TY - JOUR

T1 - Kernelization of cycle packing with relaxed disjointness constraints

AU - Agrawal, Akanksha

AU - Lokshtanov, Daniel

AU - Majumdar, Diptapriyo

AU - Mouawad, Amer E.

AU - Saurabh, Saket

N1 - Funding Information:
∗Received by the editors June 28, 2017; accepted for publication (in revised form) April 4, 2018; published electronically July 12, 2018. An extended abstract of this paper [2] has appeared in the proceedings of the 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016). http://www.siam.org/journals/sidma/32-3/M113661.html Funding: The research leading to these results received funding from the BeHard grant under the recruitment program of the of Bergen Research Foundation (D. Lokshtanov) and the European Research Council under the European Union’s Seventh Framework Programme (FP/2007-2013)/ERC Grant Agreements 306992 (S. Saurabh).
Publisher Copyright:
© 2018 Society for Industrial and Applied Mathematics.

PY - 2018/1/1

Y1 - 2018/1/1

N2 - 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.

AB - 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.

KW - Cycle packing

KW - Kernelization

KW - Lower bounds

KW - Parameterized complexity

KW - Relaxation

UR - http://www.scopus.com/inward/record.url?scp=85053875924&partnerID=8YFLogxK

U2 - 10.1137/17M1136614

DO - 10.1137/17M1136614

M3 - Article

AN - SCOPUS:85053875924

SN - 0895-4801

VL - 32

SP - 1619

EP - 1643

JO - SIAM Journal on Discrete Mathematics

JF - SIAM Journal on Discrete Mathematics

IS - 3

ER -