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17
Beyond NPCompleteness for Problems of Bounded Width: Hardness for the W Hierarchy (Extended Abstract)
 In Proceedings of the 26th Annual ACM Symposium on the Theory of Computing
, 1994
"... The parameterized computational complexity of a collection of wellknown problems including: Bandwidth, Precedence constrained kprocessor scheduling, Longest Common Subsequence, DNA physical mapping (or Intervalizing colored graphs), Perfect phylogeny (or Triangulating colored graphs), Colored cutw ..."
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Cited by 56 (21 self)
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The parameterized computational complexity of a collection of wellknown problems including: Bandwidth, Precedence constrained kprocessor scheduling, Longest Common Subsequence, DNA physical mapping (or Intervalizing colored graphs), Perfect phylogeny (or Triangulating colored graphs), Colored cutwidth, and Feasible register assignment is explored. It is shown that these problems are hard for various levels of the W hierarchy. In the case of Precedence constrained kprocessor scheduling the results can be interpreted as providing substantial new complexity lower bounds on the outcome of [OPEN 8] of the Garey and Johnson list. We also obtain the conjectured "third strike" against Perfect phylogeny.
Four Strikes against Physical Mapping of DNA
 JOURNAL OF COMPUTATIONAL BIOLOGY
, 1993
"... Physical Mapping is a central problem in molecular biology ... and the human genome project. The problem is to reconstruct the relative position of fragments of DNA along the genome from information on their pairwise overlaps. We show that four simplified models of the problem lead to NPcomplete ..."
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Cited by 55 (8 self)
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Physical Mapping is a central problem in molecular biology ... and the human genome project. The problem is to reconstruct the relative position of fragments of DNA along the genome from information on their pairwise overlaps. We show that four simplified models of the problem lead to NPcomplete decision problems: Colored unit interval graph completion, the maximum interval (or unit interval) subgraph, the pathwidth of a bipartite graph, and the kconsecutive ones problem for k >= 2. These models have been chosen to reflect various features typical in biological data, including false negative and positive errors, small width of the map and chimericism.
Tractability of Parameterized Completion Problems on Chordal, Strongly Chordal and Proper Interval Graphs
, 1994
"... We study the parameterized complexity of three NPhard graph completion problems. The MINIMUM FILLIN problem is to decide if a graph can be triangulated by adding at most k edges. We develop O(c m) and O(k mn + f(k)) algorithms for this problem on a graph with n vertices and m edges. Here f(k ..."
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Cited by 40 (5 self)
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We study the parameterized complexity of three NPhard graph completion problems. The MINIMUM FILLIN problem is to decide if a graph can be triangulated by adding at most k edges. We develop O(c m) and O(k mn + f(k)) algorithms for this problem on a graph with n vertices and m edges. Here f(k) is exponential in k and the constants hidden by the bigO notation are small and do not depend on k. In particular, this implies that the problem is fixedparameter tractable (FPT). The PROPER
Pathwidth, Bandwidth and Completion Problems to Proper Interval Graphs with Small Cliques
 SIAM Journal on Computing
, 1996
"... We study two related problems motivated by molecular biology: ffl Given a graph G and a constant k, does there exist a supergraph G of G which is a unit interval graph and has clique size at most k? ffl Given a graph G and a proper kcoloring c of G, does there exist a supergraph We show th ..."
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Cited by 29 (6 self)
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We study two related problems motivated by molecular biology: ffl Given a graph G and a constant k, does there exist a supergraph G of G which is a unit interval graph and has clique size at most k? ffl Given a graph G and a proper kcoloring c of G, does there exist a supergraph We show that those problems are polynomial for fixed k. On the other hand we prove that the first problem is equivalent to deciding if the bandwidth of G is at most k \Gamma 1. Hence, it is NPhard, and W [t]hard for all t. We also show that the second problem is W [1]hard. This implies that for fixed k, both of the problems are unlikely to have an O(n ) algorithm, where ff is a constant independent of k.
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 28 (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...
The Parameterized Complexity of Relational Database Queries and an Improved Characterization of W [1
 Combinatorics, Complexity, and Logic – Proceedings of DMTCS ’96
, 1996
"... Abstract. It is well known that for a fixed relational database query φ in m free variables, it can be determined in time polynomial in the size n of the database whether there exists an mtuple x that belongs to the relation defined by the query. For the best known algorithms, however, the exponent ..."
