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

Vertex elimination is a graph operation that turns the neighborhood of a vertex into a clique and removes the vertex itself. It has widely known applications within sparse matrix computations. We define the Elimination problem as follows: given two graphs G and H, decide whether H can be obtained from G by |V(G)|−|V(H)| vertex eliminations. We study the parameterized complexity of the Elimination problem. We show that Elimination is W[1]-hard when parameterized by |V(H)|, even if both input graphs are split graphs, and W[2]-hard when parameterized by |V(G)|−|V(H)|, even if H is a complete graph. On the positive side, we show that Elimination admits a kernel with at most 5|V(H)| vertices in the case when G is connected and H is a complete graph, which is in sharp contrast to the W[1]-hardness of the related Clique problem. We also study the case when either G or H is tree. The computational complexity of the problem depends on which graph is assumed to be a tree: we show that Elimination can be solved in polynomial time when H is a tree, whereas it remains NP-complete when G is a tree.

/pdf/how-to-eliminate-a-graph-17shqftgzg.pdf

How to eliminate a graph

We study the problem of finding small s---t separators that induce graphs having certain properties. It is known that finding a minimum clique s---t separator is polynomial-time solvable (Tarjan 1985), while for example the problems of finding a minimum s---t separator that is a connected graph or an independent set are fixed-parameter tractable (Marx, O'Sullivan and Razgon, manuscript). We extend these results the following way:

· Finding a minimum c-connected s---t separator is FPT for c=2 and W[1]-hard for any c≥3.

· Finding a minimum s---t separator with diameter at most d is W[1]-hard for any d≥2.

· Finding a minimum r-regular s---t separator is W[1]-hard for any r≥1.

· For any decidable graph property, finding a minimum s---t separator with this property is FPT parameterized jointly by the size of the separator and the maximum degree.

We also show that finding a connected s---t separator of minimum size does not have a polynomial kernel, even when restricted to graphs of maximum degree at most 3, unless NP ⊆ coNP/poly.

/pdf/on-the-parameterized-complexity-of-finding-separators-with-2n8nas7oxx.pdf

On the parameterized complexity of finding separators with non-hereditary properties

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

Scheduling unit-length jobs with precedence constraints of small height

The Induced Subtree Isomorphism problem takes as input a graph G and a tree T, and the task is to decide whether G has an induced subgraph that is isomorphic to T. This problem is known to be NP-complete on bipartite graphs, but it can be solved in polynomial time when G is a forest. We show that Induced Subtree Isomorphism can be solved in polynomial time when G is an interval graph. In contrast to this positive result, we show that the closely related Subtree Isomorphism problem is NP-complete even when G is restricted to the class of proper interval graphs, a well-known subclass of interval graphs.

/pdf/induced-subtrees-in-interval-graphs-543a1yo0f6.pdf

Induced Subtrees in Interval Graphs

The bandwidth minimization problem has a long history and a number of practical applications. In this paper we introduce a generalization of bandwidth to partially ordered layouts. We consider this generalization from two main viewpoints: graph searching and tree decompositions. 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 generalized to treewidth and fill-in, and our first viewpoint considers the analogous tree-like generalization that arises from the bandwidth variant. Bandwidth also has a definition in terms of ordered path decompositions, and our second viewpoint generalizes this in a natural way to ordered tree decompositions. In showing that both generalizations 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.

/pdf/graph-searching-elimination-trees-and-a-generalization-of-4sfl5m2nno.pdf

Pinar Heggernes

Papers

How to eliminate a graph

On the parameterized complexity of finding separators with non-hereditary properties

Scheduling unit-length jobs with precedence constraints of small height

Induced Subtrees in Interval Graphs

Graph Searching, Elimination Trees, and a Generalization of Bandwidth