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36
Parameterized Complexity: A Framework for Systematically Confronting Computational Intractability
 DIMACS Series in Discrete Mathematics and Theoretical Computer Science
, 1997
"... In this paper we give a programmatic overview of parameterized computational complexity in the broad context of the problem of coping with computational intractability. We give some examples of how fixedparameter tractability techniques can deliver practical algorithms in two different ways: (1) by ..."
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Cited by 72 (15 self)
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In this paper we give a programmatic overview of parameterized computational complexity in the broad context of the problem of coping with computational intractability. We give some examples of how fixedparameter tractability techniques can deliver practical algorithms in two different ways: (1) by providing useful exact algorithms for small parameter ranges, and (2) by providing guidance in the design of heuristic algorithms. In particular, we describe an improved FPT kernelization algorithm for Vertex Cover, a practical FPT algorithm for the Maximum Agreement Subtree (MAST) problem parameterized by the number of species to be deleted, and new general heuristics for these problems based on FPT techniques. In the course of making this overview, we also investigate some structural and hardness issues. We prove that an important naturally parameterized problem in artificial intelligence, STRIPS Planning (where the parameter is the size of the plan) is complete for W [1]. As a corollary, this implies that kStep Reachability for Petri Nets is complete for W [1]. We describe how the concept of treewidth can be applied to STRIPS Planning and other problems of logic to obtain FPT results. We describe a surprising structural result concerning the top end of the parameterized complexity hierarchy: the naturally parameterized Graph kColoring problem cannot be resolved with respect to XP either by showing membership in XP, or by showing hardness for XP without settling the P = NP question one way or the other.
On the parameterized complexity of short computation and factorization
 Archive for Mathematical Logic
, 1997
"... A completeness theory for parameterized computational complexity has been studied in a series of recent papers, and has been shown to have many applications in diverse problem domains including familiar graphtheoretic problems, VLSI layout, games, computational biology, cryptography, and computatio ..."
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Cited by 42 (20 self)
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A completeness theory for parameterized computational complexity has been studied in a series of recent papers, and has been shown to have many applications in diverse problem domains including familiar graphtheoretic problems, VLSI layout, games, computational biology, cryptography, and computational learning [ADF,DEF,DF17,FH,FHW,FK]. We here study the parameterized complexity of two kinds of problems: (1) problems concerning parameterized computations of Turing machines, such as determining whether a nondeterministic machine can reach an accept state in k steps (the Short TM Computation Problem), and (2) problems concerning derivations and factorizations, such as determining whether a word x can be derived in a grammar G in k steps, or whether a permutation has a factorization of length k over a given set of generators. These include a natural parameterized version of the famous Post Correspondence Systems. We show hardness and completeness for these problems for various levels of the W hierarchy. In particular, we show that Short TM Computation is complete for W [1]. This gives a new and useful characterization of the most important of the apparently intractable parameterized complexity classes. The result could be viewed as one analogue of Cook’s theorem and we believe provides strong evidence for the parameterized intractability of W [1]. 1.
New Upper Bounds for Maximum Satisfiability
 Journal of Algorithms
, 1999
"... The (unweighted) Maximum Satisfiability problem (MaxSat) is: given a boolean formula in conjunctive normal form, find a truth assignment that satisfies the most number of clauses. This paper describes exact algorithms that provide new upper bounds for MaxSat. We prove that MaxSat can be solved i ..."
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Cited by 36 (2 self)
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The (unweighted) Maximum Satisfiability problem (MaxSat) is: given a boolean formula in conjunctive normal form, find a truth assignment that satisfies the most number of clauses. This paper describes exact algorithms that provide new upper bounds for MaxSat. We prove that MaxSat can be solved in time O(F  1.3803 K ), where F  is the length of a formula F in conjunctive normal form and K is the number of clauses in F . We also prove the time bounds O(F 1.3995 k ), where k is the maximum number of satisfiable clauses, and O(1.1279 F  ) for the same problem. For Max2Sat this implies a bound of O(1.2722 K ). # An extended abstract of this paper was presented at the 26th International Colloquium on Automata, Languages, and Programming (ICALP'99), LNCS 1644, SpringerVerlag, pages 575584, held in Prague, Czech Republic, July 1115, 1999. + Supported by a Feodor Lynen fellowship (1998) of the Alexander von HumboldtStiftung, Bonn, and the Center for Discrete Ma...
