Parameterized complexity theory provides a framework for a refined analysis. We introduce some classical complexitytheoretic techniques to parameterized complexity. Parameterized complexity theory request pdf researchgate. Incremental problems in the parameterized complexity setting. A kernelization for a parameterized problem is an algorithm that takes an instance with parameter and maps it in polynomial time to. Scribe notes for parameterized complexity of problems in coalition resource games rajesh chitnis, tom chan and kanthi k sarpatwar department of computer science, university of maryland, college park email. The central notion of the theory, fixedparameter tractability, has led to the development of various new algorithmic techniques and a whole new theory of intractability. Oxford university press, oxford, 2006 and the other was the book. Martin grohe abstract we give machine characterizations of most parameterized complexity classes, in particular, of wp, oftheclassesofthewhierarchy, andoftheahierarchy.
Fixedparameter tractability and parameterized complexity. Parameterized complexity theory is a branch of computational complexity theory that provides a framework for a refined analysis of hard algorithmic problems. Web of science you must be logged in with an active subscription to view this. This process is experimental and the keywords may be updated as the learning algorithm improves.
A newer trend in combinatorial optimization is to analyze nphard problems from a parameterized complexity point of view see, e. Bounded nondeterminism and alternation in parameterized complexity theory. Google scholar papers containing parameterized complexity or. The big, important problems facing planet earth have structure with secondary measurements parameters, apart. Parameterized complexity theory provides a framework for a finegrain. Their early work demonstrated that xedparameter tractability is a ubiquitous phenomenon, naturally arising in ariousv contexts and applications. The fine classification of conjunctive queries and. Lfp reductions which, by a result of flum and grohe, are known to be equiv. Parameterized complexity theory provides a framework for a re. In proceedings of the 43rd annual ieee symposium on foundations of computer science. The parameterized complexity of counting problems jo. Recommended parameterized complexity theory, textbook.
In order to compare the complexity of parameterized problems, we use parameterized logspace reductions, called paralreductions in the following. Parameterized complexity theory pdf free download epdf. Cesati \the turing way to parameterized complexity jcss 67, 2003. The problem of finding a minimum vertex cover is a classical optimization problem in computer science and is a typical example of an nphard optimization problem that has an approximation algorithm. Short course complexity design and analysis of guest.
In this talk, i apply techniques from parameterized complexity theory to assess. For every classical complexity class we introduce a parameterized analogue in a natural way. Grohe, bounded nondeterminism and alternation in parameterized complexity theory, in. The focus of our attention is the class wp, which we characterise as the class of all param. Shanghai summer school on parameterized complexity. The parameterized complexity of counting problems request pdf. Parameterized complexity and the understanding, design, and analysis of heuristicst organizers. Parameterized complexity theory texts in theoretical. Our goal in this paper is to show that having machinebased characterizations of parameterized complexity classes opens up a largely unexplored, but possibly very fruitful, path toward understanding parameterized intractability. When confronted with nphard problems, there are basically two possibilities.
Parameterized complexity parameterized problem computable function linear temporal logic relation symbol these keywords were added by machine and not by the authors. The logic classes employed in various problems are fragments of propositional logic defined inductively. In particular, the analogue of polynomial time is the class of all. The gentle revolution of parameterized complexity bioinformatics, operations research, optimization, automated reasoning, etc. I with this assumption, the theory of nphardness is an excellent vehicle for mapping anouterboundary of intractability, for all practical purposes. Bounded nondeterminism and alternation in parameterized. An isomorphism between subexponential and parameterized complexity theory conference paper pdf available in proceedings of the annual ieee conference on computational complexity 374. Anyone interested in this area should take the recent work of flum and grohe. Parameterized counting complexity was introduced by flum and grohe 11. The book of flum and grohe 189 focuses to a large extent on.
There are only the barest beginnings of a structure theory of parametric intractability. Let a be a set of boolean literals, then we define the following formula classes. I indeed, assuming the reasonable current working assumption that ntm acceptance is 2n, nphardness allows for practical lower bound for. Machinebased methods in parameterized complexity theory yijia chen. To that end we extend the work of chen, flum, and grohe 6, 5, 7 in two directions. Parameterized complexity attends to the underlying structure of realworld input distributions, as well as structure in the computational objectives, such as size of solutions, noise in the data, goodness of the desired approximation, or combinations of these.
However, one difference between the two, perhaps no more than a historical accident, is that np was. In computer science, parameterized complexity is a branch of computational complexity theory that focuses on classifying computational problems according to their inherent difficulty with respect to multiple parameters of the input or output. In the mathematical discipline of graph theory, a vertex cover sometimes node cover of a graph is a set of vertices that includes at least one endpoint of every edge of the graph. Scribe notes for parameterized complexity of problems in. The parameterized view on algorithms has led to a theory that is both mathematically beautiful and practically applicable. Grohe, parameterized complexity theory, springer, 2006. Here we obtain a new and nontrivial characterization of the ahierarchy in terms of oracle machines, and parameterize a famous result of baker.
