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Register a new userA Tight Approximation Algorithm for the Cluster Vertex Deletion Problem
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Added by
Tony Huynh
Publication date
2020
fields
Informatics Engineering
and research's language is
English
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We give the first $2$-approximation algorithm for the cluster vertex deletion problem. This is tight, since approximating the problem within any constant factor smaller than $2$ is UGC-hard. Our algorithm combines the previous approaches, based on the local ratio technique and the management of true twins, with a novel construction of a good cost function on the vertices at distance at most $2$ from any vertex of the input graph. As an additional contribution, we also study cluster vertex deletion from the polyhedral perspective, where we prove almost matching upper and lower bounds on how well linear programming relaxations can approximate the problem.
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A split graph is a graph whose vertex set can be partitioned into a clique and a stable set. Given a graph $G$ and weight function $w: V(G) to mathbb{Q}_{geq 0}$, the Split Vertex Deletion (SVD) problem asks to find a minimum weight set of vertices $X$ such that $G-X$ is a split graph. It is easy to show that a graph is a split graph if and only it it does not contain a $4$-cycle, $5$-cycle, or a two edge matching as an induced subgraph. Therefore, SVD admits an easy $5$-approximation algorithm. On the other hand, for every $delta >0$, SVD does not admit a $(2-delta)$-approximation algorithm, unless P=NP or the Unique Games Conjecture fails. For every $epsilon >0$, Lokshtanov, Misra, Panolan, Philip, and Saurabh recently gave a randomized $(2+epsilon)$-approximation algorithm for SVD. In this work we give an extremely simple deterministic $(2+epsilon)$-approximation algorithm for SVD.
In the $d$-dimensional hypercube bin packing problem, a given list of $d$-dimensional hypercubes must be packed into the smallest number of hypercube bins. Epstein and van Stee [SIAM J. Comput. 35 (2005)] showed that the asymptotic performance ratio $rho$ of the online bounded space variant is $Omega(log d)$ and $O(d/log d)$, and conjectured that it is $Theta(log d)$. We show that $rho$ is in fact $Theta(d/log d)$, using probabilistic arguments.
One of the most fundamental results in combinatorial optimization is the polynomial-time 3/2-approximation algorithm for the metric traveling salesman problem. It was presented by Christofides in 1976 and is well known as the Christofides algorithm. Recently, some authors started calling it Christofides-Serdyukov algorithm, pointing out that it was published independently in the USSR in 1978. We provide some historic background on Serdyukovs findings and a translation of his article from Russian into English.
A cactus is a connected graph that does not contain $K_4 - e$ as a minor. Given a graph $G = (V, E)$ and integer $k ge 0$, Cactus Vertex Deletion (also known as Diamond Hitting Set) is the problem of deciding whether $G$ has a vertex set of size at most $k$ whose removal leaves a forest of cacti. The current best deterministic parameterized algorithm for this problem was due to Bonnet et al. [WG 2016], which runs in time $26^kn^{O(1)}$, where $n$ is the number of vertices of $G$. In this paper, we design a deterministic algorithm for Cactus Vertex Deletion, which runs in time $17.64^kn^{O(1)}$. As a straightforward application of our algorithm, we give a $17.64^kn^{O(1)}$-time algorithm for Even Cycle Transversal. The idea behind this improvement is to apply the measure and conquer analysis with a slightly elaborate measure of instances.
Given a graph $G=(V,E)$ and a positive integer $tgeq2$, the task in the vertex cover $P_t$ ($VCP_t$) problem is to find a minimum subset of vertices $Fsubseteq V$ such that every path of order $t$ in $G$ contains at least one vertex from $F$. The $VCP_t$ problem is NP-complete for any integer $tgeq2$ and has many applications in real world. Recently, the authors presented a dynamic programming algorithm running in time $4^pcdot n^{O(1)}$ for the $VCP_3$ problem on $n$-vertex graphs with treewidth $p$. In this paper, we propose an improvement of it and improved the time-complexity to $3^pcdot n^{O(1)}$. The connected vertex cover $P_3$ ($CVCP_3$) problem is the connected variation of the $VCP_3$ problem where $G[F]$ is required to be connected. Using the Cut&Count technique, we give a randomized algorithm with runtime $4^pcdot n^{O(1)}$ for the $CVCP_3$ problem on $n$-vertex graphs with treewidth $p$.
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