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**Published in:** LIPIcs, Volume 187, 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021)

We investigate the parameterized complexity of finding diverse sets of solutions to three fundamental combinatorial problems, two from the theory of matroids and the third from graph theory. The input to the Weighted Diverse Bases problem consists of a matroid M, a weight function ω:E(M)→N, and integers k ≥ 1, d ≥ 0. The task is to decide if there is a collection of k bases B_1, ..., B_k of M such that the weight of the symmetric difference of any pair of these bases is at least d. This is a diverse variant of the classical matroid base packing problem. The input to the Weighted Diverse Common Independent Sets problem consists of two matroids M₁,M₂ defined on the same ground set E, a weight function ω:E→N, and integers k ≥ 1, d ≥ 0. The task is to decide if there is a collection of k common independent sets I_1, ..., I_k of M₁ and M₂ such that the weight of the symmetric difference of any pair of these sets is at least d. This is motivated by the classical weighted matroid intersection problem. The input to the Diverse Perfect Matchings problem consists of a graph G and integers k ≥ 1, d ≥ 0. The task is to decide if G contains k perfect matchings M_1, ..., M_k such that the symmetric difference of any two of these matchings is at least d.
The underlying problem of finding one solution (basis, common independent set, or perfect matching) is known to be doable in polynomial time for each of these problems, and Diverse Perfect Matchings is known to be NP-hard for k = 2. We show that Weighted Diverse Bases and Weighted Diverse Common Independent Sets are both NP-hard. We show also that Diverse Perfect Matchings cannot be solved in polynomial time (unless P=NP) even for the case d = 1. We derive fixed-parameter tractable (FPT) algorithms for all three problems with (k,d) as the parameter.
The above results on matroids are derived under the assumption that the input matroids are given as independence oracles. For Weighted Diverse Bases we present a polynomial-time algorithm that takes a representation of the input matroid over a finite field and computes a poly(k,d)-sized kernel for the problem.

Fedor V. Fomin, Petr A. Golovach, Fahad Panolan, Geevarghese Philip, and Saket Saurabh. Diverse Collections in Matroids and Graphs. In 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 187, pp. 31:1-31:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{fomin_et_al:LIPIcs.STACS.2021.31, author = {Fomin, Fedor V. and Golovach, Petr A. and Panolan, Fahad and Philip, Geevarghese and Saurabh, Saket}, title = {{Diverse Collections in Matroids and Graphs}}, booktitle = {38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021)}, pages = {31:1--31:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-180-1}, ISSN = {1868-8969}, year = {2021}, volume = {187}, editor = {Bl\"{a}ser, Markus and Monmege, Benjamin}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2021.31}, URN = {urn:nbn:de:0030-drops-136769}, doi = {10.4230/LIPIcs.STACS.2021.31}, annote = {Keywords: Matroids, Diverse solutions, Fixed-parameter tractable algorithms} }

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**Published in:** LIPIcs, Volume 181, 31st International Symposium on Algorithms and Computation (ISAAC 2020)

We initiate the study of the Diverse Pair of (Maximum/ Perfect) Matchings problems which given a graph G and an integer k, ask whether G has two (maximum/perfect) matchings whose symmetric difference is at least k. Diverse Pair of Matchings (asking for two not necessarily maximum or perfect matchings) is NP-complete on general graphs if k is part of the input, and we consider two restricted variants. First, we show that on bipartite graphs, the problem is polynomial-time solvable, and second we show that Diverse Pair of Maximum Matchings is FPT parameterized by k. We round off the work by showing that Diverse Pair of Matchings has a kernel on 𝒪(k²) vertices.

