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Vertex Cover: Further Observations and Further Improvements
- Journal of Algorithms
, 1999
"... Recently, there have been increasing interests and progresses in lowering the worst case time complexity for well-known NP-hard problems, in particular for the Vertex Cover problem. In this paper, new properties for the Vertex Cover problem are indicated and several simple and new techniques are int ..."
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Cited by 186 (19 self)
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Recently, there have been increasing interests and progresses in lowering the worst case time complexity for well-known NP-hard problems, in particular for the Vertex Cover problem. In this paper, new properties for the Vertex Cover problem are indicated and several simple and new techniques are introduced, which lead to an improved algorithm of time O(kn + 1:271 k k 2 ) for the problem. Our algorithm also induces improvement on previous algorithms for the Independent Set problem on graphs of small degree. 1 Introduction Many optimization problems from industrial applications are NP-hard. According to the NPcompleteness theory [10], these problems cannot be solved in polynomial time unless P = NP. However, this fact does not obviate the need for solving these problems for their practical importance. There has been a number of approaches to attacking the NP-hardness of optimization problems, including approximation algorithms, heuristic algorithms, and average time analysis. Recent...
Conjunctive-Query Containment and Constraint Satisfaction
- Journal of Computer and System Sciences
, 1998
"... Conjunctive-query containment is recognized as a fundamental problem in database query evaluation and optimization. At the same time, constraint satisfaction is recognized as a fundamental problem in artificial intelligence. What do conjunctive-query containment and constraint satisfaction have in c ..."
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Cited by 164 (14 self)
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Conjunctive-query containment is recognized as a fundamental problem in database query evaluation and optimization. At the same time, constraint satisfaction is recognized as a fundamental problem in artificial intelligence. What do conjunctive-query containment and constraint satisfaction have in common? Our main conceptual contribution in this paper is to point out that, despite their very different formulation, conjunctive-query containment and constraint satisfaction are essentially the same problem. The reason is that they can be recast as the following fundamental algebraic problem: given two finite relational structures A and B, is there a homomorphism h : A ! B? As formulated above, the homomorphism problem is uniform in the sense that both relational structures A and B are part of the input. By fixing the structure B, one obtains the following non-uniform problem: given a finite relational structure A, is there a homomorphism h : A ! B? In general, non-uniform tractability results do not uniformize. Thus, it is natural to ask: which tractable cases of non-uniform tractability results for constraint satisfaction and conjunctive-query containment do uniformize? Our main technical contribution in this paper is to show that several cases of tractable non-uniform constraint satisfaction problems do indeed uniformize. We exhibit three non-uniform tractability results that uniformize and, thus, give rise to polynomial-time solvable cases of constraint satisfaction and conjunctive-query containment.
Exact algorithms for NP-hard problems: a survey, in: Combinatorial Optimization—Eureka, you shrink!”,
- Lecture Notes in Computer Science,
, 2003
"... Abstract. We discuss fast exponential time solutions for NP-complete problems. We survey known results and approaches, we provide pointers to the literature, and we discuss several open problems in this area. The list of discussed NP-complete problems includes the travelling salesman problem, sched ..."
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Cited by 151 (4 self)
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Abstract. We discuss fast exponential time solutions for NP-complete problems. We survey known results and approaches, we provide pointers to the literature, and we discuss several open problems in this area. The list of discussed NP-complete problems includes the travelling salesman problem, scheduling under precedence constraints, satisfiability, knapsack, graph coloring, independent sets in graphs, bandwidth of a graph, and many more.
On problems without polynomial kernels
- LECT. NOTES COMPUT. SCI
, 2007
"... Kernelization is a strong and widely-applied technique in parameterized complexity. In a nutshell, a kernelization algorithm, or simply a kernel, is a polynomial-time transformation that transforms any given parameterized instance to an equivalent instance of the same problem, with size and parame ..."