Subexponential Parameterized Algorithms Collapse the Whierarchy (Extended Abstract)
, 2001
"... It is shown that for essentially all MAX SNPhard optimization problems finding exact solutions in subexponential time is not possible unless W [1] = FPT . In particular, we show that O(2 o(k) p(n)) parameterized algorithms do not exist for Vertex Cover, Max Cut, Max cSat, and a number of pr ..."
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Cited by 36 (2 self)
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It is shown that for essentially all MAX SNPhard optimization problems finding exact solutions in subexponential time is not possible unless W [1] = FPT . In particular, we show that O(2 o(k) p(n)) parameterized algorithms do not exist for Vertex Cover, Max Cut, Max cSat, and a number of problems on bounded degree graphs such as Dominating Set and Independent Set, unless W [1] = FPT . Our results are derived via an approach that uses an extended parameterization of optimization problems and associated techniques to relate the parameterized complexity of problems in FPT to the parameterized complexity of extended versions that are W [1]hard.
Bidimensionality: New Connections between FPT Algorithms and PTASs
"... We demonstrate a new connection between fixedparameter tractability and approximation algorithms for combinatorial optimization problems on planar graphs and their generalizations. Specifically, we extend the theory of socalled “bidimensional” problems to show that essentially all such problems ha ..."
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Cited by 36 (5 self)
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We demonstrate a new connection between fixedparameter tractability and approximation algorithms for combinatorial optimization problems on planar graphs and their generalizations. Specifically, we extend the theory of socalled “bidimensional” problems to show that essentially all such problems have both subexponential fixedparameter algorithms and PTASs. Bidimensional problems include e.g. feedback vertex set, vertex cover, minimum maximal matching, face cover, a series of vertexremoval problems, dominating set, edge dominating set, rdominating set, diameter, connected dominating set, connected edge dominating set, and connected rdominating set. We obtain PTASs for all of these problems in planar graphs and certain generalizations; of particular interest are our results for the two wellknown problems of connected dominating set and general feedback vertex set for planar graphs and their generalizations, for which PTASs were not known to exist. Our techniques generalize and in some sense unify the two main previous approaches for designing PTASs in planar graphs, namely, the LiptonTarjan separator approach [FOCS’77] and the Baker layerwise decomposition approach [FOCS’83]. In particular, we replace the notion of separators with a more powerful tool from the bidimensionality theory, enabling the first approach to apply to a much broader class of minimization problems than previously possible; and through the use of a structural backbone and thickening of layers we demonstrate how the second approach can be applied to problems with a “nonlocal” structure.
Parameterized Complexity for the Skeptic
 In Proc. 18th IEEE Annual Conference on Computational Complexity
, 2003
"... The goal of this article is to provide a tourist guide, with an eye towards structural issues, to what I consider some of the major highlights of parameterized complexity. ..."
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Cited by 36 (1 self)
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The goal of this article is to provide a tourist guide, with an eye towards structural issues, to what I consider some of the major highlights of parameterized complexity.
The parameterized complexity of counting problems
 SIAM Journal on Computing
, 2002
"... We develop a parameterized complexity theory for counting problems. As the basis of this theory, we introduce a hierarchy of parameterized counting complexity classes #W[t], for t ≥ 1, that corresponds to Downey and Fellows’s Whierarchy [13] and show that a few central Wcompleteness results for de ..."
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Cited by 32 (0 self)
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We develop a parameterized complexity theory for counting problems. As the basis of this theory, we introduce a hierarchy of parameterized counting complexity classes #W[t], for t ≥ 1, that corresponds to Downey and Fellows’s Whierarchy [13] and show that a few central Wcompleteness results for decision problems translate to #Wcompleteness results for the corresponding counting problems. Counting complexity gets interesting with problems whose decision version is tractable, but whose counting version is hard. Our main result states that counting cycles and paths of length k in both directed and undirected graphs, parameterized by k, is #W[1]complete. This makes it highly unlikely that any of these problems is fixedparameter tractable, even though their decision versions are fixedparameter tractable. More explicitly, our result shows that most likely there is no f(k) · n calgorithm for counting cycles or paths of length k in a graph of size n for any computable function f: N → N and constant c, even though there is a 2 O(k) · n 2.376 algorithm for finding a cycle or path of length k [2]. 1
Fixedparameter tractability, definability, and model checking
 SIAM Journal on Computing
, 2001
"... In this article, we study parameterized complexity theory from the perspective of logic, or more specifically, descriptive complexity theory. We propose to consider parameterized modelchecking problems for various fragments of firstorder logic as generic parameterized problems and show how this ap ..."