Fedor V. Fomin, Petr A. Golovach, Lars Jaffke, Geevarghese Philip, and Danil Sagunov. Diverse Pairs of Matchings. In 31st International Symposium on Algorithms and Computation (ISAAC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 181, pp. 26:1-26:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{fomin_et_al:LIPIcs.ISAAC.2020.26, author = {Fomin, Fedor V. and Golovach, Petr A. and Jaffke, Lars and Philip, Geevarghese and Sagunov, Danil}, title = {{Diverse Pairs of Matchings}}, booktitle = {31st International Symposium on Algorithms and Computation (ISAAC 2020)}, pages = {26:1--26:12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-173-3}, ISSN = {1868-8969}, year = {2020}, volume = {181}, editor = {Cao, Yixin and Cheng, Siu-Wing and Li, Minming}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2020.26}, URN = {urn:nbn:de:0030-drops-133706}, doi = {10.4230/LIPIcs.ISAAC.2020.26}, annote = {Keywords: Matching, Solution Diversity, Fixed-Parameter Tractability} }

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**Published in:** LIPIcs, Volume 170, 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)

A clique coloring of a graph is an assignment of colors to its vertices such that no maximal clique is monochromatic. We initiate the study of structural parameterizations of the Clique Coloring problem which asks whether a given graph has a clique coloring with q colors. For fixed q ≥ 2, we give an 𝒪^⋆(q^{tw})-time algorithm when the input graph is given together with one of its tree decompositions of width tw. We complement this result with a matching lower bound under the Strong Exponential Time Hypothesis. We furthermore show that (when the number of colors is unbounded) Clique Coloring is XP parameterized by clique-width.

Lars Jaffke, Paloma T. Lima, and Geevarghese Philip. Structural Parameterizations of Clique Coloring. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 49:1-49:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{jaffke_et_al:LIPIcs.MFCS.2020.49, author = {Jaffke, Lars and Lima, Paloma T. and Philip, Geevarghese}, title = {{Structural Parameterizations of Clique Coloring}}, booktitle = {45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)}, pages = {49:1--49:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-159-7}, ISSN = {1868-8969}, year = {2020}, volume = {170}, editor = {Esparza, Javier and Kr\'{a}l', Daniel}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.49}, URN = {urn:nbn:de:0030-drops-127157}, doi = {10.4230/LIPIcs.MFCS.2020.49}, annote = {Keywords: clique coloring, treewidth, clique-width, structural parameterization, Strong Exponential Time Hypothesis} }

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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 168, 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)

In the Split Vertex Deletion (SVD) problem, the input is an n-vertex undirected graph G and a weight function w: V(G) → ℕ, and the objective is to find a minimum weight subset S of vertices such that G-S is a split graph (i.e., there is bipartition of V(G-S) = C ⊎ I such that C is a clique and I is an independent set in G-S). This problem is a special case of 5-Hitting Set and consequently, there is a simple factor 5-approximation algorithm for this. On the negative side, it is easy to show that the problem does not admit a polynomial time (2-δ)-approximation algorithm, for any fixed δ > 0, unless the Unique Games Conjecture fails.
We start by giving a simple quasipolynomial time (n^O(log n)) factor 2-approximation algorithm for SVD using the notion of clique-independent set separating collection. Thus, on the one hand SVD admits a factor 2-approximation in quasipolynomial time, and on the other hand this approximation factor cannot be improved assuming UGC. It naturally leads to the following question: Can SVD be 2-approximated in polynomial time? In this work we almost close this gap and prove that for any ε > 0, there is a n^O(log 1/(ε))-time 2(1+ε)-approximation algorithm.

Daniel Lokshtanov, Pranabendu Misra, Fahad Panolan, Geevarghese Philip, and Saket Saurabh. A (2 + ε)-Factor Approximation Algorithm for Split Vertex Deletion. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 80:1-80:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{lokshtanov_et_al:LIPIcs.ICALP.2020.80, author = {Lokshtanov, Daniel and Misra, Pranabendu and Panolan, Fahad and Philip, Geevarghese and Saurabh, Saket}, title = {{A (2 + \epsilon)-Factor Approximation Algorithm for Split Vertex Deletion}}, booktitle = {47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)}, pages = {80:1--80:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-138-2}, ISSN = {1868-8969}, year = {2020}, volume = {168}, editor = {Czumaj, Artur and Dawar, Anuj and Merelli, Emanuela}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2020.80}, URN = {urn:nbn:de:0030-drops-124879}, doi = {10.4230/LIPIcs.ICALP.2020.80}, annote = {Keywords: Approximation Algorithms, Graph Algorithms, Split Vertex Deletion} }

