Tournaments and Optimality: New Results in Parameterized Complexity Micha lPilipczuk Dissertation for the degree of Philosophiae Doctor (PhD) University of Bergen, Norway August 2013 Contents I Introduction1 1 A gentle introduction to parameterized complexity3 1.1 The phenomenon of NP-completeness..........................3 1.2 Parameterized complexity in a nutshell..........................4 1.3 Goals of the parameterized analysis............................6 1.4 Overview of the techniques................................7 1.4.1 Upper bounds....................................7 1.4.2 Lower bounds.................................... 12 1.5 Highlights of this thesis.................................. 16 1.5.1 Survey of the content............................... 16 1.5.2 Sparse graphs and dense graphs.......................... 17 2 Preliminaries 19 2.1 Notation........................................... 19 2.1.1 General notation.................................. 19 2.1.2 Notation for graphs................................ 19 2.2 Algorithmic frameworks.................................. 23 2.2.1 Parameterized complexity............................. 23 2.2.2 Kernelization.................................... 23 2.2.3 ETH and SETH.................................. 24 2.2.4 Kernelization lower bounds............................ 24 2.3 Width measures and the topological theory for graphs................. 25 2.3.1 Containment relations............................... 25 2.3.2 Treewidth and pathwidth............................. 26 2.3.3 Cliquewidth..................................... 26 2.3.4 Branch decompositions, branchwidth, rankwidth, and carving-width..... 27 2.3.5 Linear width measures and cutwidth....................... 27 2.3.6 Models of logic on graphs............................. 27 2.3.7 Well-quasi orderings................................ 29 3 Width parameters of graphs 31 3.1 The Graph Minors project................................. 31 3.1.1 An express overview of the proof......................... 31 3.1.2 The Excluded Grid Minor Theorem....................... 32 3.1.3 The Decomposition Theorem........................... 33 iii 3.1.4 Algorithmic problems............................... 35 3.1.5 The theory of graph minors for directed graphs................. 36 3.2 Treewidth and its applications.............................. 37 3.2.1 Algorithms for treewidth............................. 37 3.2.2 Treewidth and MSO2 ............................... 38 3.2.3 Model checking FO on sparse graph classes................... 39 3.2.4 WIN/WIN approaches............................... 41 3.3 Other undirected width measures............................. 43 3.3.1 Cliquewidth and rankwidth............................ 44 3.3.2 Branchwidth and carving-width.......................... 45 3.3.3 Pathwidth and cutwidth.............................. 46 3.4 Width measures of directed graphs............................ 47 3.4.1 Directed treewidth................................. 47 3.4.2 DAG-width and Kelly-width........................... 48 3.4.3 Directed pathwidth................................. 49 3.4.4 No good width measures for directed graphs?.................. 50 4 The optimality program 51 4.1 Results following immediately from ETH......................... 51 4.1.1 Classical complexity................................ 51 4.1.2 Parameterized complexity............................. 52 4.2 The mysterious class SUBEPT.............................. 53 4.3 Slightly superexponential parameterized time...................... 54 4.4 Dynamic programming on treewidth........................... 55 4.5 Lower bounds for XP algorithms............................. 57 4.6 Linking brute-force and dynamic programming to SETH................ 59 II Topological problems in semi-complete digraphs 61 5 Introduction to tournaments and semi-complete digraphs 63 5.1 Introduction......................................... 63 5.1.1 The theory of graph minors for digraphs..................... 63 5.1.2 The containment theory for tournaments.................... 64 5.1.3 Tournaments and parameterized complexity................... 66 5.1.4 Our results and techniques............................ 67 5.2 Preliminaries........................................ 72 5.2.1 Folklore and simple facts............................. 72 5.2.2 Definitions of containment relations....................... 74 5.2.3 Width parameters................................. 76 5.3 MSO and semi-complete digraphs............................. 82 6 Computing width measures of semi-complete digraphs 85 6.1 The obstacle zoo...................................... 85 6.1.1 Jungles and short jungles............................. 85 6.1.2 Triples........................................ 