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Edge-Disjoint Packing of Stars and Cycles

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Combinatorial Optimization and Applications

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9486))

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Abstract

We study the parameterized complexity of two graph packing problems, Edge-Disjoint \(k\)-Packing of \(s\)-Stars and Edge-Disjoint \(k\)-Packing of \(s\)-Cycles. With respect to the choice of parameters, we show that although the two problems are FPT with both k and s as parameters, they are unlikely to be fixed-parameter tractable when parameterized by only k or only s. In terms of kernelization complexity, we show that Edge-Disjoint \(k\)-Packing of \(s\)-Stars has a kernel with size polynomial in both k and s, but in contrast, unless NP \(\subseteq \) coNP/poly, Edge-Disjoint \(k\)-Packing of \(s\)-Cycles does not have a kernel with size polynomial in both k and s, and moreover does not have a kernel with size polynomial in s for any fixed k. Specifically, (1) from the negative direction, we show that Edge-Disjoint \(k\)-Packing of \(s\)-Stars is W[1]-hard with parameter k in general graphs, and that Edge-Disjoint \(k\)-Packing of \(s\)-Cycles is W[1]-hard with parameter k and NP-hard for any even \(s \ge 4\) in bipartite graphs; (2) from the positive direction, we show that Edge-Disjoint \(k\)-Packing of \(s\)-Stars admits a \(ks^2\) kernel, and that Edge-Disjoint \(k\)-Packing of 4-Cycles admits a \(96k^2\) kernel in general graphs and a 96k kernel in planar graphs.

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Notes

  1. 1.

    For the general problem of deciding whether a graph G contains k edge-disjoint copies of a graph H, the running time of this algorithm [5, Corollary 5.7] is \(4^{rk + O(\log ^3(rk))} n^{r+1}\), where n and r are the numbers of vertices in G and H, respectively. The \(n^{r+1}\) factor in the running time accounts for the time for solving the subproblem of determining whether G contains H as a subgraph, by brute force. The brute-force algorithm for this subproblem can be replaced by faster algorithms in our setting, with polynomial running time when H is an s-star, or \(2^{O(s)}\mathrm {poly}(n)\) time when H is an s-cycle [1].

  2. 2.

    We remark that in our setting of Edge-Disjoint \(k\)-Packing of \(s\)-Cycles, since any two different s-cycles share at most \(s-2\) edges, the base case of \(f(0,k) = 1\) in the analysis of [13, page 170] can be upgraded to \(f(1,k) = 1\), which saves 1 in the exponent and yields an \(O(k^{s-1})\) kernel.

  3. 3.

    We remark that the construction in our proof of Theorem 2 can be easily adapted to prove the hardness of all four variants of related problems on vertex/edge-disjoint cycles/paths.

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Author information

Authors and Affiliations

  1. Department of Computer Science, Utah State University, Logan, UT, 84322, USA

    Minghui Jiang

  2. Department of Computer Science, Lafayette College, Easton, PA, 18042, USA

    Ge Xia

  3. Department of Computer Science and Information Technology, Kutztown University, Kutztown, PA, 19530, USA

    Yong Zhang

Authors
  1. Minghui Jiang
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  2. Ge Xia
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  3. Yong Zhang
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Correspondence to Yong Zhang .

Editor information

Editors and Affiliations

  1. Dept. of Mathematics and Computer Science, Marywood University, Scranton, Pennsylvania, USA

    Zaixin Lu

  2. North Carolina Central University, Durham, North Carolina, USA

    Donghyun Kim

  3. University of Texas at Dallas, Richardson, Texas, USA

    Weili Wu

  4. Texas Southern University, Houston, Texas, USA

    Wei Li

  5. University of Texas at Dallas, Richardson, Texas, USA

    Ding-Zhu Du

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Jiang, M., Xia, G., Zhang, Y. (2015). Edge-Disjoint Packing of Stars and Cycles. In: Lu, Z., Kim, D., Wu, W., Li, W., Du, DZ. (eds) Combinatorial Optimization and Applications. Lecture Notes in Computer Science(), vol 9486. Springer, Cham. https://doi.org/10.1007/978-3-319-26626-8_49

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  • DOI: https://doi.org/10.1007/978-3-319-26626-8_49

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