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"Some Contributions to Parameterized Complexity"

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Laboratoire d'Informatique, de Robotique et de Micro´electronique de Montpellier Universite´ de Montpellier Speciality: Computer Science Habilitation `aDiriger des Recherches (HDR) Ignasi Sau Valls Some Contributions to Parameterized Complexity Quelques Contributions en Complexit´eParam´etr´ee Version of June 21, 2018 { Defended on June 25, 2018 Committee: Reviewers: Michael R. Fellows - University of Bergen (Norway) Fedor V. Fomin - University of Bergen (Norway) Rolf Niedermeier - Technische Universit¨at Berlin (Germany) Examinators: Jean-Claude Bermond - CNRS, U. de Nice-Sophia Antipolis (France) Marc Noy - Univ. Polit`ecnica de Catalunya (Catalonia) Dimitrios M. Thilikos - CNRS, Universit´ede Montpellier (France) Gilles Trombettoni - Universit´ede Montpellier (France) Contents 0 R´esum´eet projet de recherche7 1 Introduction 13 1.1 Contextualization................................. 13 1.2 Scientific collaborations ............................. 15 1.3 Organization of the manuscript......................... 17 2 Curriculum vitae 19 2.1 Education and positions............................. 19 2.2 Full list of publications.............................. 20 2.3 Supervised students ............................... 32 2.4 Awards, grants, scholarships, and projects................... 33 2.5 Teaching activity................................. 34 2.6 Committees and administrative duties..................... 35 2.7 Research visits .................................. 36 2.8 Research talks................................... 38 2.9 Journal and conference refereeing........................ 43 3 Preliminaries 45 3.1 Graphs....................................... 45 3.1.1 Basic notation .............................. 45 3.1.2 Graph minors............................... 47 3.1.3 Treewidth................................. 48 3.1.4 (Counting) Monadic Second Order Logic................ 49 3.2 Parameterized complexity............................ 50 3.3 Some classical problems ............................. 53 4 Summary of my contributions 57 4.1 FPT algorithms.................................. 57 4.2 Kernelization................................... 70 4.3 Combinatorial results .............................. 74 4.4 Problems arising from applications....................... 77 5 Linear kernels and single-exponential algorithms via protrusion de- compositions 81 5.1 Introduction.................................... 82 5.2 Protrusions, t-boundaried graphs, and finite integer index.......... 87 5.3 Constructing protrusion decompositions.................... 92 5.4 Linear kernels on graphs excluding a topological minor............ 96 5.4.1 Proof of Theorem 5.1 .......................... 96 5.4.2 Problems affected by our result.....................102 5.4.3 A comparison with earlier results....................104 5.4.4 The limits of our approach .......................105 5.4.5 An illustrative example: Edge Dominating Set . 106 5.5 Single-exponential algorithm for Planar- -Deletion . 108 F 5.5.1 Analysis of the bag marking algorithm . 110 5.5.2 Branching step and linear protrusion decomposition . 111 5.5.3 Solving Planar- -Deletion with a linear protrusion decomposition112 F 5.5.4 Proof of Theorem 5.2 ..........................116 5.6 Some deferred results...............................117 5.6.1 Edge modification problems are not minor-closed . 117 5.6.2 Disconnected planar obstructions....................117 5.6.3 Disconnected Planar- -Deletion has not finite integer index . 118 F 5.6.4 MSO formula for topological minor containment . 119 5.7 Concluding remarks ...............................119 6 Explicit linear kernels via dynamic programming 123 6.1 Introduction....................................124 6.2 Preliminaries ...................................125 6.3 An explicit protrusion replacer .........................128 6.3.1 Encoders .................................129 6.3.2 Equivalence relations and representatives . 132 6.3.3 Explicit protrusion replacer.......................138 6.4 An explicit linear kernel for r-Dominating Set . 140 6.4.1 Description of the encoder........................140 6.4.2 Construction of the kernel........................144 6.5 An explicit linear kernel for r-Scattered Set . 145 6.5.1 Description of the encoder........................145 6.5.2 Construction of the kernel........................148 6.6 An explicit linear kernel for Planar- -Deletion . 149 F 6.6.1 The encoder for -Deletion and the index of ;t . 150 F ∼G 6.6.2 Construction of the kernel on H-minor-free graphs . 152 6.6.3 Linear kernels on H-topological-minor-free graphs . 