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Register a new userImproved Kernels for Edge Modification Problems
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Added by
Yixin Cao
Publication date
2021
fields
Informatics Engineering
and research's language is
English
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In an edge modification problem, we are asked to modify at most $k$ edges to a given graph to make the graph satisfy a certain property. Depending on the operations allowed, we have the completion problems and the edge deletion problems. A great amount of efforts have been devoted to understanding the kernelization complexity of these problems. We revisit several well-studied edge modification problems, and develop improved kernels for them: begin{itemize} item a $2 k$-vertex kernel for the cluster edge deletion problem, item a $3 k^2$-vertex kernel for the trivially perfect completion problem, item a $5 k^{1.5}$-vertex kernel for the split completion problem and the split edge deletion problem, and item a $5 k^{1.5}$-vertex kernel for the pseudo-split completion problem and the pseudo-split edge deletion problem. end{itemize} Moreover, our kernels for split completion and pseudo-split completion have only $O(k^{2.5})$ edges. Our results also include a $2 k$-vertex kernel for the strong triadic closure problem, which is related to cluster edge deletion.
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Let $H$ be a fixed graph. Given a graph $G$ and an integer $k$, the $H$-free edge modification problem asks whether it is possible to modify at most $k$ edges in $G$ to make it $H$-free. Sandeep and Sivadasan (IPEC 2015) asks whether the paw-free completion problem and the paw-free edge deletion problem admit polynomial kernels. We answer both questions affirmatively by presenting, respectively, $O(k)$-vertex and $O(k^4)$-vertex kernels for them. This is part of an ongoing program that aims at understanding compressibility of $H$-free edge modification problems.
In a (parameterized) graph edge modification problem, we are given a graph $G$, an integer $k$ and a (usually well-structured) class of graphs $mathcal{G}$, and ask whether it is possible to transform $G$ into a graph $G in mathcal{G}$ by adding and/or removing at most $k$ edges. Parameterized graph edge modification problems received considerable attention in the last decades. In this paper, we focus on finding small kernels for edge modification problems. One of the most studied problems is the Cluster Editing problem, in which the goal is to partition the vertex set into a disjoint union of cliques. Even if this problem admits a $2k$ kernel [Cao, 2012], this kernel does not reduce the size of most instances. Therefore, we explore the question of whether linear kernels are a theoretical limit in edge modification problems, in particular when the target graphs are very structured (such as a partition into cliques for instance). We prove, as far as we know, the first sublinear kernel for an edge modification problem. Namely, we show that Clique + Independent Set Deletion, which is a restriction of Cluster Deletion, admits a kernel of size $O(k/log k)$. We also obtain small kernels for several other edge modification problems. We prove that Split Addition (and the equivalent Split Deletion) admits a linear kernel, improving the existing quadratic kernel of Ghosh et al. [Ghosh et al., 2015]. We complement this result by proving that Trivially Perfect Addition admits a quadratic kernel (improving the cubic kernel of Guo [Guo, 2007]), and finally prove that its triangle-free version (Starforest Deletion) admits a linear kernel, which is optimal under ETH.
We give new approximation algorithms for the submodular joint replenishment problem and the inventory routing problem, using an iterative rounding approach. In both problems, we are given a set of $N$ items and a discrete time horizon of $T$ days in which given demands for the items must be satisfied. Ordering a set of items incurs a cost according to a set function, with properties depending on the problem under consideration. Demand for an item at time $t$ can be satisfied by an order on any day prior to $t$, but a holding cost is charged for storing the items during the intermediate period; the goal is to minimize the sum of the ordering and holding cost. Our approximation factor for both problems is $O(log log min(N,T))$; this improves exponentially on the previous best results.
In the Metric Capacitated Covering (MCC) problem, given a set of balls $mathcal{B}$ in a metric space $P$ with metric $d$ and a capacity parameter $U$, the goal is to find a minimum sized subset $mathcal{B}subseteq mathcal{B}$ and an assignment of the points in $P$ to the balls in $mathcal{B}$ such that each point is assigned to a ball that contains it and each ball is assigned with at most $U$ points. MCC achieves an $O(log |P|)$-approximation using a greedy algorithm. On the other hand, it is hard to approximate within a factor of $o(log |P|)$ even with $beta < 3$ factor expansion of the balls. Bandyapadhyay~{et al.} [SoCG 2018, DCG 2019] showed that one can obtain an $O(1)$-approximation for the problem with $6.47$ factor expansion of the balls. An open question left by their work is to reduce the gap between the lower bound $3$ and the upper bound $6.47$. In this current work, we show that it is possible to obtain an $O(1)$-approximation with only $4.24$ factor expansion of the balls. We also show a similar upper bound of $5$ for a more generalized version of MCC for which the best previously known bound was $9$.
A graph is said to be a Konig graph if the size of its maximum matching is equal to the size of its minimum vertex cover. The Konig Edge Deletion problem asks if in a given graph there exists a set of at most k edges whose deletion results in a Konig graph. While the vertex version of the problem (Konig vertex deletion) has been shown to be fixed-parameter tractable more than a decade ago, the fixed-parameter-tractability of the Konig Edge Deletion problem has been open since then, and has been conjectured to be W[1]-hard in several papers. In this paper, we settle the conjecture by proving it W[1]-hard. We prove that a variant of this problem, where we are given a graph G and a maximum matching M and we want a k-sized Konig edge deletion set that is disjoint from M, is fixed-parameter-tractable.
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