoretical Computer Science (FST&TCS 2011 31st) |
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Abbreviated title | FST&TCS |

Country/Territory | India |

Period | 12/12/11 → 14/12/11 |