1,099 research outputs found
Data Reductions and Combinatorial Bounds for Improved Approximation Algorithms
Kernelization algorithms in the context of Parameterized Complexity are often
based on a combination of reduction rules and combinatorial insights. We will
expose in this paper a similar strategy for obtaining polynomial-time
approximation algorithms. Our method features the use of
approximation-preserving reductions, akin to the notion of parameterized
reductions. We exemplify this method to obtain the currently best approximation
algorithms for \textsc{Harmless Set}, \textsc{Differential} and
\textsc{Multiple Nonblocker}, all of them can be considered in the context of
securing networks or information propagation
New Algorithms for Mixed Dominating Set
A mixed dominating set is a collection of vertices and edges that dominates
all vertices and edges of a graph. We study the complexity of exact and
parameterized algorithms for \textsc{Mixed Dominating Set}, resolving some open
questions. In particular, we settle the problem's complexity parameterized by
treewidth and pathwidth by giving an algorithm running in time
(improving the current best ), as well as a lower bound showing
that our algorithm cannot be improved under the Strong Exponential Time
Hypothesis (SETH), even if parameterized by pathwidth (improving a lower bound
of ). Furthermore, by using a simple but so far
overlooked observation on the structure of minimal solutions, we obtain
branching algorithms which improve both the best known FPT algorithm for this
problem, from to , and the best known
exponential-time exact algorithm, from and exponential space, to
and polynomial space.Comment: This paper has been accepted to IPEC 202
Twin-Width and Polynomial Kernels
We study the existence of polynomial kernels for parameterized problems without a polynomial kernel on general graphs, when restricted to graphs of bounded twin-width. It was previously observed in [Bonnet et al., ICALP\u2721] that the problem k-Independent Set allows no polynomial kernel on graph of bounded twin-width by a very simple argument, which extends to several other problems such as k-Independent Dominating Set, k-Path, k-Induced Path, k-Induced Matching. In this work, we examine the k-Dominating Set and variants of k-Vertex Cover for the existence of polynomial kernels.
As a main result, we show that k-Dominating Set does not admit a polynomial kernel on graphs of twin-width at most 4 under a standard complexity-theoretic assumption. The reduction is intricate, especially due to the effort to bring the twin-width down to 4, and it can be tweaked to work for Connected k-Dominating Set and Total k-Dominating Set with a slightly worse bound on the twin-width.
On the positive side, we obtain a simple quadratic vertex kernel for Connected k-Vertex Cover and Capacitated k-Vertex Cover on graphs of bounded twin-width. These kernels rely on that graphs of bounded twin-width have Vapnik-Chervonenkis (VC) density 1, that is, for any vertex set X, the number of distinct neighborhoods in X is at most c?|X|, where c is a constant depending only on the twin-width. Interestingly the kernel applies to any graph class of VC density 1, and does not require a witness sequence. We also present a more intricate O(k^{1.5}) vertex kernel for Connected k-Vertex Cover.
Finally we show that deciding if a graph has twin-width at most 1 can be done in polynomial time, and observe that most graph optimization/decision problems can be solved in polynomial time on graphs of twin-width at most 1
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