Abstract
In the Maximum Degree Contraction problem, the input is a graph G on n vertices, and integers k, d, and the objective is to check whether G can be transformed into a graph of maximum degree at most d, using at most k edge contractions. A simple brute-force algorithm that checks all possible sets of edges for a solution runs in time \(n^{\mathcal {O}(k)}\). As our first result, we prove that this algorithm is asymptotically optimal, upto constants in the exponents, under Exponential Time Hypothesis (ETH). Belmonte, Golovach, van’t Hof, and Paulusma studied the problem in the realm of parameterized complexity and proved, among other things, that it admits an FPT algorithm running in time \((d + k)^{2k} \cdot n^{\mathcal {O}(1)} = 2^{\mathcal {O}(k \log (k+d) )} \cdot n^{\mathcal {O}(1)}\), and remains NP-hard for every constant \(d \ge 2\) (Acta Informatica (2014)). We present a different FPT algorithm that runs in time \(2^{\mathcal {O}(dk)} \cdot n^{\mathcal {O}(1)}\). In particular, our algorithm runs in time \(2^{\mathcal {O}(k)} \cdot n^{\mathcal {O}(1)}\), for every fixed d. In the same article, the authors asked whether the problem admits a polynomial kernel, when parameterized by \(k + d\). We answer this question in the negative and prove that it does not admit a polynomial compression unless \(\textsf {NP}\subseteq \textsf {coNP}/poly\).
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Notes
The Exponential Time Hypothesis (ETH) is a conjecture stating that, roughly speaking, 3-SAT has no algorithm subexponential in the number of variables. We refer the readers to Chapter 14 in [14] for formal definition and known lower bounds results under this conjecture.
The algorithm colors vertices in the reduced instance with two colors and contracts each connected component in the colored subgraphs.
Since we are looking for an independent set, it is intuitive to add all missing edges in a row or a column of the table. But to simplify our reduction, we remove edges that have both their endpoints in the same row or column. It is easy to verify that this operation is safe.
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Acknowledgements
Some part of this project was completed when the second author was a Postdoctoral Fellow at Max Planck Institute for Informatics, Saarland Informatics Campus, Saarbrücken, Germany.
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An extended abstract of this article has been accepted in International Symposium on Parameterized and Exact Computation (IPEC) 2020. Corresponding author: Prafullkumar Tale (prafullkumar.tale@cispa.saarland).
The first author has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant Agreement No 819416), and Swarnajayanti Fellowship (No DST/SJF/MSA01/2017-18).
The second author has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme under Grant Agreement SYSTEMATICGRAPH (No. 725978).
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Saurabh, S., Tale, P. On the Parameterized Complexity of Maximum Degree Contraction Problem. Algorithmica 84, 405–435 (2022). https://doi.org/10.1007/s00453-021-00897-6
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DOI: https://doi.org/10.1007/s00453-021-00897-6