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 $$\textsf {NP}\subseteq \textsf {coNP}/poly$$ .

Highlights

  • For any graph class H, the H-Modification problem takes as input a graph G and an integer k, and asks whether one can make at most k modifications in G such that the resulting graph is in H

  • If allowed modification operation is vertex deletion we know the problem as Bounded Degree Deletion (BDD) and if it is edge contraction as Maximum Degree Contraction (MDC)

  • We studied Maximum Degree Contraction problem

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Summary

Introduction

For any graph class H, the H-Modification problem takes as input a graph G and an integer k, and asks whether one can make at most k modifications in G such that the resulting graph is in H. Brouwer proved that it is NP-Hard even to decide whether a graph can be contracted to a path of length four [11] Note that this problem admits a simple polynomial time algorithm if we consider any other modification operation. For the same target graph class, edge contraction problem tends to more difficult than their counterparts where modification operation is vertex/edge addition/deletion This difficulty is evident even in the realm of the Parameterized Complexity and Exact Exponential Algorithms. Belmonte et al [8] proved that MDC problem admits linear vertex kernels on connected graphs when d = 2 This linear kernel leads to an FPT algorithm running in time 2O(k) · nO(1).

Preliminaries
A Lower Bound for the Algorithm
A Different FPT Algorithm
No Polynomial Kernel
Conclusion

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