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Cited by 30 (12 self)
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In this article, we study parameterized complexity theory from the perspective of logic, or more specifically, descriptive 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. For example, we establish the equivalence between the modelchecking for existential firstorder logic, the homomorphism problem for relational structures, and the substructure isomorphism problem. Our main tractability result shows that modelchecking for firstorder formulas is fixedparameter tractable when restricted to a class of input structures with an excluded minor. On the intractability side, for everyØ�we prove an equivalence between modelchecking for firstorder formulas withØquantifier alternations and the parameterized halting problem for alternating Turing machines withØalternations. We discuss the close connection between this alternation hierarchy and Downey and Fellows ’ Whierarchy. On a more abstract level, we consider two forms of definability, called Fagin definability and slicewise definability, that are appropriate for describing parameterized problems. We give a characterization of the class FPT of all fixedparameter tractable problems in terms of slicewise definability in finite variable least fixedpoint logic, which is reminiscent of the ImmermanVardi Theorem characterizing the class PTIME in terms of definability in least fixedpoint logic. 1
The Parameterized Complexity of Some Problems in Logic and Linguistics (Extended Abstract)
 Proceedings Symposium on Logical Foundations of Computer Science (LFCS), SpringerVerlag, Lecture Notes in Computer Science
, 2002
"... March 1, 2002 Rodney G. Downey Department of Mathematics, Victoria University P.O. Box 600, Wellington, New Zealand downey@math.vuw.ac.nz Michael R. Fellows, Bruce M. Kapron and Michael T. Hallett Department of Computer Science, University of Victoria Victoria, British Columbia V8W 3P6 Canada ..."
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Cited by 29 (19 self)
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March 1, 2002 Rodney G. Downey Department of Mathematics, Victoria University P.O. Box 600, Wellington, New Zealand downey@math.vuw.ac.nz Michael R. Fellows, Bruce M. Kapron and Michael T. Hallett Department of Computer Science, University of Victoria Victoria, British Columbia V8W 3P6 Canada contact author: mfellows@csr.uvic.ca H. Todd Wareham Department of Computer Science Memorial University of Newfoundland St. Johns, Newfoundland A1C 5S7 Canada harold@odie.cs.mun.ca Summary The theory of parameterized computational complexity introduced in [DF13] appears to be of wide applicability in the study of the complexity of concrete problems [ADF,BFH,DEF,FHW,FK]. We believe the theory may be of particular importance to practical applications of logic formalisms in programming language design and in system specification. The reason for this relevance is that while many computational problems in logic are extremely intractable generally, realistic applications often involve a "hidden parameter" according to which the computational problem may be feasible according to the more sensitive criteria of fixedparameter tractability that is the central issue in parameterized computational complexity. We illustrate how this theory may apply to problems in logic, programming languages and linguistics by describing some examples of both tractability and intractability results in these areas. It is our strong expectation that these results are just the tip of the iceberg of interesting applications of parameterized complexity theory to logic and linguistics. The main results described in this abstract are as follows. (1) The problem of determining whether a word x can be derived in k steps in a contextsensitive grammar G (Short CSL Derivation) is complete for the paramet...
On the efficiency of polynomial time approximation schemes
, 1997
"... A polynomial time approximation scheme (PTAS) for an optimization problem A is an algorithm that given in input an instance of A and E> 0 find;,; (1 + E)approximate solution in time that is polynomial for each fixed E. Typical running times are no(+) or 2” ’ n. While algorithms of the former kind t ..."
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Cited by 27 (0 self)
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A polynomial time approximation scheme (PTAS) for an optimization problem A is an algorithm that given in input an instance of A and E> 0 find;,; (1 + E)approximate solution in time that is polynomial for each fixed E. Typical running times are no(+) or 2” ’ n. While algorithms of the former kind tend to be impractical, the latter ones are more interesting. In several cases, the development of algorithms of the second type required considerably new, and sometimes harder, techniques. For some interesting problems, only n”(“E) approximation schemes are known. Under likely assumptions, we prove that for some problems (including natural ones) there cannot be approximation schemes running in time f ( l/n)nO(‘), no matter how fast function f grows. Our result relies on a connection with Parameterized Complexity Theory, and we show that this connection is necessary.