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**Published in:** LIPIcs, Volume 202, 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)

Let G be a graph on n vertices. For i ∈ {0,1} and a connected graph G, a spanning forest F of G is called an i-perfect forest if every tree in F is an induced subgraph of G and exactly i vertices of F have even degree (including zero). An i-perfect forest of G is proper if it has no vertices of degree zero. Scott (2001) showed that every connected graph with even number of vertices contains a (proper) 0-perfect forest. We prove that one can find a 0-perfect forest with minimum number of edges in polynomial time, but it is NP-hard to obtain a 0-perfect forest with maximum number of edges. We also prove that for a prescribed edge e of G, it is NP-hard to obtain a 0-perfect forest containing e, but we can find a 0-perfect forest not containing e in polynomial time. It is easy to see that every graph with odd number of vertices has a 1-perfect forest. It is not the case for proper 1-perfect forests. We give a characterization of when a connected graph has a proper 1-perfect forest.

Gregory Gutin and Anders Yeo. Perfect Forests in Graphs and Their Extensions. In 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 202, pp. 54:1-54:13, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)

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@InProceedings{gutin_et_al:LIPIcs.MFCS.2021.54, author = {Gutin, Gregory and Yeo, Anders}, title = {{Perfect Forests in Graphs and Their Extensions}}, booktitle = {46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)}, pages = {54:1--54:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-201-3}, ISSN = {1868-8969}, year = {2021}, volume = {202}, editor = {Bonchi, Filippo and Puglisi, Simon J.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2021.54}, URN = {urn:nbn:de:0030-drops-144947}, doi = {10.4230/LIPIcs.MFCS.2021.54}, annote = {Keywords: graphs, odd degree subgraphs, perfect forests, polynomial algorithms} }

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**Published in:** LIPIcs, Volume 180, 15th International Symposium on Parameterized and Exact Computation (IPEC 2020)

A directed graph D is semicomplete if for every pair x,y of vertices of D, there is at least one arc between x and y. Thus, a tournament is a semicomplete digraph. In the Directed Component Order Connectivity (DCOC) problem, given a digraph D = (V,A) and a pair of natural numbers k and 𝓁, we are to decide whether there is a subset X of V of size k such that the largest strong connectivity component in D-X has at most 𝓁 vertices. Note that DCOC reduces to the Directed Feedback Vertex Set problem for 𝓁 = 1. We study parameterized complexity of DCOC for general and semicomplete digraphs with the following parameters: k, 𝓁, 𝓁+k and n-𝓁. In particular, we prove that DCOC with parameter k on semicomplete digraphs can be solved in time O^*(2^(16k)) but not in time O^*(2^o(k)) unless the Exponential Time Hypothesis (ETH) fails. The upper bound O^*(2^(16k)) implies the upper bound O^*(2^(16(n-𝓁))) for the parameter n-𝓁. We complement the latter by showing that there is no algorithm of time complexity O^*(2^o(n-𝓁)) unless ETH fails. Finally, we improve (in dependency on 𝓁) the upper bound of Göke, Marx and Mnich (2019) for the time complexity of DCOC with parameter 𝓁+k on general digraphs from O^*(2^O(k𝓁 log (k𝓁))) to O^*(2^O(klog (k𝓁))). Note that Drange, Dregi and van 't Hof (2016) proved that even for the undirected version of DCOC on split graphs there is no algorithm of running time O^*(2^o(klog 𝓁)) unless ETH fails and it is a long-standing problem to decide whether Directed Feedback Vertex Set admits an algorithm of time complexity O^*(2^o(klog k)).

Jørgen Bang-Jensen, Eduard Eiben, Gregory Gutin, Magnus Wahlström, and Anders Yeo. Component Order Connectivity in Directed Graphs. In 15th International Symposium on Parameterized and Exact Computation (IPEC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 180, pp. 2:1-2:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)

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@InProceedings{bangjensen_et_al:LIPIcs.IPEC.2020.2, author = {Bang-Jensen, J{\o}rgen and Eiben, Eduard and Gutin, Gregory and Wahlstr\"{o}m, Magnus and Yeo, Anders}, title = {{Component Order Connectivity in Directed Graphs}}, booktitle = {15th International Symposium on Parameterized and Exact Computation (IPEC 2020)}, pages = {2:1--2:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-172-6}, ISSN = {1868-8969}, year = {2020}, volume = {180}, editor = {Cao, Yixin and Pilipczuk, Marcin}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2020.2}, URN = {urn:nbn:de:0030-drops-133058}, doi = {10.4230/LIPIcs.IPEC.2020.2}, annote = {Keywords: Parameterized Algorithms, component order connectivity, directed graphs, semicomplete digraphs} }

