1,184 research outputs found
Graph Algorithms and Applications
The mixture of data in real-life exhibits structure or connection property in nature. Typical data include biological data, communication network data, image data, etc. Graphs provide a natural way to represent and analyze these types of data and their relationships. Unfortunately, the related algorithms usually suffer from high computational complexity, since some of these problems are NP-hard. Therefore, in recent years, many graph models and optimization algorithms have been proposed to achieve a better balance between efficacy and efficiency. This book contains some papers reporting recent achievements regarding graph models, algorithms, and applications to problems in the real world, with some focus on optimization and computational complexity
Cliques, colouring and satisfiability : from structure to algorithms
We examine the implications of various structural restrictions on the computational
complexity of three central problems of theoretical computer science
(colourability, independent set and satisfiability), and their relatives. All problems
we study are generally NP-hard and they remain NP-hard under various restrictions.
Finding the greatest possible restrictions under which a problem is computationally
difficult is important for a number of reasons. Firstly, this can make it easier to
establish the NP-hardness of new problems by allowing easier transformations. Secondly,
this can help clarify the boundary between tractable and intractable instances
of the problem.
Typically an NP-hard graph problem admits an infinite sequence of narrowing
families of graphs for which the problem remains NP-hard. We obtain a number
of such results; each of these implies necessary conditions for polynomial-time
solvability of the respective problem in restricted graph classes. We also identify
a number of classes for which these conditions are sufficient and describe explicit
algorithms that solve the problem in polynomial time in those classes. For the
satisfiability problem we use the language of graph theory to discover the very first
boundary property, i.e. a property that separates tractable and intractable instances
of the problem. Whether this property is unique remains a big open problem
The Parameterized Complexity of Degree Constrained Editing Problems
This thesis examines degree constrained editing problems within the framework of parameterized complexity. A degree constrained editing problem takes as input a graph and a set of constraints and asks whether the graph can be altered in at most k editing steps such that the degrees of the remaining vertices are within the given constraints. Parameterized complexity gives a framework for examining
problems that are traditionally considered intractable and developing efficient exact algorithms for them, or showing that it is unlikely that they have such algorithms, by introducing an additional component to the input, the parameter, which gives additional information about the structure of the problem. If the problem has an algorithm that is exponential in the parameter, but polynomial, with constant degree, in the size of the input, then it is considered to be fixed-parameter tractable.
Parameterized complexity also provides an intractability framework for identifying problems that are likely to not have such an algorithm.
Degree constrained editing problems provide natural parameterizations in terms of the total cost k of vertex deletions, edge deletions and edge additions allowed, and
the upper bound r on the degree of the vertices remaining after editing. We define a class of degree constrained editing problems, WDCE, which generalises several well know problems, such as Degree r Deletion, Cubic Subgraph, r-Regular Subgraph, f-Factor and General Factor. We show that in general if both k and r are part of the parameter, problems in the WDCE class are fixed-parameter tractable, and if parameterized by k or r alone, the problems are intractable in a parameterized sense.
We further show cases of WDCE that have polynomial time kernelizations, and in particular when all the degree constraints are a single number and the editing
operations include vertex deletion and edge deletion we show that there is a kernel with at most O(kr(k + r)) vertices. If we allow vertex deletion and edge addition,
we show that despite remaining fixed-parameter tractable when parameterized by k and r together, the problems are unlikely to have polynomial sized kernelizations, or
polynomial time kernelizations of a certain form, under certain complexity theoretic assumptions.
We also examine a more general case where given an input graph the question is whether with at most k deletions the graph can be made r-degenerate. We show that in this case the problems are intractable, even when r is a constant
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