## Parametric duality and kernelization: lower bounds and upper bounds on kernel size (2005)

### Cached

### Download Links

- [facweb.cs.depaul.edu]
- [www.informatik.uni-trier.de]
- DBLP

### Other Repositories/Bibliography

Venue: | In Proc. 22nd STACS, volume 3404 of LNCS |

Citations: | 35 - 4 self |

### BibTeX

@INPROCEEDINGS{Chen05parametricduality,

author = {Jianer Chen and Henning Fernau and Iyad A. Kanj and Ge Xia},

title = {Parametric duality and kernelization: lower bounds and upper bounds on kernel size},

booktitle = {In Proc. 22nd STACS, volume 3404 of LNCS},

year = {2005},

pages = {269--280},

publisher = {Springer}

}

### Years of Citing Articles

### OpenURL

### Abstract

Abstract. Determining whether a parameterized problem is kernelizable and has a small kernel size has recently become one of the most interesting topics of research in the area of parameterized complexity and algorithms. Theoretically, it has been proved that a parameterized problem is kernelizable if and only if it is fixed-parameter tractable. Practically, applying a data reduction algorithm to reduce an instance of a parameterized problem to an equivalent smaller instance (i.e., a kernel) has led to very efficient algorithms and now goes hand-in-hand with the design of practical algorithms for solving NP-hard problems. Well-known examples of such parameterized problems include the vertex cover problem, which is kernelizable to a kernel of size bounded by 2k, and the planar dominating set problem, which is kernelizable to a kernel of size bounded by 335k. In this paper we develop new techniques to derive upper and lower bounds on the kernel size for certain parameterized problems. In terms of our lower bound results, we show, for example, that unless P = NP, planar vertex cover does not have a problem kernel of size smaller than 4k/3, and planar independent set and planar dominating set do not have kernels of size smaller than 2k. In terms of our upper bound results, we further reduce the upper bound on the kernel size for the planar dominating set problem to 67k, improving significantly the 335k previous upper bound given by Alber, Fellows, and Niedermeier [J. ACM, 51 (2004), pp. 363–384]. This latter result is obtained by introducing a new set of reduction and coloring rules, which allows the derivation of nice combinatorial properties in the kernelized graph leading to a tighter bound on the size of the kernel. The paper also shows how this improved upper bound yields a simple and competitive algorithm for the planar dominating set problem.

### Citations

868 | Parametrized Complexity
- Downey, Fellows
- 1999
(Show Context)
Citation Context ... is a pair (I, k), where the second component k is called the parameter. The language L(P ) is the set of all YES-instances of P . We say that the parameterized problem P is fixed-parameter tractable =-=[7]-=- if there is an algorithm that decides whether an input (I, k) is a member of L(P ) in time f(k)|I| c , where c is a fixed constant and f(k) is a recursive function independent of the input length |I|... |

307 |
Approximation algorithms for NP-complete problems on planar graphs
- Baker
- 1994
(Show Context)
Citation Context ...es in the family Fμ to split the graph into chunks, then to compute a minimum dominating set for the resulting chunks using the algorithm introduced in [33], which is a variation of Baker’s algorithm =-=[8]-=-. To do this, for each vertex v in the Fμ, we “guess” whether v is in the minimum dominating set for G or not (basically, what we mean by guessing is enumerating all sequences corresponding to the dif... |

155 |
A local-ratio theorem for approximating the weighted vertex cover problem
- Bar-Yehuda, Even
- 1985
(Show Context)
Citation Context ... in time O(n 3 ) we can construct a graph G ′ from G such that: (1) G ′ is reduced, (2) γ(G ′ ) = γ(G), (3) there exists a minimum dominating set for G ′ that excludes all white vertices of G ′ , and =-=(4)-=- from a minimum dominating set for G ′ a minimum dominating set for G can be constructed in linear time. 5 A problem kernel for planar dominating set Let G be a reduced graph, and let D be a minimum d... |

151 | Vertex cover: further observations and further improvements
- Chen, Kanj, et al.
- 2001
(Show Context)
Citation Context ...hat improved kernelization can lead to improved parameterized algorithms. Many efforts havesbeen made towards obtaining smaller kernels for well-known N P-hard parameterized problems (see for example =-=[1, 5, 8]-=-). A natural question to ask along this line of research, is about the limit of polynomial time kernelization. In this section we develop techniques for deriving lower bounds on the kernel size for ce... |

