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

Graph homomorphism, also called H-coloring, is a natural generalization of graph coloring: There is a homomorphism from a graph G to a complete graph on k vertices if and only if G is k-colorable. During recent years the topic of exact (exponential-time) algorithms for NP-hard problems in general, and for graph coloring in particular, has led to extensive research. Consequently, it is natural to ask how the techniques developed for exact graph coloring algorithms can be extended to graph homomorphisms. By the celebrated result of Hell and Nesetril, for each fixed simple graph H, deciding whether a given simple graph G has a homomorphism to H is polynomial-time solvable if H is a bipartite graph, and NP-complete otherwise. The case where H is the cycle of length 5, is the first NP-hard case different from graph coloring. We show that for an odd integer $k\ge 5$, whether an input graph G with n vertices is homomorphic to the cycle of length k, can be decided in time $\min \{ \binom{n}{\lfloor n/k\rfloor}, 2^{n/2}\}\cdot n^{{\mathcal{O}(1)}}$. We extend the results obtained for cycles, which are graphs of treewidth two, to graphs of bounded treewidth as follows: if H is of treewidth at most t, then whether input graph G with n vertices is homomorphic to H can be decided in time $(t +3)^{n}\cdot n^{{\mathcal{O}(1)}}$.

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Exact Algorithms for Graph Homomorphisms

The elimination tree plays an important role in many aspects of sparse matrix factorization. The height of the elimination tree presents a rough, but usually effective, measure of the time needed to perform parallel elimination. Finding orderings that produce low elimination is therefore important. As the problem of finding minimum height elimination tree orderings is NP-hard, it is interesting to concentrate on limited classes of graphs and find minimum height elimination trees for these efficiently. In this paper, we use clique trees to find an efficient algorithm for interval graphs which make an important subclass of chordal graphs. We first illustrate this method through an algorithm that finds minimum height elimination for chordal graphs. This algorithm, although of exponential time complexity, is conceptionally simple and leads to a polynomial-time algorithm for finding minimum height elimination trees for interval graphs.

Finding minimum height elimination trees for interval graphs in polynomial time

The Minimum Degree Algorithm, one of the classical algorithms of sparse matrix computations, is a heuristic for computing a minimum triangulation of a graph. It is widely used as a component in every sparse matrix package, and it is known to produce triangulations with few fill edges in practice, although no theoretical bound or guarantee has been shown on its quality. Another interesting behavior of Minimum Degree observed in practice is that it often results in a minimal triangulation. Our goal in this paper is to examine the theoretical reasons behind this good performance. We give new invariants which partially explain the mechanisms underlying this heuristic. We show that Minimum Degree is in fact resilient to error, as even when an undesirable triangulating edge with respect to minimal triangulation is added at some step of the algorithm, at later steps the chances of adding only desirable edges remain intact. We also use our new insight to propose an improvement of this heuristic, which introduces at most as many fill edges as Minimum Degree but is guaranteed to yield a minimal triangulation.

The Minimum Degree heuristic and the minimal triangulation process

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 also computes a minimal triangulation of an arbitrary graph. MCS simplified the fundamental concept used in Lex-BFS, resulting in a simpler algorithm for recognizing chordal graphs. The new simpler 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-1af0neba4s.pdf

Maximum Cardinality Search for Computing Minimal Triangulations

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(n3) 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 α log n) time algorithm for computing minimal triangulations, where O(nα) 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.

Pinar Heggernes

Papers

Exact Algorithms for Graph Homomorphisms

Finding minimum height elimination trees for interval graphs in polynomial time

The Minimum Degree heuristic and the minimal triangulation process

Maximum Cardinality Search for Computing Minimal Triangulations

Computing minimal triangulations in time O(nα log n) = o(n2.376)