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Cited by 25 (8 self)
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Abstract. It is well known that for a fixed relational database query φ in m free variables, it can be determined in time polynomial in the size n of the database whether there exists an mtuple x that belongs to the relation defined by the query. For the best known algorithms, however, the exponent of the polynomial is proportional to the size of the query. We study the data complexity of this problem parameterized by the size k = φ  of the query, and answer a question recently raised by Yannakakis [Yan95]. Our main results show: (1) the general problem is complete for the parametric complexity class AW[∗], and (2) when restricted to monotone queries, the problem is complete for the fundamental parametric complexity class W[1]. The practical significance of these results is that unless the parameterized complexity hierarchy collapses, there are unlikely to be algorithms that solve this problem (even under the restriction to monotone queries) in time f(k)n c where f is an arbitrary function of k and c is a constant independent of k. An important consequence of the proof of (2) is a significantly improved characterization of the parameterized complexity class W[1]. Previous results by Downey and Fellows characterize W[1] in terms of the kWeighted Circuit Satisfiability problem, for families of circuits that satisfy: (1) the depth of the circuits is bounded by a constant c, (2) on any inputoutput path there is at most one gate having unbounded fanin (termed a large gate), with all other gates having fanin bounded by c (that is, small gates). We show that the definition can be broadened by allowing circuits of depth bounded by an arbitrary function f(k). If we denote this parameterized complexity class W ∗ [1], then our corollary
Sharp Tractability Borderlines for Finding Connected Motifs in VertexColored Graphs
 34TH INTERNATIONAL COLLOQUIUM ON AUTOMATA, LANGUAGES AND PROGRAMMING (ICALP 2007), WROCLAW: POLAND
, 2007
"... We study the problem of finding occurrences of motifs in vertexcolored graphs, where a motif is a multiset of colors, and an occurrence of a motif is a subset of connected vertices with a bijection between its colors and the colors of the motif. This problem has applications in metabolic network an ..."
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Cited by 17 (8 self)
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We study the problem of finding occurrences of motifs in vertexcolored graphs, where a motif is a multiset of colors, and an occurrence of a motif is a subset of connected vertices with a bijection between its colors and the colors of the motif. This problem has applications in metabolic network analysis, an important area in bioinformatics. We give two positive results and three negative results that together draw sharp borderlines between tractable and intractable instances of the problem.
Parameterized Complexity Analysis in Computational Biology
 Comput. Appl. Biosci
, 1995
"... Many computational problems in biology involve parameters for which a small range of values cover important applications. We argue that for many problems in this setting, parameterized computational complexity rather than NPcompleteness is the appropriate tool for studying apparent intractability. ..."
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Cited by 9 (4 self)
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Many computational problems in biology involve parameters for which a small range of values cover important applications. We argue that for many problems in this setting, parameterized computational complexity rather than NPcompleteness is the appropriate tool for studying apparent intractability. At issue in the theory of parameterized complexity is whether a problem can be solved in time O(n ff ) for each fixed parameter value, where ff is a constant independent of the parameter. In addition to surveying this complexity framework, we describe a new result for the Longest common subsequence problem. In particular, we show that the problem is hard for W [t] for all t when parameterized by the number of strings and the size of the alphabet. Lower bounds on the complexity of this basic combinatorial problem imply lower bounds on more general sequence alignment and consensus discovery problems. We also describe a number of open problems pertaining to the parameterized complexity of pro...
On Physical Mapping and the Consecutive Ones Property for Sparse Matrices
 Discrete Appl. Math
, 1996
"... this paper we give a simplified model for Physical Mapping with probes that tend to occur very rarely along the DNA and show that the problem is NPcomplete even for sparse matrices. Moreover, we show that Physical Mapping with chimeric clones (a clone is chimeric if it stems from a concatenatio ..."
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Cited by 8 (0 self)
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this paper we give a simplified model for Physical Mapping with probes that tend to occur very rarely along the DNA and show that the problem is NPcomplete even for sparse matrices. Moreover, we show that Physical Mapping with chimeric clones (a clone is chimeric if it stems from a concatenation of several fragments of the DNA) is NPcomplete even for sparse matrices. Both problems are modeled as variants of the Consecutive Ones Problem which makes our results interesting for other application areas. 1 Supported by the Applied Mathematical Sciences program, U.S. Dept. of Energy, Office of Energy Research, and the work was performed at Sandia National Labs, operated for the U.S. DOE under contract No. DEAC0476DP00789. Preprint submitted to Elsevier Preprint 19 January 1996 1 Introduction In order to study a long DNA molecule it is necessary to break several copies of the molecule into smaller fragments. For further investigation copies
On Computing Graph Minor Obstruction Sets
 THEORETICAL COMPUTER SCIENCE A
, 1997
"... The Graph Minor Theorem of Robertson and Seymour establishes nonconstructively that many natural graph properties are characterized by a finite set of forbidden substructures, the obstructions for the property. We prove several general theorems regarding the computation of obstruction sets from o ..."
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Cited by 7 (4 self)
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The Graph Minor Theorem of Robertson and Seymour establishes nonconstructively that many natural graph properties are characterized by a finite set of forbidden substructures, the obstructions for the property. We prove several general theorems regarding the computation of obstruction sets from other information about a family of graphs. The methods can be adapted to other partial orders on graphs, such as the immersion and topological orders. The algorithms are in some cases practical and have been implemented. Two new technical ideas are introduced. The first is a method of computing a stopping signal for search spaces of increasing pathwidth. This allows obstruction sets to be computed without the necessity of a prior bound on maximum obstruction width. The second idea is that of a second order congruence for a graph property. This is an equivalence relation defined on finite sets of graphs that generalizes the recognizability congruence that is defined on single graphs. I...