140 | Theory of Graphs - Ore - 1962 |

103 | Fixed parameter algorithms for DOMINATING SET and related problems on planar graphs
- Alber, Bodlaender, et al.
- 2002
(Show Context)
Citation Context ...tices that lie on the outer face of G − � i−1 j=1 Lj for 1 <i≤ r. It is well known that a layer decomposition of a planar graph G can be computed in linear time in the number of vertices in the graph =-=[4]-=-. A separator in a graph G is a set of vertices S whose removal disconnects G. If (L1,...,Lr) is a layer decomposition of G, then clearly the vertices in any layer Li form a separator in G, separating... |

67 | Parameterized complexity: a framework for systematically confronting computational intractability
- Downey, Fellows, et al.
- 1997
(Show Context)
Citation Context ...I ′ , k ′ ) such that: (1) s(I ′ ) ≤ g(k) (g is a recursive function), (2) k ′ ≤ k, and (3) (I, k) ∈ L(P ) if and only if (I ′ , k ′ ) ∈ L(P ). I ′ is called the problem kernel of I. It is known (see =-=[8]-=-) that a parameterized problem is fixed-parameter tractable if and only if it has a kernelization. Of special interest to us in this paper are problems with linear kernels in which g(k) = αk for some ... |

65 |
Ein dreifarbensatz für dreikreisfreie netze auf der kugal
- Grötzsch
(Show Context)
Citation Context ...ver restricted to triangle-free planar graphs (this problem is still N Phard [15, Chapter 7]). Proof. Based on a theorem by Grötzsch (which can be turned into a polynomialtime coloring algorithm; see =-=[11]-=-) it is known that planar triangle-free graphs are 3-colorable. This implies a 3k kernel for independent set restricted to this graph class, which gives the result. Observe that the 2k-kernelization f... |

62 | Parameterized complexity: exponential speedup for planar graph problems
- Alber, Fernau, et al.
- 1995
(Show Context)
Citation Context ...for a parameterized problem P with size function s is a polynomial-time computable reduction which maps an instance (I, k) onto (I ′ , k ′ ) such that: (1) s(I ′ ) ≤ g(k) (g is a recursive function), =-=(2)-=- k ′ ≤ k, and (3) (I, k) ∈ L(P ) if and only if (I ′ , k ′ ) ∈ L(P ). I ′ is called the problem kernel of I. It is known (see [8]) that a parameterized problem is fixed-parameter tractable if and only... |

38 | Polynomial-time data reduction for dominating set
- Alber, Fellows, et al.
(Show Context)
Citation Context ...oblems, respectively, the lower bound derived for planar dominating set is still very far from the 335k upper bound on the problem kernel (computable in O(n 3 ) time), which was given by Alber et al. =-=[1]-=-. To bridge this gap, we derive better upper bounds on the problem kernel for planar dominating set. We improve the reduction rules proposed in [1], and introduce new rules that color the vertices of ... |

37 | Improved parameterized upper bounds for Vertex Cover - Chen, Kanj, et al. - 2006 |

29 | Graph separators: a parameterized view
- Alber, Fernau, et al.
(Show Context)
Citation Context ...ed problem P with size function s is a polynomial-time computable reduction which maps an instance (I, k) onto (I ′ , k ′ ) such that: (1) s(I ′ ) ≤ g(k) (g is a recursive function), (2) k ′ ≤ k, and =-=(3)-=- (I, k) ∈ L(P ) if and only if (I ′ , k ′ ) ∈ L(P ). I ′ is called the problem kernel of I. It is known (see [8]) that a parameterized problem is fixed-parameter tractable if and only if it has a kern... |

22 |
Coordinatized kernels and catalytic reductions: An improved FPT algorithm for max leaf spanning tree and other problems
- Fellows, McCartin, et al.
- 2000
(Show Context)
Citation Context ..., there is no (2 − ɛ)k kernel for planar dominating set. This remains true when further restricting the graph class to planar graphs of maximum degree three (the problem is still N P-hard). Proof. In =-=[10]-=-, a 2k-kernelization for nonblocker on general graphs which preserves planarity and degree bounds, was derived (see also [13, Theorem 13.1.3]). The above results open a new line of research, and promp... |

21 | Parameterized complexity: the main ideas and connections to practical computing
- Fellows
(Show Context)
Citation Context ...r bound by Alber et al.. Keywords. kernelization, parameterized complexity 1 Introduction Many problems which are parameterized tractable become intractable when the parameter is “turned around” (see =-=[9, 12, 14]-=-). As an example, consider the vertex cover and independent set problems. If n denotes the number of vertices in the whole graph G, then it is well-known that (G, k) is a YES-instance of vertex cover ... |