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**Published in:** LIPIcs, Volume 45, 35th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2015)

Problems of the following kind have been the focus of much recent research in the realm of parameterized complexity: Given an input graph (digraph) on n vertices and a positive integer parameter k, find if there exist k edges(arcs) whose deletion results in a graph that satisfies some specified parity constraints. In particular, when the objective is to obtain a connected graph in which all the
vertices have even degrees--where the resulting graph is Eulerian,the problem is called Undirected Eulerian Edge Deletion. The corresponding
problem in digraphs where the resulting graph should be strongly connected and every vertex should have the same in-degree as its
out-degree is called Directed Eulerian Edge Deletion. Cygan et al.[Algorithmica, 2014] showed that these problems are fixed parameter tractable (FPT), and gave algorithms with the running time
2^O(k log k)n^O(1). They also asked, as an open problem, whether there exist FPT algorithms which solve these problems in time
2^O(k)n^O(1). It was also posed as an open problem at the School on Parameterized Algorithms and Complexity 2014, Bedlewo, Poland.
In this paper we answer their question in the affirmative: using the technique of computing representative families of co-graphic matroids we design algorithms which solve these problems in time 2^O(k)n^O(1). The crucial insight we bring to these problems is to view the solution as an independent set of a co-graphic matroid. We believe that this view-point/approach will be useful in other problems where one of the constraints that need to be satisfied is that of connectivity.

Prachi Goyal, Pranabendu Misra, Fahad Panolan, Geevarghese Philip, and Saket Saurabh. Finding Even Subgraphs Even Faster. In 35th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 45, pp. 434-447, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{goyal_et_al:LIPIcs.FSTTCS.2015.434, author = {Goyal, Prachi and Misra, Pranabendu and Panolan, Fahad and Philip, Geevarghese and Saurabh, Saket}, title = {{Finding Even Subgraphs Even Faster}}, booktitle = {35th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2015)}, pages = {434--447}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-97-2}, ISSN = {1868-8969}, year = {2015}, volume = {45}, editor = {Harsha, Prahladh and Ramalingam, G.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2015.434}, URN = {urn:nbn:de:0030-drops-56336}, doi = {10.4230/LIPIcs.FSTTCS.2015.434}, annote = {Keywords: Eulerian Edge Deletion, FPT, Representative Family.} }

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**Published in:** LIPIcs, Volume 43, 10th International Symposium on Parameterized and Exact Computation (IPEC 2015)

The b-chromatic number of a graph G, chi_b(G), is the largest integer k such that G has a k-vertex coloring with the property that each color class has a vertex which is adjacent to at least one vertex in each of the other color classes. In the B-Chromatic Number problem, the objective is to decide whether chi_b(G) >= k. Testing whether chi_b(G)=Delta(G)+1, where Delta(G) is the maximum degree of a graph, itself is NP-complete even for connected bipartite graphs (Kratochvil, Tuza and Voigt, WG 2002). In this paper we study B-Chromatic Number in the realm of parameterized complexity and exact exponential time algorithms. We show that B-Chromatic Number is W[1]-hard when parameterized by k, resolving the open question posed by Havet and Sampaio (Algorithmica 2013). When k=Delta(G)+1, we design an algorithm for B-Chromatic Number running in time 2^{O(k^2 * log(k))}*n^{O(1)}. Finally, we show that B-Chromatic Number for an n-vertex graph can be solved in time O(3^n * n^{4} * log(n)).

