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

Given a permutation π of {1,,n} and a positive integer k, we give an algorithm with running time $2^{O(k^2 \log k)}n^{O(1)}$ that decides whether π can be partitioned into at most k increasing or decreasing subsequences Thus we resolve affirmatively the open question of whether the problem is fixed parameter tractable This NP-complete problem is equivalent to deciding whether the cochromatic number (the minimum number of cliques and independent sets the vertices of the graph can be partitioned into) of a given permutation graph on n vertices is at most k In fact, we give a more general result: within the mentioned running time, one can decide whether the cochromatic number of a given perfect graph on n vertices is at most k

To obtain our result we use a combination of two well-known techniques within parameterized algorithms, namely greedy localization and iterative compression We further demonstrate the power of this combination by giving a $2^{O(k^2 \log k)}n \log n$ time algorithm for deciding whether a given set of n non-overlapping axis-parallel rectangles can be stabbed by at most k of the given set of horizontal and vertical lines Whether such an algorithm exists was mentioned as an open question in several papers

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Fixed-Parameter algorithms for cochromatic number and disjoint rectangle stabbing

The problem of determining the cutwidth of a graph is a notoriously hard problem which remains NP-complete under severe restrictions on input graphs. Until recently, non-trivial polynomial-time cutwidth algorithms were known only for subclasses of graphs of bounded treewidth. In WG 2008, Heggernes et al. initiated the study of cutwidth on graph classes containing graphs of unbounded treewidth, and showed that a greedy algorithm computes the cutwidth of threshold graphs. We continue this line of research and present the first polynomial-time algorithm for computing the cutwidth of bipartite permutation graphs. Our algorithm runs in linear time. We stress that the cutwidth problem is NP-complete on bipartite graphs and its computational complexity is open even on small subclasses of permutation graphs, such as trivially perfect graphs.

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Computing the cutwidth of bipartite permutation graphs in linear time

We show that the mixed search number and the linear-width of interval graphs and of split graphs can be computed in linear time and in polynomial time, respectively. © 2009 Wiley Periodicals, Inc. NETWORKS, 2010

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Mixed search number and linear-width of interval and split graphs

We initiate a multivariate analysis of two well-known NPhard decision problems on DFAs: the problem of finding a short synchronizing word and that of finding a DFA on few states consistent with a given sample of the intended language and its complement. For both problems, we study natural parameterizations and classify them with the tools provided by Parameterized Complexity. Somewhat surprisingly, in both cases, rather simple FPT algorithms can be shown to be optimal, mostly assuming the (Strong) Exponential Time Hypothesis.

A Multivariate Analysis of Some DFA Problems

The Induced Subgraph Isomorphism problem on two input graphs G and H is to decide whether G has an induced subgraph isomorphic to H. Already for the restricted case where H is a complete graph the problem is NP-complete, as it is then equivalent to the Clique problem. In a recent paper [7] Marx and Schlotter show that Induced Subgraph Isomorphism is NP-complete when G and H are restricted to be interval graphs. They also show that the problem is W[1]-hard with this restriction when parametrised by the number of vertices in H. In this paper we show that when G is an interval graph and H is a connected proper interval graph, the problem is solvable in polynomial time. As a more general result, we show that when G is an interval graph and H is an arbitrary proper interval graph, the problem is fixed parameter tractable when parametrised by the number of connected components of H. To complement our results, we prove that the problem remains NP-complete when G and H are both proper interval graphs and H is disconnected.

/pdf/induced-subgraph-isomorphism-on-interval-and-proper-interval-4s2pyz2zj2.pdf

Pinar Heggernes

Papers

Fixed-Parameter algorithms for cochromatic number and disjoint rectangle stabbing

Computing the cutwidth of bipartite permutation graphs in linear time

Mixed search number and linear-width of interval and split graphs

A Multivariate Analysis of Some DFA Problems

Induced Subgraph Isomorphism on Interval and Proper Interval Graphs