Randomized parameterized algorithms for $$P_2$$ P 2 -Packing and Co-Path Packing problems

Qi, Feng; Wang, Jianxin; Li, Shaohua; Chen, Jianer

doi:10.1007/s10878-013-9691-z

Cited by 25 publications

(15 citation statements)

References 10 publications

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“…The stages in the algorithm correspond to the small universes-we will have 1 + 1 stages, such that at stage 2 ≤ i ≤ 1 + 1, the element that we are "given" is the largest element in the piece U i−1 . 17 We still iterate over U in an ascending order (but now this is performed according to the new order-corresponding to the currently examined option that cut the universe), such that at stage i, when inserting a 3-set to partial solutions, we can remove its smallest element. However, at stage i, we can also remove from our partial solutions all the elements that are smaller than the given element; there should be enough such elements at each partial solution (according to a recursive formula that we discuss later), since if there are not, we simply discard the partial solution.…”

Section: C1 Intuitionmentioning

confidence: 99%

Mixing Color Coding-Related Techniques

Zehavi

2015

Algorithms - ESA 2015

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Abstract. Narrow sieves, representative sets and divide-and-color are three breakthrough color coding-related techniques, which led to the design of extremely fast parameterized algorithms. We present a novel family of strategies for applying mixtures of them. This includes: (a) a mix of representative sets and narrow sieves; (b) a faster computation of representative sets under certain separateness conditions, mixed with divide-and-color and a new technique, "balanced cutting"; (c) two mixtures of representative sets, iterative compression and a new technique, "unbalanced cutting". We demonstrate our strategies by obtaining, among other results, significantly faster algorithms for k-Internal Out-Branching and Weighted 3-Set k-Packing, and a framework for speeding-up the previous best deterministic algorithms for k-Path, kTree, r-Dimensional k-Matching, Graph Motif and Partial Cover. IntroductionA problem is fixed-parameter tractable (FPT) with respect to a parameter k if it can be solved in time O * (f (k)) for some function f , where O * hides factors polynomial in the input size. The color coding technique, introduced by Alon et al. [1], led to the discovery of the first single exponential time FPT algorithms for many subcases of Subgraph Isomorphism. In the past decade, three breakthrough techniques improved upon it, and led to the development of extremely fast FPT algorithms for many fundamental problems. This includes the combinatorial divide-and-color technique [8], the algebraic multilinear detection technique [26,27,43] (which was later improved to the more powerful narrow sieves technique [2,3]), and the combinatorial representative sets technique [22].Divide-and-color was the first technique that resulted in (both randomized and deterministic) FPT algorithms for weighted problems that are faster than those relying on color coding. Later, representative sets led to the design of deterministic FPT algorithms for weighted problems that are faster than the randomized ones based on divide-and-color. The fastest FPT algorithms, however, rely on narrow sieves. Unfortunately, narrow sieves is only known to be relevant to the design of randomized algorithms for unweighted problems. arXiv:1410.5062v3 [cs.DS] 20 Apr 2015We present novel strategies for applying these techniques, combining the following elements (see Section 3).• Mixing narrow sieves and representative sets, previously considered to be two independent color coding-related techniques.• Under certain "separateness conditions", speeding-up the best known computation of representative sets.• Mixing divide-and-color-based preprocessing with the computation in the previous item, speeding-up any standard representative sets-based algorithm.• Cutting the universe into small pieces in two special manners, one used in the mix in the previous item, and the other mixed with a non-standard representative sets-based algorithm to improve its running time (by decreasing the size of the partial solutions it computes).To demonstrate our strategies, we consider th...

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Section: C1 Intuitionmentioning

confidence: 99%

Mixing Color Coding-Related Techniques

Zehavi

2015

Algorithms - ESA 2015

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“…By doing O(6.75 k ) iterations of the process of random partition and constructing the bipartite graph, a P 2 -Packing of size k in G can be found with a high probability. Therefore, by putting the vertices of V into V 1 and V 2 with the probability 2/3 and 1/3, respectively, an O * (6.75 k )-time randomized algorithm can be obtained for the P 2 -packing problem [28][29] . We remark that the (2/3, 1/3) random partition for the P 2 -packing problem is the best possible for this simple strategy of partitioning the vertex set V of the input graph G into two disjoint subsets, one containing all 2k end-points of the P 2 's in P and the other containing all k mid-points of the P 2 's in P. Let f (x) = x 2k (1 − x) k , where 0 < x < 1.…”

Section: Dealing With a Small Unknown Subsetmentioning

confidence: 99%

On Unknown Small Subsets and Implicit Measures: New Techniques for Parameterized Algorithms

Chen

2014

J. Comput. Sci. Technol.

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Parameterized computation is a recently proposed alternative approach to dealing with NP-hard problems. Developing efficient parameterized algorithms has become a very active research area in the current research in theoretical computer science. In this paper, we investigate a number of new algorithmic techniques that were proposed and initiated by ourselves in our research in parameterized computation. The techniques have proved to be very useful and promising, and have led to improved parameterized algorithms for many well-known NP-hard problems.

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“…Recently, Chen et al (2011) gave an approximation algorithm with ratio 10/7 using local search and dynamic programming. Recently, randomized methods have been applied to get parameterized algorithms for many NP-hard problems Feng et al (2015), Chen and Feng (2014), Heggernes et al (2014). In this paper, we use random techniques to design parameterized algorithm for the Parameterized Co-path Set problem, which is defined as follows:…”

Section: Introductionmentioning

confidence: 99%

Kernelization and randomized Parameterized algorithms for Co-path Set problem

Zhou

Wang

2015

J Comb Optim

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Co-path Set problem is of important applications in mapping unique DNA sequences onto chromosomes and whole genomes. Given a graph G, the parameterized version of Co-path Set problem is to find a subset F of edges with size at most k such that each connected component in G[E\F] is a path. In this paper, we give a kernel of size 6k and a randomized algorithm of running time O * (2.17 k ) for the Parameterized Co-path Set problem. The randomized methods developed in the paper are of great promising to be applied to other branch-based algorithms.

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Randomized parameterized algorithms for $$P_2$$ P 2 -Packing and Co-Path Packing problems

Cited by 25 publications

References 10 publications

Mixing Color Coding-Related Techniques

Mixing Color Coding-Related Techniques

On Unknown Small Subsets and Implicit Measures: New Techniques for Parameterized Algorithms

Kernelization and randomized Parameterized algorithms for Co-path Set problem

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