Fahad Panolan, Geevarghese Philip, and Saket Saurabh. B-Chromatic Number: Beyond NP-Hardness. In 10th International Symposium on Parameterized and Exact Computation (IPEC 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 43, pp. 389-401, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{panolan_et_al:LIPIcs.IPEC.2015.389, author = {Panolan, Fahad and Philip, Geevarghese and Saurabh, Saket}, title = {{B-Chromatic Number: Beyond NP-Hardness}}, booktitle = {10th International Symposium on Parameterized and Exact Computation (IPEC 2015)}, pages = {389--401}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-92-7}, ISSN = {1868-8969}, year = {2015}, volume = {43}, editor = {Husfeldt, Thore and Kanj, Iyad}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2015.389}, URN = {urn:nbn:de:0030-drops-55997}, doi = {10.4230/LIPIcs.IPEC.2015.389}, annote = {Keywords: b-chromatic number, exact algorithm, parameterized complexity} }

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**Published in:** LIPIcs, Volume 29, 34th International Conference on Foundation of Software Technology and Theoretical Computer Science (FSTTCS 2014)

Given a graph G and a function f:V(G) -> [V(G)], an f-factor is a subgraph H of G such that deg_H(v)=f(v) for every vertex v in V(G); we say that H is a connected f-factor if, in addition, the subgraph H is connected. Tutte (1954) showed that one can check whether a given graph has a specified f-factor in polynomial time. However, detecting a connected f-factor is NP-complete, even when f is a constant function - a foremost example is the problem of checking whether a graph has a Hamiltonian cycle; here f is a function which maps every vertex to 2. The current best algorithm for this latter problem is due to Björklund (FOCS 2010), and runs in randomized O^*(1.657^n) time (the O^*() notation hides polynomial factors). This was the first superpolynomial improvement, in nearly fifty years, over the previous best algorithm of Bellman, Held and Karp (1962) which checks for a Hamiltonian cycle in deterministic O(2^n*n^2) time.
In this paper we present the first vertex-exponential algorithms for the more general problem of finding a connected f-factor. Our first result is a randomized algorithm which, given a graph G on n vertices and a function f:V(G) -> [n], checks whether G has a connected f-factor in O^*(2^n) time. We then extend our result to the case when f is a mapping from V(G) to {0,1} and the degree of every vertex v in the subgraph H is required to be f(v)(mod 2). This generalizes the problem of checking whether a graph has an Eulerian subgraph; this is a connected subgraph whose degrees are all even (f(v) equiv 0). Furthermore, we show that the min-cost editing and edge-weighted versions of these problems can be solved in randomized O^*(2^n) time as long as the costs/weights are bounded polynomially in n.

Geevarghese Philip and M. S. Ramanujan. Vertex Exponential Algorithms for Connected f-Factors. In 34th International Conference on Foundation of Software Technology and Theoretical Computer Science (FSTTCS 2014). Leibniz International Proceedings in Informatics (LIPIcs), Volume 29, pp. 61-71, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2014)

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@InProceedings{philip_et_al:LIPIcs.FSTTCS.2014.61, author = {Philip, Geevarghese and Ramanujan, M. S.}, title = {{Vertex Exponential Algorithms for Connected f-Factors}}, booktitle = {34th International Conference on Foundation of Software Technology and Theoretical Computer Science (FSTTCS 2014)}, pages = {61--71}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-77-4}, ISSN = {1868-8969}, year = {2014}, volume = {29}, editor = {Raman, Venkatesh and Suresh, S. P.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2014.61}, URN = {urn:nbn:de:0030-drops-48337}, doi = {10.4230/LIPIcs.FSTTCS.2014.61}, annote = {Keywords: Exact Exponential Time Algorithms, f-Factors} }

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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.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} }

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

Poljak and Turzík (Discrete Math. 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)/2 (n-1) edges. The property of being bipartite is lambda-extendible for lambda=1/2, and thus the Poljak-Turzík bound generalizes the well-known Edwards-Erdos bound for MAXCUT.
We define a variant, namely strong lambda-extendibility, to which the Poljak-Turzík bound applies. For a strong lambda-extendible graph property \Pi, we define the parameterized Above Poljak-Turzík problem as follows: Given a connected graph G on n vertices and m edges and an integer parameter k, does there exist a spanning subgraph H of G such that H in Pi and H has at least lambda m+ (1-lambda)/2 (n-1)+k edges? The parameter is k, the surplus over the number of edges guaranteed by the Poljak-Turzík bound.
We consider properties Pi for which the Above Poljak-Turzík problem is fixed-parameter tractable (FPT) on graphs which are O(k) vertices away from being a graph in which each block is a clique. We show that for all such properties, Above Poljak-Turzík is FPT for all 0< lambda <1. Our results hold for properties of oriented graphs and graphs with edge labels.
Our results generalize the recent result of Crowston et al. (ICALP 2012) on MAXCUT parameterized above the Edwards-Erdos, and yield FPT algorithms for several graph problems parameterized above lower bounds. For instance, we get that the above-guarantee Max q-Colorable Subgraph problem is FPT. Our results also imply that the parameterized above-guarantee Oriented Max Acyclic Digraph problem thus solving an open question of Raman and Saurabh (Theor. Comput. Sci. 2006).

