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**Published in:** LIPIcs, Volume 24, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2013)

Poljak and Turzik (Discrete Mathematics 1986) introduced the notion of
lambda-extendible properties of graphs as a generalization of the property of being bipartite. They showed that for any 0<lambda<1 and
lambda-extendible property Pi, any connected graph G on n vertices and m edges contains a spanning subgraph H in Pi with at least lambda*m+(1-lambda)(n-1)/2 edges. The property of being bipartite is
lambda-extendible for lambda =1/2, and so the Poljak-Turzik bound
generalizes the well-known Edwards-Erdos bound for Max Cut. Other examples of lambda-extendible properties include: being an acyclic oriented graph, a balanced signed graph, or a q-colorable graph for some q in N.
Mnich et al. (FSTTCS 2012) defined the closely related notion of strong lambda-extendibility. They showed that the problem of finding a subgraph satisfying a given strongly lambda-extendible property Pi is fixed-parameter tractable (FPT) when parameterized above the Poljak-Turzik bound---does there exist a spanning subgraph H of a connected graph G such that H in Pi and H has at least lambda*m+(1-lambda)(n-1)/2+k edges?---subject to the condition that the problem is FPT on a certain simple class of graphs called almost-forests of cliques. This generalized an earlier result of Crowston et al. (ICALP 2012) for Max Cut, to all strongly lambda-extendible properties which satisfy the additional criterion.
In this paper we settle the kernelization complexity of nearly all problems parameterized above Poljak-Turzik bounds, in the affirmative. We show that these problems admit quadratic kernels (cubic when lambda=1/2), without using the assumption that the problem is FPT on almost-forests of cliques. Thus our results not only remove the technical condition of being FPT on almost-forests of cliques from previous results, but also unify and extend previously known kernelization results in this direction. Our results add to the select list of generic kernelization results known in the literature.

Robert Crowston, Mark Jones, Gabriele Muciaccia, Geevarghese Philip, Ashutosh Rai, and Saket Saurabh. Polynomial Kernels for lambda-extendible Properties Parameterized Above the Poljak-Turzik Bound. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2013). Leibniz International Proceedings in Informatics (LIPIcs), Volume 24, pp. 43-54, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2013)

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@InProceedings{crowston_et_al:LIPIcs.FSTTCS.2013.43, author = {Crowston, Robert and Jones, Mark and Muciaccia, Gabriele and Philip, Geevarghese and Rai, Ashutosh and Saurabh, Saket}, title = {{Polynomial Kernels for lambda-extendible Properties Parameterized Above the Poljak-Turzik Bound}}, booktitle = {IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2013)}, pages = {43--54}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-64-4}, ISSN = {1868-8969}, year = {2013}, volume = {24}, editor = {Seth, Anil and Vishnoi, Nisheeth K.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2013.43}, URN = {urn:nbn:de:0030-drops-43599}, doi = {10.4230/LIPIcs.FSTTCS.2013.43}, annote = {Keywords: Kernelization, Lambda Extension, Above-Guarantee Parameterization, MaxCut} }

Document

**Published in:** LIPIcs, Volume 18, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2012)

An oriented graph is a directed graph without directed 2-cycles. Poljak and Turzik (1986) proved that every connected oriented graph G on n vertices and m arcs contains an acyclic subgraph with at least m/2+(n-1)/4 arcs. Raman and Saurabh (2006) gave another proof of this result and left it as an open question to establish the parameterized complexity of the following problem: does G have an acyclic subgraph with least m/2 + (n-1)/4 + k arcs, where k is the parameter? We answer this question by showing that the problem can be solved by an algorithm of runtime (12k)!n^{O(1)}. Thus, the problem is fixed-parameter tractable. We also prove that there is a polynomial time algorithm that either establishes that the input instance of the problem is a Yes-instance or reduces the input instance to an equivalent one of size O(k^2).

Robert Crowston, Gregory Gutin, and Mark Jones. Directed Acyclic Subgraph Problem Parameterized above the Poljak-Turzik Bound. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2012). Leibniz International Proceedings in Informatics (LIPIcs), Volume 18, pp. 400-411, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)

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@InProceedings{crowston_et_al:LIPIcs.FSTTCS.2012.400, author = {Crowston, Robert and Gutin, Gregory and Jones, Mark}, title = {{Directed Acyclic Subgraph Problem Parameterized above the Poljak-Turzik Bound}}, booktitle = {IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2012)}, pages = {400--411}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-47-7}, ISSN = {1868-8969}, year = {2012}, volume = {18}, editor = {D'Souza, Deepak and Radhakrishnan, Jaikumar and Telikepalli, Kavitha}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2012.400}, URN = {urn:nbn:de:0030-drops-38765}, doi = {10.4230/LIPIcs.FSTTCS.2012.400}, annote = {Keywords: Acyclic Subgraph, Fixed-parameter tractable, Polynomial Kernel} }

Document

**Published in:** LIPIcs, Volume 13, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2011)

In the parameterized problem MaxLin2-AA[$k$], we are given a system with variables x_1,...,x_n consisting of equations of the form Product_{i in I}x_i = b, where x_i,b in {-1, 1} and I is a nonempty subset of {1,...,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 that maps {-1,1}^n to the set of reals and 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-Erdös 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.

Robert Crowston, Michael Fellows, Gregory Gutin, Mark Jones, Frances Rosamond, Stéphan Thomassé, and Anders Yeo. Simultaneously Satisfying Linear Equations Over F_2: MaxLin2 and Max-r-Lin2 Parameterized Above Average. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2011). Leibniz International Proceedings in Informatics (LIPIcs), Volume 13, pp. 229-240, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2011)

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@InProceedings{crowston_et_al:LIPIcs.FSTTCS.2011.229, author = {Crowston, Robert and Fellows, Michael and Gutin, Gregory and Jones, Mark and Rosamond, Frances and Thomass\'{e}, St\'{e}phan and Yeo, Anders}, title = {{Simultaneously Satisfying Linear Equations Over F\underline2: MaxLin2 and Max-r-Lin2 Parameterized Above Average}}, booktitle = {IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2011)}, pages = {229--240}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-34-7}, ISSN = {1868-8969}, year = {2011}, volume = {13}, editor = {Chakraborty, Supratik and Kumar, Amit}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2011.229}, URN = {urn:nbn:de:0030-drops-33416}, doi = {10.4230/LIPIcs.FSTTCS.2011.229}, annote = {Keywords: MaxLin, fixed-parameter tractability, kernelization, pseudo-boolean functions} }

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