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Cited by 143 (17 self)
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Kernelization is a strong and widely-applied technique in parameterized complexity. In a nutshell, a kernelization algorithm, or simply a kernel, is a polynomial-time transformation that transforms any given parameterized instance to an equivalent instance of the same problem, with size and parameter bounded by a function of the parameter in the input. A kernel is polynomial if the size and parameter of the output are polynomially-bounded by the parameter of the input. In this paper we develop a framework which allows showing that a wide range of FPT problems do not have polynomial kernels. Our evidence relies on hypothesis made in the classical world (i.e. nonparametric complexity), and evolves around a new type of algorithm for classical decision problems, called a distillation algorithm, which might be of independent interest. Using the notion of distillation algorithms, we develop a generic lower-bound engine which allows us to show that a variety of FPT problems, fulfilling certain criteria, cannot have polynomial kernels unless the polynomial hierarchy collapses. These problems include k-Path, k-Cycle, k-Exact Cycle, k-Short Cheap Tour, k-Graph Minor Order Test, k-Cutwidth, k-Search Number, k-Pathwidth, k-Treewidth, k-Branchwidth, and several optimization problems parameterized by treewidth or cliquewidth.
Subtree Transfer Operations and their Induced Metrics on Evolutionary Trees
- ANNALS OF COMBINATORICS
, 2000
"... Leaf-labelled trees are widely used to describe evolutionary relationships, particularly in biology. In this setting, extant species label the leaves of the tree, while the internal vertices correspond to ancestral species. Various techniques exist for reconstructing these evolutionary trees from da ..."
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Cited by 119 (2 self)
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Leaf-labelled trees are widely used to describe evolutionary relationships, particularly in biology. In this setting, extant species label the leaves of the tree, while the internal vertices correspond to ancestral species. Various techniques exist for reconstructing these evolutionary trees from data, and an important problem is to determine how "far apart" two such reconstructed trees are from each other, or indeed from the true historical tree. To investigate this question requires tree metrics, and these can be induced by operations that rearrange trees locally. Here we investigate three such operations: nearest neighbour interchange (NNI), subtree prune and regraft (SPR), and tree bisection and reconnection (TBR). The SPR operation is of particular interest as it can be used to model biological processes such as horizontal gene transfer and recombination. We count the number of unrooted binary trees one SPR from any given unrooted binary tree, as well as providing new upper and lower bounds for the diameter of the adjacency graph of trees under SPR and TBR. We also show that the problem of computing the minimum number of TBR operations required to transform one tree to another can be reduced to a problem whose size is a function just of the distance between the trees (and not of the size of the two trees), and thereby establish that the problem is fixed-parameter tractable.
A Survey on Knowledge Compilation
, 1998
"... this paper we survey recent results in knowledge compilation of propositional knowledge bases. We first define and limit the scope of such a technique, then we survey exact and approximate knowledge compilation methods. We include a discussion of compilation for non-monotonic knowledge bases. Keywor ..."
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Cited by 119 (4 self)
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this paper we survey recent results in knowledge compilation of propositional knowledge bases. We first define and limit the scope of such a technique, then we survey exact and approximate knowledge compilation methods. We include a discussion of compilation for non-monotonic knowledge bases. Keywords: Knowledge Representation, Efficiency of Reasoning
Deciding First-Order Properties of Locally Tree-Decomposable Graphs
- In Proc. 26th ICALP
, 1999
"... . We introduce the concept of a class of graphs being locally tree-decomposable. There are numerous examples of locally treedecomposable classes, among them the class of planar graphs and all classes of bounded valence or of bounded tree-width. We show that for each locally tree-decomposable cl ..."
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Cited by 98 (14 self)
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. We introduce the concept of a class of graphs being locally tree-decomposable. There are numerous examples of locally treedecomposable classes, among them the class of planar graphs and all classes of bounded valence or of bounded tree-width. We show that for each locally tree-decomposable class C of graphs and for each property ' of graphs that is denable in rst-order logic, there is a linear time algorithm deciding whether a given graph G 2 C has property '. 1 Introduction It is an important task in the theory of algorithms to nd feasible instances of otherwise intractable algorithmic problems. A notion that has turned out to be extremely useful in this context is that of tree-width of a graph. 3-Colorability, Hamiltonicity, and many other NP-complete properties of graphs can be decided in linear time when restricted to graphs whose tree-width is bounded by a xed constant (see [Bod97] for a survey). Courcelle [Cou90] proved a meta-theorem, which easily implies numer...
The complexity of first-order and monadic second-order logic revisited
- ANNALS OF PURE AND APPLIED LOGIC
, 2004
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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 fixed-parameter tractability techniques can deliver practical algorithms in two different ways: (1) by ..."
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Cited by 87 (16 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 fixed-parameter 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 k-Step 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 k-Coloring 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.