Matthias Mnich, Geevarghese Philip, Saket Saurabh, and Ondrej Suchy. Beyond Max-Cut: lambda-Extendible Properties 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. 412-423, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)

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@InProceedings{mnich_et_al:LIPIcs.FSTTCS.2012.412, author = {Mnich, Matthias and Philip, Geevarghese and Saurabh, Saket and Suchy, Ondrej}, title = {{Beyond Max-Cut: lambda-Extendible Properties Parameterized Above the Poljak-Turzik Bound}}, booktitle = {IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2012)}, pages = {412--423}, 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.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2012.412}, URN = {urn:nbn:de:0030-drops-38776}, doi = {10.4230/LIPIcs.FSTTCS.2012.412}, annote = {Keywords: Algorithms and data structures; fixed-parameter algorithms; bipartite graphs; above-guarantee parameterization} }

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

The Minimum Fill-in problem is to decide if a graph can be triangulated by adding at most k edges. The problem has important applications in numerical algebra, in particular in sparse matrix computations. We develop kernelization algorithms for the problem on several classes of sparse graphs. We obtain linear kernels on planar graphs, and kernels of size O(k^{3/2}) in graphs excluding some fixed graph as a minor and in graphs of bounded degeneracy. As a byproduct of our results, we obtain approximation algorithms with approximation
ratios O(log{k}) on planar graphs and O(sqrt{k} log{k}) on H-minor-free graphs. These results significantly improve the previously known kernelization and approximation results for Minimum Fill-in on sparse graphs.

Fedor V. Fomin, Geevarghese Philip, and Yngve Villanger. Minimum Fill-in of Sparse Graphs: Kernelization and Approximation. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2011). Leibniz International Proceedings in Informatics (LIPIcs), Volume 13, pp. 164-175, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2011)

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@InProceedings{fomin_et_al:LIPIcs.FSTTCS.2011.164, author = {Fomin, Fedor V. and Philip, Geevarghese and Villanger, Yngve}, title = {{Minimum Fill-in of Sparse Graphs: Kernelization and Approximation}}, booktitle = {IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2011)}, pages = {164--175}, 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.2011.164}, URN = {urn:nbn:de:0030-drops-33451}, doi = {10.4230/LIPIcs.FSTTCS.2011.164}, annote = {Keywords: Minimum Fill-In, Approximation, Kernelization, Sparse graphs} }

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**Published in:** LIPIcs, Volume 9, 28th International Symposium on Theoretical Aspects of Computer Science (STACS 2011)

We study a general class of problems called F-Deletion problems. In an F-Deletion problem, we are asked whether a subset of at most k
vertices can be deleted from a graph G such that the resulting graph does not contain as a minor any graph from the family F of forbidden minors. We obtain a number of algorithmic results on the F-Deletion problem when F contains a planar graph. We give
- a linear vertex kernel on graphs excluding t-claw K_(1,t), the star with t leaves, as an induced subgraph, where t is a fixed integer.
- an approximation algorithm achieving an approximation ratio of O(log^(3/2) OPT), where $OPT$ is the size of an optimal solution on general undirected graphs.
Finally, we obtain polynomial kernels for the case when F only contains graph theta_c as a minor for a fixed integer c. The graph theta_c consists of two vertices connected by $c$ parallel edges. Even though this may appear to be a very restricted class of problems it already encompasses well-studied problems such as Vertex Cover, Feedback Vertex Set and Diamond Hitting Set. The generic kernelization algorithm is based on a non-trivial application of protrusion techniques, previously used only for problems on topological graph classes.

