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18
Fixedparameter tractability and completeness II: On completeness for W[1]
, 1995
"... For many fixedparameter problems that are trivially solvable in polynomialtime, such as kDOMINATING SET, essentially no better algorithm is presently known than the one which tries all possible solutions. Other problems, such as FEEDBACK VERTEX SET, exhibit fixedparameter tractability: for each ..."
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Cited by 89 (9 self)
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For many fixedparameter problems that are trivially solvable in polynomialtime, such as kDOMINATING SET, essentially no better algorithm is presently known than the one which tries all possible solutions. Other problems, such as FEEDBACK VERTEX SET, exhibit fixedparameter tractability: for each fixed k the problem is solvable in time bounded by a polynomial of degree c, where c is a constant independent of k. In a previous paper, the W Hierarchy of parameterixed problems was defined, and complete problems were identified for the classes W[t] for t >= 2. Our main result shows that INDEPENDENT SET is complete for W[1].
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.
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.
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.
On Limited versus Polynomial Nondeterminism
, 1997
"... In this paper, we show that efficient algorithms for some problems that require limited nondeterminism imply the subexponential simulation of nondeterministic computation by deterministic computation. In particular, if cliques of size O(log n) can be found in polynomial time, then nondeterministic t ..."
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Cited by 26 (1 self)
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In this paper, we show that efficient algorithms for some problems that require limited nondeterminism imply the subexponential simulation of nondeterministic computation by deterministic computation. In particular, if cliques of size O(log n) can be found in polynomial time, then nondeterministic time f(n) is contained in deterministic time 2 O( p f(n)polylogf(n)) .
On The Amount Of Nondeterminism And The Power Of Verifying
 SIAM Journal on Computing
, 1997
"... . The relationship between nondeterminism and other computational resources is investigated based on the "guessthencheck" model GC. Systematic techniques are developed to construct natural complete languages for the classes defined by this model. This improves a number of previous results in the s ..."
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Cited by 23 (5 self)
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. The relationship between nondeterminism and other computational resources is investigated based on the "guessthencheck" model GC. Systematic techniques are developed to construct natural complete languages for the classes defined by this model. This improves a number of previous results in the study of limited nondeterminism. Connections of the model GC to computational optimization problems are exhibited. Key words. computational complexity, nondeterminism, complete languages, computational optimization AMS subject classifications. 68Q05, 68Q10, 68Q15, 68Q25 PII. S0097539793258295 1. Introduction. The study of the power of nondeterminism is central to complexity theory. The relationship between nondeterminism and other computational resources still remains unclear. Two fundamental questions are those of how much computational resource we should pay in order to eliminate nondeterminism and how much computational resource we can save if we are granted nondeterminism. A computation ...
A Duality between Clause Width and Clause Density for SAT
 In IEEE Conference on Computational Complexity (CCC
"... We consider the relationship between the complexities of and those of restricted to formulas of constant density. Let be the infimum of those such that on variables can be decided in time and be the infimum of those such that on variables and clauses can be decided in time. We show that. So, for a ..."
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Cited by 22 (3 self)
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We consider the relationship between the complexities of and those of restricted to formulas of constant density. Let be the infimum of those such that on variables can be decided in time and be the infimum of those such that on variables and clauses can be decided in time. We show that. So, for any, can be solved in time independent of if and only if the same is true for with any fixed density of clauses to variables. We derive some interesting consequences from this. For example, assuming thatis exponentially hard (that is,), of any fixed density can be solved in time whose exponent is strictly less than that for general. We also give an improvement to the sparsification lemma of [12] showing that instances of of density slightly more than exponential in are almost the hardest instances of. The previous result showed this for densities doubly exponential in. 1.
Fixedparameter approximation: Conceptual framework and approximability results
 Parameterized and Exact Computation, Second International Workshop, IWPEC 2006, volume 4169 of Lecture Notes in Computer Science
, 2006
"... Abstract. The notion of fixedparameter approximation is introduced to investigate the approximability of optimization problems within the framework of fixedparameter computation. This work partially aims at enhancing the world of fixedparameter computation in parallel with the conventional theory ..."
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Cited by 21 (0 self)
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Abstract. The notion of fixedparameter approximation is introduced to investigate the approximability of optimization problems within the framework of fixedparameter computation. This work partially aims at enhancing the world of fixedparameter computation in parallel with the conventional theory of computation that includes both exact and approximate computations. In particular, it is proved that fixedparameter approximability is closely related to the approximation of smallcost solutions in polynomial time. It is also demonstrated that many fixedparameter intractable problems are not fixedparameter approximable. On the other hand, fixedparameter approximation appears to be a viable approach to solving some inapproximable yet important optimization problems. For instance, all problems in the class MAX SNP admit fixedparameter approximation schemes in time O(2 O((1−ɛ/O(1))k) p(n)) for any small ɛ>0. 1
Colorcoding: a new method for finding simple paths, cycles and other small subgraphs within large graphs (Extended Abstract)
"... We describe a novel randomized method, the method of colorcoding for finding simple paths and cycles of a specified length k, and other small subgraphs, within a given graph G = (V, E). The randomized algorithms obtained using this method can be derandomized using families of perfect hash functions ..."
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Cited by 20 (1 self)
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We describe a novel randomized method, the method of colorcoding for finding simple paths and cycles of a specified length k, and other small subgraphs, within a given graph G = (V, E). The randomized algorithms obtained using this method can be derandomized using families of perfect hash functions. Using the colorcoding method we obtain, among others, the following new results: • For every fixed k, if a graph G = (V, E) contains a simple cycle of size exactly k, then such a cycle can be found in either O(V ω) expected time or O(V ω log V) worstcase time, where ω < 2.376 is the exponent of matrix multiplication. (Here and in what follows we use V and E instead of V  and E  whenever no confusion may arise.) • For every fixed k, if a planar graph G = (V, E) contains a simple cycle of size exactly k, then