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**Published in:** Dagstuhl Follow-Ups, Volume 7, The Constraint Satisfaction Problem: Complexity and Approximability (2017)

We consider constraint satisfaction problems parameterized above or below guaranteed values. One example is MaxSat parameterized above m/2: given a CNF formula F with m clauses, decide whether there is a truth assignment that satisfies at least m/2 + k clauses, where k is the parameter. Among other problems we deal with are MaxLin2-AA (given a system of linear equations over F_2 in which each equation has a positive integral weight, decide whether there is an assignment to the variables that satisfies equations of total weight at least W/2+k, where W is the total weight of all equations), Max-r-Lin2-AA (the same as MaxLin2-AA, but each equation has at most r variables, where r is a constant) and Max-r-Sat-AA (given a CNF formula F with m clauses in which each clause has at most r literals, decide whether there is a truth assignment satisfying at least sum_{i=1}^m (1-2^{r_i})+k clauses, where k is the parameter, r_i is the number of literals in clause i, and r is a constant). We also consider Max-r-CSP-AA, a natural generalization of both Max-r-Lin2-AA and Max-r-Sat-AA, order (or, permutation) constraint satisfaction problems parameterized above the average value and some other problems related to MaxSat. We discuss results, both polynomial kernels and parameterized algorithms, obtained for the problems mainly in the last few years as well as some open questions.

Gregory Gutin and Anders Yeo. Parameterized Constraint Satisfaction Problems: a Survey. In The Constraint Satisfaction Problem: Complexity and Approximability. Dagstuhl Follow-Ups, Volume 7, pp. 179-203, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)

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@InCollection{gutin_et_al:DFU.Vol7.15301.179, author = {Gutin, Gregory and Yeo, Anders}, title = {{Parameterized Constraint Satisfaction Problems: a Survey}}, booktitle = {The Constraint Satisfaction Problem: Complexity and Approximability}, pages = {179--203}, series = {Dagstuhl Follow-Ups}, ISBN = {978-3-95977-003-3}, ISSN = {1868-8977}, year = {2017}, volume = {7}, editor = {Krokhin, Andrei and Zivny, Stanislav}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DFU.Vol7.15301.179}, URN = {urn:nbn:de:0030-drops-69641}, doi = {10.4230/DFU.Vol7.15301.179}, annote = {Keywords: Constraint satisfaction problems, Fixed-parameter tractability} }

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**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.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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**Published in:** LIPIcs, Volume 3, 26th International Symposium on Theoretical Aspects of Computer Science (2009)

The {\sc Multicut In Trees} problem consists in deciding, given a tree, a set of requests (i.e. paths in the tree) and an integer $k$, whether there exists a set of $k$ edges cutting all the requests. This problem was shown to be FPT by Guo and Niedermeyer (2005). They also provided an exponential kernel. They asked whether this problem has a polynomial kernel. This question was also raised by Fellows (2006).
We show that {\sc Multicut In Trees} has a polynomial kernel.

Nicolas Bousquet, Jean Daligault, Stephan Thomasse, and Anders Yeo. A Polynomial Kernel for Multicut in Trees. In 26th International Symposium on Theoretical Aspects of Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 3, pp. 183-194, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2009)

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@InProceedings{bousquet_et_al:LIPIcs.STACS.2009.1824, author = {Bousquet, Nicolas and Daligault, Jean and Thomasse, Stephan and Yeo, Anders}, title = {{A Polynomial Kernel for Multicut in Trees}}, booktitle = {26th International Symposium on Theoretical Aspects of Computer Science}, pages = {183--194}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-09-5}, ISSN = {1868-8969}, year = {2009}, volume = {3}, editor = {Albers, Susanne and Marion, Jean-Yves}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2009.1824}, URN = {urn:nbn:de:0030-drops-18247}, doi = {10.4230/LIPIcs.STACS.2009.1824}, annote = {Keywords: } }

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