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

We show that the geodetic number of proper interval graphs can be computed in polynomial time. This problem is NP-hard on chordal graphs and on bipartite weakly chordal graphs. Only an upper bound on the geodetic number of proper interval graphs has been known prior to our result.

Computing minimum geodetic sets of proper interval graphs

The problem of computing minimal triangulations of graphs, also called minimal fill, was introduced and solved in 1976 by Rose, Tarjan, and Lueker [17] in time O(nm), thus O(n) for dense graphs. Although the topic has received increasing attention since then, and several new results on characterizing and computing minimal triangulations have been presented, this first time bound has remained the best. In this paper we introduce an O(n logn) time algorithm for computing minimal triangulations, where O(n) is the time required to multiply two n×n matrices. The current best known α is less than 2.376, and thus our result breaks the long standing asymptotic time complexity bound for this problem. To achieve this result, we introduce and combine several techniques that are new to minimal triangulation algorithms, like working on the complement of the input graph, graph search for a vertex set A that bounds the size of the connected components when A is removed, and matrix multiplication.

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Computing Minimal Triangulations in Time

The bandwidth minimization problem has a long history and a number of practical applications. In this paper we introduce a natural extension of bandwidth to partially ordered layouts. We consider this extension from three main viewpoints: graph searching, tree decompositions, and elimination orderings. The three graph parameters pathwidth, profile, and bandwidth related to linear layouts can be defined by variants of graph searching using a standard fugitive. Switching to an inert fugitive, the two former parameters are extended to treewidth and fill-in, and our first viewpoint considers the analogous tree-like extension that arises from the bandwidth variant. Bandwidth also has a definition in terms of ordered path decompositions, and our second viewpoint extends this in a natural way to ordered tree decompositions. In showing that both extensions are equivalent we employ the third viewpoint of elimination trees, as used in the field of sparse matrix computations. We call the resulting parameter the treespan of a graph and prove some of its combinatorial and algorithmic properties.

Pinar Heggernes

Papers

Computing minimum geodetic sets of proper interval graphs

Computing Minimal Triangulations in Time

Graph Searching, Elimination Trees, and a Generalization of Bandwidth

The Firefighter problem on graph classes

A multi-parameter analysis of hard problems on deterministic finite automata