153 6.7 Concluding remarks ...............................155 7 On the number of labeled graphs of bounded treewidth 157 7.1 Introduction....................................157 7.2 The construction.................................159 7.2.1 Notation and definitions.........................159 7.2.2 Description of the construction.....................160 7.2.3 Bounding the treewidth.........................162 7.3 Proof of the main result.............................163 7.3.1 Number of constructible triples (σ; f; N) . 163 7.3.2 Bounding the number of duplicates...................163 7.3.3 Choosing the parameter s . 165 7.4 Concluding remarks ...............................166 8 Finding a spanning tree with minimum reload cost diameter 169 8.1 Introduction....................................170 8.2 Preliminaries ...................................173 8.3 Para-NP-hardness results.............................175 8.4 A polynomial-time algorithm on cactus graphs . 180 8.5 FPT algorithm parameterized by k + tw + ∆ . 187 8.6 Polynomially bounded costs...........................191 8.7 Concluding remarks ...............................193 9 Further research 195 Bibliography 197 Chapter 0 R´esum´eet projet de recherche Nous pr´esentons ici un petit r´esum´ede ce manuscrit, ainsi que quelques pistes de recherche pour les ann´ees `avenir, en relation avec le contenu scientifique expos´edans le manuscrit. R´esum´e Ce document contient une synth`ese de mes principales contributions scientifiques apr`es ma soutenance de th`ese, qui a eu lieu en octobre 2009, et surtout pendant le temps que j'ai pass´eau LIRMM (Montpellier, France) en tant que Charg´ede Recherche au CNRS. Au lieu d'une compilation exhaustive de toutes mes contributions, ce qui serait certaine- ment trop long et trop h´et´erog`ene, ce manuscrit contient deux parties principales. Dans la premi`ere, on pr´esente une description courte de toutes mes contributions, en les met- tant en contexte et en ´enon¸cant les r´esultats principaux, mais sans rentrer dans les d´etails scientifiques. Cette description permet d'avoir une vision globale sur l'ensemble de mes contributions. Dans la seconde partie, on pr´esente quatre de ces contributions avec tous les d´etails. Ces quatre contributions ont ´et´echoisies avec le crit`ere d'^etre repr´esentatives `ala fois sur les th´ematiques sur lequelles elles portent, et aussi sur le type de techniques utilis´ees pour obtenir les r´esultats. En rentrant sur la th´ematique de mes recherches, la grande plupart de mes contributions se placent dans le cadre de la complexit´eparam´etr´ee. Cette th´eorie a ´et´ed´evelopp´ee par Downey et Fellows dans les ann´ees 90, et aujourd'hui est d´ej`aun domaine de recherche compl`etement ´etabli avec, par exemple, quatre bouquins publi´essur le sujet, et de centaines d'articles accept´esannuellement dans les plus prestigieuses conf´erences internationales en informatique th´eorique. Nous allons maintenant introduire de mani`ere succincte les id´ees principales de la complexit´eparam´etr´ee,pour pouvoir mettre en contexte mes contribu- tions scientifiques. La th´eorie de la NP-compl´etude \classique" consid`ere qu'un probl`eme est facile s'il admet un algorithme polynomial pour le r´esoudre. Mais beaucoup de probl`emes sont NP-difficiles, c'est-`a-dire, ils n'admettent pas de tels algorithmes `amoins que P = NP, une hypoth`ese que, au jour d'aujourd'hui, est consid´er´eede mani`ere presque unanime comme essez peu probable dans la communaut´e. 7 8 R´esum´eet projet de recherche Cependant, une fois qu'un probl`eme est montr´e NP-difficile, que faire si on doit malgr´e tout le r´esoudre? Plusieurs m´ethodologies peuvent ^etre envisag´ees, les plus connues ´etant les suivantes: algorithmes d'approximation, algorithmes randomis´es, heuristiques, ou bien algorithmes exponentiels exacts. Logiquement, chacune de ces m´ethodes a des avantages et d´esavantages. L'algorithmique `aparam`etre fix´ee se place dans l'approche des algorithmes exponentiels exacts, notamment en proposant une analyse de la complexit´e bidimensionnelle, et donc plus fine que l'analyse classique. La motivation principale est d'essayer d'obtenir une meilleure vision de la complexit´ed'un probl`eme en explorant comment certains param`etres, sp´ecifiques du probl`eme ou des instances, influencent cette complexit´e. Id´ealement, on cherche `aidentifier un param`etre k tel que si k est petit alors le probl`eme peut ^etre r´esolu efficacement. Mais qu'est-ce qu'on veut dire par “efficacement"? La notion fondamentale de la th´eorie param´etr´eeest la d´efinition suivante d'algorithme “efficace": un algorithme tel que son temps d'ex´ecution peut ^etre born´epar f(k) x (1), o`u f est une · j jO fonction quelconque qui ne d´epend que du param`etre k, et x repr´esente la taille de j j l'entr´ee du probl`eme. Les probl`emes param´etr´es qui poss`edent un tel algorithme sont appel´es solubles a param`etre fix´e (en anglais, fixed-parameter tractable),
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