Fedor V. Fomin, Daniel Lokshtanov, Neeldhara Misra, Geevarghese Philip, and Saket Saurabh. Hitting forbidden minors: Approximation and Kernelization. In 28th International Symposium on Theoretical Aspects of Computer Science (STACS 2011). Leibniz International Proceedings in Informatics (LIPIcs), Volume 9, pp. 189-200, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2011)

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@InProceedings{fomin_et_al:LIPIcs.STACS.2011.189, author = {Fomin, Fedor V. and Lokshtanov, Daniel and Misra, Neeldhara and Philip, Geevarghese and Saurabh, Saket}, title = {{Hitting forbidden minors: Approximation and Kernelization}}, booktitle = {28th International Symposium on Theoretical Aspects of Computer Science (STACS 2011)}, pages = {189--200}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-25-5}, ISSN = {1868-8969}, year = {2011}, volume = {9}, editor = {Schwentick, Thomas and D\"{u}rr, Christoph}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2011.189}, URN = {urn:nbn:de:0030-drops-30103}, doi = {10.4230/LIPIcs.STACS.2011.189}, annote = {Keywords: kernelization} }

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**Published in:** LIPIcs, Volume 8, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2010)

In the Connected Dominating Set problem we are given as input a graph $G$ and a positive integer $k$, and are asked if there is a set $S$ of at most $k$ vertices of $G$ such that $S$ is a dominating set of $G$ and the subgraph induced by $S$ is connected. This is a basic connectivity problem that is known to be NP-complete, and it has been extensively studied using several algorithmic approaches. In this paper we study the effect of excluding short cycles, as a subgraph, on the kernelization complexity of Connected Dominating Set.
Kernelization algorithms are polynomial-time algorithms that take an input and a positive integer $k$ (the parameter) and output an equivalent instance where the size of the new instance and the new parameter are both bounded by some function $g(k)$. The new instance is called a $g(k)$ kernel for the problem. If $g(k)$ is a polynomial in $k$ then we say that the problem admits polynomial kernels. The girth of a graph $G$ is the length of a shortest cycle in $G$. It turns out that Connected Dominating Set is ``hard'' on graphs with small cycles, and becomes progressively easier as the girth increases. More specifically, we obtain the following interesting trichotomy: Connected Dominating Set (a) does not have a kernel of any size on graphs of girth $3$ or $4$ (since the problem is W[2]-hard); (b) admits a $g(k)$ kernel, where $g(k)$ is $k^{O(k)}$, on graphs of girth $5$ or $6$ but has no polynomial kernel (unless the Polynomial Hierarchy (PH) collapses to the third level) on these graphs; (c) has a cubic ($O(k^3)$) kernel on graphs of girth at least $7$.
While there is a large and growing collection of parameterized complexity results available for problems on graph classes characterized by excluded minors, our results add to the very few known in the field for graph classes characterized by excluded subgraphs.

Neeldhara Misra, Geevarghese Philip, Venkatesh Raman, and Saket Saurabh. The effect of girth on the kernelization complexity of Connected Dominating Set. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2010). Leibniz International Proceedings in Informatics (LIPIcs), Volume 8, pp. 96-107, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2010)

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@InProceedings{misra_et_al:LIPIcs.FSTTCS.2010.96, author = {Misra, Neeldhara and Philip, Geevarghese and Raman, Venkatesh and Saurabh, Saket}, title = {{The effect of girth on the kernelization complexity of Connected Dominating Set}}, booktitle = {IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2010)}, pages = {96--107}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-23-1}, ISSN = {1868-8969}, year = {2010}, volume = {8}, editor = {Lodaya, Kamal and Mahajan, Meena}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2010.96}, URN = {urn:nbn:de:0030-drops-28567}, doi = {10.4230/LIPIcs.FSTTCS.2010.96}, annote = {Keywords: Connected Dominating Set, parameterized complexity, kernelization, girth} }

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