NDLI: Kernelization of Cycle Packing with Relaxed Disjointness Constraints

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Lokshtanov, Daniel Mouawad, Amer E. Agrawal, Akanksha Majumdar, Diptapriyo Saurabh, Saket
Copyright Year	2018
Abstract	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 ${\sf NP} \subseteq {\sf coNP} / {\sf poly}$. 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 $t \geq 1$, the kernelization complexity of Almost Disjoint Cycle Packing reveals an interesting spectrum of upper and lower bounds. In particular, for $t = \frac{k}{c}$, where $c$ could be a function of $k$, we obtain a kernel of size $\mathcal{O}(2^{c^2}k^{7 + c}\log^3 k)$ whenever $c\in o(\sqrt k)$. Thus the kernel size varies from being subexponential when $c\in o(\sqrt k)$, to quasi-polynomial when $c\in o(\log^{\ell} k)$, $\ell \in \mathbb{R}_+$, and polynomial when $c\in \mathcal{O}(1)$. We complement these results for Almost Disjoint Cycle Packing by showing that the problem does not admit a polynomial kernel whenever $t \in \mathcal{O}(k^{\epsilon})$ for any $0 \leq \epsilon < 1$, unless ${\sf NP} \subseteq {\sf coNP} / {\sf poly}$.
Sponsorship	H2020 European Research Council. Bergens Forskningsstiftelse
Starting Page	1619
Ending Page	1643
Page Count	25
File Format	PDF
ISSN	08954801
DOI	10.1137/17M1136614
e-ISSN	10957146
Journal	SIAM Journal on Discrete Mathematics (SJDMEC)
Issue Number	3
Volume Number	32
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2018-07-12
Access Restriction	Subscribed
Subject Keyword	Computational difficulty of problems Analysis of algorithms and problem complexity cycle packing relaxation parameterized complexity Graph theory kernelization lower bounds Complexity classes
Content Type	Text
Resource Type	Article
Subject	Mathematics

Sl.	Authority	Responsibilities	Communication Details
1	Ministry of Education (GoI), Department of Higher Education	Sanctioning Authority	https://www.education.gov.in/ict-initiatives
2	Indian Institute of Technology Kharagpur	Host Institute of the Project: The host institute of the project is responsible for providing infrastructure support and hosting the project	https://www.iitkgp.ac.in
3	National Digital Library of India Office, Indian Institute of Technology Kharagpur	The administrative and infrastructural headquarters of the project	Dr. B. Sutradhar bsutra@ndl.gov.in
4	Project PI / Joint PI	Principal Investigator and Joint Principal Investigators of the project	Dr. B. Sutradhar bsutra@ndl.gov.in Prof. Saswat Chakrabarti will be added soon
5	Website/Portal (Helpdesk)	Queries regarding NDLI and its services	support@ndl.gov.in
6	Contents and Copyright Issues	Queries related to content curation and copyright issues	content@ndl.gov.in
7	National Digital Library of India Club (NDLI Club)	Queries related to NDLI Club formation, support, user awareness program, seminar/symposium, collaboration, social media, promotion, and outreach	clubsupport@ndl.gov.in
8	Digital Preservation Centre (DPC)	Assistance with digitizing and archiving copyright-free printed books	dpc@ndl.gov.in
9	IDR Setup or Support	Queries related to establishment and support of Institutional Digital Repository (IDR) and IDR workshops	idr@ndl.gov.in

Kernelization Lower Bounds by Cross-Composition

Fractals for Kernelization Lower Bounds

The Parameterized Complexity of the Minimum Shared Edges Problem

Kernelization Lower Bounds for Finding Constant-Size Subgraphs

Almost Optimal Lower Bounds for Problems Parameterized by Clique-Width

The Minimum Shared Edges Problem on Planar Graphs

Dual parameterization and parameterized approximability of subset graph problems

Parametric Duality and Kernelization: Lower Bounds and Upper Bounds on Kernel Size

The Complexity of Routing with Few Collisions

Kernelization of Cycle Packing with Relaxed Disjointness Constraints

Similar Documents

Kernelization Lower Bounds by Cross-Composition

Fractals for Kernelization Lower Bounds

The Parameterized Complexity of the Minimum Shared Edges Problem

Kernelization Lower Bounds for Finding Constant-Size Subgraphs

Almost Optimal Lower Bounds for Problems Parameterized by Clique-Width

The Minimum Shared Edges Problem on Planar Graphs

Dual parameterization and parameterized approximability of subset graph problems

Parametric Duality and Kernelization: Lower Bounds and Upper Bounds on Kernel Size

The Complexity of Routing with Few Collisions

Kernelization of Cycle Packing with Relaxed Disjointness Constraints