There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

다중혈관 관상동맥 환자에서 y-문합을 이용하여 양쪽 내흉동맥만을 사용한 우회술의 조기 성적

This comprehensive textbook presents a clean and coherent account of most fundamental tools and techniques in Parameterized Algorithms and is a self-contained guide to the area. The book covers many of the recent developments of the field, including application of important separators, branching based on linear programming, Cut & Count to obtain faster algorithms on tree decompositions, algorithms based on representative families of matroids, and use of the Strong Exponential Time Hypothesis. A number of older results are revisited and explained in a modern and didactic way. The book provides a toolbox of algorithmic techniques. Part I is an overview of basic techniques, each chapter discussing a certain algorithmic paradigm. The material covered in this part can be used for an introductory course on fixed-parameter tractability. Part II discusses more advanced and specialized algorithmic ideas, bringing the reader to the cutting edge of current research. Part III presents complexity results and lower bounds, giving negative evidence by way of W[1]-hardness, the Exponential Time Hypothesis, and kernelization lower bounds. All the results and concepts are introduced at a level accessible to graduate students and advanced undergraduate students. Every chapter is accompanied by exercises, many with hints, while the bibliographic notes point to original publications and related work.

Parameterized Algorithms

A graph is just a bunch of points with lines between some of them, like a map of cities linked by roads. A rather simple notion. Nevertheless, the theory of graphs has broad and important applications, because so many things can be modeled by graphs. For example, planar graphs — graphs in which none of the lines cross are— important in designing computer chips and other electronic circuits. Also, various puzzles and games are solved easily if a little graph theory is applied.

A Theory of Graphs

Improved upper bounds for vertex cover

http://www.ii.uib.no/~pinar/MinTriSurvey.pdf

Minimal triangulations of graphs: A survey

We study the computational complexity of partitioning the vertices of a graph into generalized dominating sets. Generalized dominating sets are parameterized by two sets of nonnegative integers σ and ρ which constrain the neighborhood N(υ) of vertices. A set S of vertices of a graph is said to be a (σ, ρ)-set if ∀υ ∈ S : |N(υ) ∩ S| ∈ σ and ∀υ n ∈ S : |N(υ) ∩ S| ∈ ρ. The (k, σ, ρ)-partition problem asks for the existence of a partition V1, V2, ..., Vk of vertices of a given graph G such that Vi, i = 1, 2, ...,k is a (σ, ρ)-set of G. We study the computational complexity of this problem as the parameters σ, ρ and k vary.

Partitioning graphs into generalized dominating sets

We present a new algorithm, called MCS-M, for computing minimal triangulations of graphs. Lex-BFS, a seminal algorithm for recognizing chordal graphs, was the genesis for two other classical algorithms: LEX M and MCS. LEX M extends the fundamental concept used in Lex-BFS, resulting in an algorithm that not only recognizes chordality, but also computes a minimal triangulation of an arbitrary graph. MCS simplifies the fundamental concept used in Lex-BFS, resulting in a simpler algorithm for recognizing chordal graphs. The new algorithm MCS-M combines the extension of LEX M with the simplification of MCS, achieving all the results of LEX M in the same time complexity.

/pdf/maximum-cardinality-search-for-computing-minimal-50eu8e3wun.pdf

Maximum Cardinality Search for Computing Minimal Triangulations of Graphs

We present a new algorithm, called MCS-M, for computing minimal triangulations of graphs. Lex- BFS, a seminal algorithm for recognizing chordal graphs, was the genesis for two other classical algorithms: LEX M and MCS. LEX M extends the fundamental concept used in Lex-BFS, resulting in an algorithm that not only recognizes chordality, but also computes a minimal triangulation of an arbitrary graph. MCS simplifies the fundamental concept used in Lex-BFS, resulting in a simpler algorithm for recognizing chordal graphs. The new algorithm MCS-M combines the extension of LEX M with the simplification of MCS, achieving all the results of LEX M in the same time complexity.

Maximum Cardinality Search for Computing Minimal Triangulations of Graphs 1

We present an algorithm with runtime $O(k^{2k}n^3m)$ for the following NP-complete problem [M. Garey and D. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness, W. H. Freeman and Co., San Francisco, 1979, problem GT35]: Given an arbitrary graph $G$ on $n$ vertices and $m$ edges, can we obtain an interval graph by adding at most $k$ new edges to $G$? This resolves the long-standing open question [H. Kaplan, R. Shamir, and R. E. Tarjan, SIAM J. Comput., 28 (1999), pp. 1906-1922; R. G. Downey and M. R. Fellows, Parameterized Complexity, Springer-Verlag, New York, 1999; M. Serna and D. Thilikos, Bull. Eur. Assoc. Theory Comput. Sci. EATCS, 86 (2005), pp. 41-65; G. Gutin, S. Szeider, and A. Yeo, in Proceedings IWPEC 2006, Lecture Notes in Comput. Sci. 4169, Springer-Verlag, Berlin, 2006, pp. 60-71], first posed by Kaplan, Shamir, and Tarjan, of whether this problem was fixed parameter tractable. The problem has applications in profile minimization for sparse matrix computations [J. A. George and J. W. H. Liu, Computer Solution of Large Sparse Positive Definite Systems, Prentice-Hall, Englewood Cliffs, NJ, 1981; R. E. Tarjan, in Sparse Matrix Computations, J. R. Bunch and D. J. Rose, eds., Academic Press, 1976, pp. 3-22], and our results show tractability for the case of a small number $k$ of zero elements in the envelope. Our algorithm performs bounded search among possible ways of adding edges to a graph to obtain an interval graph and combines this with a greedy algorithm when graphs of a certain structure are reached by the search.

/pdf/interval-completion-is-fixed-parameter-tractable-2b0734q4s0.pdf

Pinar Heggernes

Papers

Minimal triangulations of graphs: A survey

Partitioning graphs into generalized dominating sets

Maximum Cardinality Search for Computing Minimal Triangulations of Graphs

Maximum Cardinality Search for Computing Minimal Triangulations of Graphs 1

Interval Completion Is Fixed Parameter Tractable