Results 1  10
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28
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 115 (11 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 68 (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 58 (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
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 48 (23 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.
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 37 (7 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.
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 32 (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.
Fixedparameter approximation: Conceptual framework and approximability results
, 2008
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On the amount of nondeterminism and the power of verifying
 SIAM J. Comput
, 1997
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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 28 (0 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)) .
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 22 (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