Kernelization of cycle packing with relaxed disjointness constraints

Akanksha Agrawal, Daniel Lokshtanov, Diptapriyo Majumdar, Amer E. Mouawad, Saket Saurabh

Research output: Contribution to journalArticlepeer-review

3 Scopus citations


A key result in the field of kernelization, a subfield of parameterized complexity, states that the classic Disjoint Cycle Packing problem, i.e., finding k vertex disjoint cycles in a given graph G, admits no polynomial kernel unless NP ⊆ coNP/poly. However, very little is known about this problem beyond the aforementioned kernelization lower bound (within the parameterized complexity framework). In the hope of clarifying the picture and better understanding the types of constraints that separate kernelizable from nonkernelizable variants of Disjoint Cycle Packing, we investigate two relaxations of the problem. The first variant, which we call Almost Disjoint Cycle Packing, introduces a global relaxation parameter t. That is, given a graph G and integers k and t, the goal is to find at least k distinct cycles such that every vertex of G appears in at most t of the cycles. The second variant, Pairwise Disjoint Cycle Packing, introduces a local relaxation parameter, and we seek at least k distinct cycles such that every two cycles intersect in at most t vertices. While the Pairwise Disjoint Cycle Packing problem admits a polynomial kernel for all t ≥ 1, the kernelization complexity of Almost Disjoint Cycle Packing reveals an interesting spectrum of upper and lower bounds. In particular, for t = k c , where c could be a function of k, we obtain a kernel of size O(2c 2k7+c log3 k) whenever c ∈ o( √ k). Thus the kernel size varies from being subexponential when c ∈ o( √ k), to quasi-polynomial when c ∈ o(logℓ k), ℓ ∈ R+, and polynomial when c ∈ O(1). We complement these results for Almost Disjoint Cycle Packing by showing that the problem does not admit a polynomial kernel whenever t ∈ O(kε) for any 0 ≤ ε < 1, unless NP ⊆ coNP/poly.

Original languageEnglish
Pages (from-to)1619-1643
Number of pages25
JournalSIAM Journal on Discrete Mathematics
Issue number3
StatePublished - 1 Jan 2018
Externally publishedYes


  • Cycle packing
  • Kernelization
  • Lower bounds
  • Parameterized complexity
  • Relaxation

ASJC Scopus subject areas

  • General Mathematics


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