87 iv 6.1.3 Degree tangles................................... 89 6.1.4 Matching tangles.................................. 91 6.1.5 Backward tangles.................................. 92 6.2 Algorithms for pathwidth................................. 93 6.2.1 Subset selectors for bipartite graphs....................... 94 6.2.2 The algorithms................................... 100 6.3 Algorithms for cutwidth and other ordering problems.................. 104 6.3.1 Approximation of cutwidth............................ 104 6.3.2 Additional problem definitions.......................... 104 6.3.3 k-cuts of semi-complete digraphs......................... 106 6.3.4 The algorithms................................... 111 6.4 Conclusions and open problems.............................. 114 7 Solving topological problems 117 7.1 Algorithms for containment testing............................ 117 7.2 Containment testing and meta-theorems......................... 119 7.3 The algorithm for Rooted Immersion........................... 120 7.3.1 Irrelevant vertex in a triple............................ 120 7.3.2 Applying the irrelevant vertex rule........................ 125 7.4 Dynamic programming routines for containment relations............... 126 7.4.1 Terminology.................................... 127 7.4.2 The algorithm for topological containment.................... 129 7.4.3 The algorithm for Rooted Immersion....................... 133 7.5 Conclusions and open problems.............................. 136 III In search for optimality 139 8 Tight bounds for parameterized complexity of Cluster Editing 141 8.1 Introduction......................................... 141 8.2 Preliminaries........................................ 144 8.3 A subexponential algorithm................................ 145 8.3.1 Reduction for large p ................................ 145 8.3.2 Bounds on binomial coefficients.......................... 149 8.3.3 Small cuts...................................... 150 8.3.4 The algorithm................................... 151 8.4 Multivariate lower bound................................. 152 8.4.1 Preprocessing of the formula........................... 152 8.4.2 Construction.................................... 154 8.4.3 Completeness.................................... 156 8.4.4 Soundness...................................... 158 8.5 General clustering under ETH............................... 162 8.6 Conclusion and open questions.............................. 164 v 9 Tight bounds for parameterized complexity of Edge Clique Cover 167 9.1 Introduction......................................... 167 9.2 Double-exponential lower bound............................. 171 9.2.1 Cocktail party graph................................ 172 9.2.2 Construction.................................... 173 9.2.3 Completeness.................................... 177 9.2.4 Soundness...................................... 178 vi Part I Introduction 1 Chapter 1 A gentle introduction to parameterized complexity 1.1 The phenomenon of NP-completeness Since Cook's seminal proof of NP-hardness of the SAT problem [75], the pursuit of understanding the border between efficiently tractable problems (i.e., languages in P) and problems that seem to require exponential-time computations (i.e., NP-hard languages) has driven most of the research in theoretical computer science. Despite the fact that a formal proof that P6=NP is still out of our reach, arguably we have a fairly good intuitive understanding of what features make a particular problem NP-hard. Starting with Karp's list of 21 problems [203], the net of known NP-complete problems has quickly reached volumes of books, see for reference the monograph of Garey and Johnson [161]. Today's research on NP-hardness focuses mostly on (i) understanding the source of intractability of a given problem, and (ii) identifying the most efficient ways of circumventing it by examining the problem's structure. Arguably the most natural way of coping with an NP-hard problem is via approximation. In this framework, given an NP-hard optimization problem we relax the request of finding an optimal solution to finding a solution that is provably not far from being optimal; for instance, we may ask for a solution whose cost is at most twice as large as the optimal one. Generally, for a minimization problem an algorithm returning a value at most c times larger than the optimal one is called a c-approximation; this definition can be easily translated to maximization problems as well. Relax- ation of the request of finding an optimal solution enables us to perform faster computations; the classical definition of an approximation algorithm assumes that it works in polynomial time. It appears that the landscape of NP-complete problems that is uniformly flat when simple