Downey and fellows wrote a monograph about this isbn 3540299521. In the theory of parameterized complexity, thewhierarchy plays a role similar tonp in classical complexity theory in that many natural parameterized problems are shown intractable by being complete for some levelwt of the hierarchy. The book of flum and grohe 189 focuses to a large extent on complexity aspects of parameterized algorithmics from the viewpoint of logic, while the material we wanted to cover in the school is primarily algorithmic, viewing complexity as a tool for proving that certain kinds of algorithms do not exist. Bounded nondeterminism and alternation in parameterized complexity theory yijia chen. Parameterized complexity theory is a fairly new branch of complexity the ory. Beyond that, at the core of the multivariate algorithmics project, there is a richer vision of the workflow between theory and practice. I feel that this book covers the basics in more detail than the famous book by flum and grohe. Invitation to fixed parameter algorithms parameterized complexity theory. We propose to consider parameterized modelchecking problems for various fragments of firstorder logic as generic parameterized problems and show how this approach can be useful in studying both fixedparameter tractability and intractability. The name of the book is fundamentals of parameterized complexity. In parameterized complexity theory, it is often possible to prove that a kernel with guaranteed bounds on the size of a kernel as a function of some parameter associated to the problem can be found in polynomial time. Parameterized complexity theory is a recent branch of computational complexity. Instead of studying the complexity just as a function of the input length n or some other size measure, parameterized complexity considers an additional parameter k. We describe parameterized complexity classes by means of classical complexity theory and descriptive complexity theory.
Joerg flum freiburg, germany, parameterized complexity julyaugust 2008. Parameterized complexity theory is a recent branch of computational complexity theory that provides a framework for a refined analysis of hard algorithmic problems. Constraint satisfaction with succinctly specified relations. We develop a basic complexity theory for the parameterized analogues of classical complexity classes and give, among other things, complete problems. Siam journal on computing society for industrial and. Siam journal on computing siam society for industrial and. Martin grohe january 22, 2004 abstract we develop a parameterized complexity theory for counting problems. Parameterized complexity results for symbolic model.
Machinebased methods in parameterized complexity theory yijia chen a. In this article, we study parameterized complexity theory from the perspective of logic, or more specifically, descriptive complexity theory. Flum and grohe, 2006 has received considerable attention in recent years, for both theoretical research and practical computation. Parameterized complexity theory texts in theoretical computer science. Theorem 12 seese 20, courcelle 3, frick and grohe 12, flum and grohe 10. Parameterized complexity and the understanding, design. Describing parameterized complexity classes sciencedirect. Parameterized complexity theory is a recent branch of computational complexity theory that provides a. Furthermore, the vertex cover problem is fixedparameter tractable and a central problem in parameterized complexity theory. For instance, ltl model checking parameterized by the size of the logic formula features as a textbook example for. Here we obtain a new and nontrivial characterization of the ahierarchy in terms of oracle machines, and parameterize a famous result of. Since the early pioneering work of downey, fellows, and other researchers this area has established plenty of. Relativization and interactive proof systems in parameterized.
The complexity of homomorphism and constraint satisfaction problems seen from the other side. However, it was not until about 10 years ago, flum and grohe 12, and independently mccartin , formalized the intractable theory of counting problems in parameterized complexity. Parameterized complexity for the database theorist martin grohe. Pdf an isomorphism between subexponential and parameterized. A parameterized complexity tutorial isaac newton institute.
Grohe is primarily about the theory of parameterized complexity, meaning by this the mathematical techniques and parameterized complexit y classes analogous to npcompleteness that can be used. In proceedings of the 1st international colloquium on structural information. The running time is then measured as a function of n and k with the understanding that an exponential or even worse dependence. Proceedings of the 18th ieee conference on computational complexity. First, we study relativization for the machine models that were used by chen, flum, and grohe 2005 to characterize a number of parameterized complexity classes. Machinebased methods in parameterized complexity theory. In particular, the analogue of polynomial time is the class of all fixedparameter tractable problems. The core idea of parameterized complexity is to single out a speci c part of the input as the parameter and ask. The complexity of firstorder and monadic secondorder logic revisited. This book provides an introduction to both algorithmic techniques for fixedparameter tractability and the structural theory of parameterized complexity classes. The complexity of a problem is then measured as a function of those parameters. This allows the classification of nphard problems on a finer scale than.
Request pdf parameterized complexity theory parameterized complexity theory is a recent branch of computational. The central innovation of the parameterized approach to algorithms and complexity is, of course, simply the explicitly multivariate view of complexity analysis and algorithm design. In contrast, the book by flum and grohe is primarily about the theory of parameterized complexity, meaning by this the mathematical techniques and parameterized complexity classes analogous to npcompleteness that can be used to establish evidence that problems are intractable in the parameterized framework, and to classify them with regards. Its decision version, the vertex cover problem, was one of karps 21 npcomplete problems and is therefore a classical npcomplete problem in computational complexity theory. Fundamentals of parameterized complexity,springer, 20 3 r. The parameterized complexity of counting problems siam. In contrast, the book by flum and grohe is primarily about the theory of parameterized complexity, meaning by this the mathematical techniques and parameterized complexity classes analogous to npcompleteness that can be used to establish evidence that problems are. The complexity of homomorphism and constraint satisfaction. For the temporal logic ctl, parameterized complexity has also been used to study the problems of model checking and. The idea was that to try to devise a complexity theory more attuned to the. The central notion of the theory, fixed parameter tractability, has led to the development of various new algorithmic techniques and a whole new theory of intractability. Kernelization is often achieved by applying a set of reduction rules that cut away parts of the instance that are easy to handle. Parameterized complexity classes under logical reductions. We establish a dichotomy theorem that splits the exponential case of riiss complexity gap theorem into two subcases, one that admits proofs of size fkno1 and one that does not.
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