21 |
An improved fixed parameter algorithm for vertex cover
- Balasubramanian, Fellows, et al.
- 1998
(Show Context)
Citation Context ...nt parameterized algorithms has provided a new approach for practically solving problems that are theoretically intractable. For example, parameterized algorithms for the NP-hard problem vertex cover =-=[9, 13]-=- have found applications in biochemistry [10], and variants thereof are applicable to problems arising in chip manufacturing [11, 21, 24], while parameterized algorithms in computational logic [35] ha... |

18 | Experiments on data reduction for optimal domination in networks
- Alber, Betzler, et al.
- 2006
(Show Context)
Citation Context ...lowing: “In many cases, reduction was so effective that it eliminated the core completely, and with it the need for decomposition and search.” Similar success was reported with dominating set as well =-=[3]-=-. On the other hand, many applications seek solutions of very small sizes to fairly large input instances of NP-hard problems. This has been the main concern for the area of parameterized computation.... |

17 | Solving large FPT problems on coarse grained parallel machines
- Cheetham, Dehne, et al.
(Show Context)
Citation Context ...pproach for practically solving problems that are theoretically intractable. For example, parameterized algorithms for the NP-hard problem vertex cover [9, 13] have found applications in biochemistry =-=[10]-=-, and variants thereof are applicable to problems arising in chip manufacturing [11, 21, 24], while parameterized algorithms in computational logic [35] have provided an effective method for solving p... |

16 | Improved exact algorithms for max-sat
- Chen, Kanj
(Show Context)
Citation Context ...ch in the area of parameterized computation. More specifically, constructing a problem kernel has become one of the main components in the design of an efficient parameterized algorithm for a problem =-=[9, 11, 12, 13]-=-, and designing efficient parameterized algorithms for a parameterized problem now goes hand-in-hand with the construction of a problem kernel of a moderate size for the problem. Two of the most celeb... |

15 | Parameterized complexity of finding subgraphs with hereditary properties
- Khot, Raman
(Show Context)
Citation Context ...r bound by Alber et al.. Keywords. kernelization, parameterized complexity 1 Introduction Many problems which are parameterized tractable become intractable when the parameter is “turned around” (see =-=[9, 12, 14]-=-). As an example, consider the vertex cover and independent set problems. If n denotes the number of vertices in the whole graph G, then it is well-known that (G, k) is a YES-instance of vertex cover ... |

11 | C.: Either/or: using vertex cover structure in designing FPTalgorithms—the case of k-Internal SpanningTree
- Prieto, Sloper
- 2003
(Show Context)
Citation Context ...r bound by Alber et al.. Keywords. kernelization, parameterized complexity 1 Introduction Many problems which are parameterized tractable become intractable when the parameter is “turned around” (see =-=[9, 12, 14]-=-). As an example, consider the vertex cover and independent set problems. If n denotes the number of vertices in the whole graph G, then it is well-known that (G, k) is a YES-instance of vertex cover ... |

10 | Constrained minimum vertex cover in bipartite graphs: complexity and parameterized algorithms
- Chen, Kanj
(Show Context)
Citation Context ...mple, parameterized algorithms for the NP-hard problem vertex cover [9, 13] have found applications in biochemistry [10], and variants thereof are applicable to problems arising in chip manufacturing =-=[11, 21, 24]-=-, while parameterized algorithms in computational logic [35] have provided an effective method for solving practical instances of the ml type-checking problem, which is complete for the class exptime ... |

9 | On the importance of being biased (1.36 hardness of approximating Vertex-Cover - Dinur, Safra - 2002 |

4 | A direct algorithm for the parameterized face cover problem - Abu-Khzam, Langston - 2004 |

3 | Nonblocker: Parameterized algorithmics for Minimum Dominating Set
- Dehne, Fellows, et al.
- 2006
(Show Context)
Citation Context ...eneral graphs, independent set is not [17]. Similarly, while dominating set is fixed-parameter intractable on general graphs, its parametric dual, called nonblocker, is fixed-parameter tractable; see =-=[15]-=-. The landscape changes when we turn our attention towards special graph classes, e.g., problems on planar graphs [6]. Here, for example, both independent set and dominating set are fixed-parameter tr... |

2 | Probabilistic Algorithms and Complexity Classes - Uehara - 1998 |

1 | A High-Performance Toolkit for Fast Exact Algorithms, invited talk to the Workshop on Fixed-Parameter Tractability
- Abu-Khzam, Langston, et al.
- 2003
(Show Context)
Citation Context ...iation of the dominating set problem, called the red/blue dominating set problem, resulted in breaking up input instances of the problem into much smaller instances. Abu-Khzam, Langston, and Shanbhag =-=[2]-=-, in their implementation of algorithms for the vertex cover problem, ∗Received by the editors November 30, 2005; accepted for publication (in revised form) April 11, 2007; published electronically No... |