We study the generalized min sum set cover (GMSSC) problem, wherein given a collection of hyperedges $E$ with arbitrary covering requirements $k_e$, the goal is to find an ordering of the vertices to minimize the total cover time of the hyperedges; a hyperedge $e$ is considered covered by the first time when $k_e$ many of its vertices appear in the ordering. We give a $4.642$ approximation algorithm for GMSSC, coming close to the best possible bound of $4$, already for the classical special case (with all $k_e=1$) of min sum set cover (MSSC) studied by Feige, Lovász and Tetali, and improving upon the previous best known bound of $12.4$ due to Im, Sviridenko and van der Zwaan. Our algorithm is based on transforming the LP solution by a suitable kernel and applying randomized rounding. This also gives an LP-based $4$ approximation for MSSC. As part of the analysis of our algorithm, we also derive an inequality on the lower tail of a sum of independent Bernoulli random variables, which might be of independent interest and broader utility.
Another well-known special case is the min sum vertex cover (MSVC) problem, in which the input hypergraph is a graph and $k_e = 1$, for every edge. We give a $16/9$ approximation for MSVC, and show a matching integrality gap for the natural LP relaxation. This improves upon the previous best $1.999946$ approximation of Barenholz, Feige and Peleg. (The claimed $1.79$ approximation result of Iwata, Tetali and Tripathi for the MSVC turned out have an unfortunate, seemingly unfixable, mistake in it.) Finally, we revisit MSSC and consider the $\ell_p$ norm of cover-time of the hyperedges. Using a dual fitting argument, we show that the natural greedy algorithm achieves tight, up to NP-hardness, approximation guarantees of $(p+1)^{1+1/p}$, for all $p\ge 1$. For $p=1$, this gives yet another proof of the $4$ approximation for MSSC.
This paper presents a fast and simple new 2-approximation algorithm for minimum weighted vertex cover. The unweighted version of this algorithm is equivalent to a well-known greedy maximal independent set algorithm. We prove that this independent set algorithm produces a 2-approximate vertex cover, and we provide a principled new way to generalize it to node-weighted graphs. Our analysis is inspired by connections to a clustering objective called correlation clustering. To demonstrate the relationship between these problems, we show how a simple Pivot algorithm for correlation clustering impli…
This paper proposes a novel branch-and-bound(BMWVC) algorithm to exactly solve the minimum weight vertex cover problem (MWVC) in large graphs. The original contribution is several new graph reduction rules, allowing to reduce a graph G and the time needed to find a minimum weight vertex cover in G. Experiments on large graphs from real-world applications show that the reduction rules are effective and the resulting BMWVC algorithm outperforms relevant exact and heuristic MWVC algorithms.
We give a characterization result for the integrality gap of the natural linear programming relaxation for the vertex cover problem. We show that integrality gap of the standard linear programming relaxation for any graph G equals $\left(2-\frac{2}{χ^f(G)}\right)$ where $χ^f(G)$ denotes the fractional chromatic number of G.
In any attempt at designing an efficient algorithm for the minimum vertex cover problem, obtaining good upper and lower bounds for the vertex cover number could be crucial. In this article we present a modified greedy algorithm of worst-case time complexity O(n3) to obtain bounds for the vertex cover number of an input graph of order n. Using simple facts, the proposed algorithm computes a lower bound for the vertex cover number. Then using this lower bound it outputs a minimal vertex cover and hence gives an upper bound. The algorithm ensures the output vertex cover is always minimal, which f…
The eternal vertex cover problem is a dynamic variant of the classical vertex cover problem. It is NP-hard to compute the eternal vertex cover number of graphs and known algorithmic results for the problem are very few. This paper presents a linear time recursive algorithm for computing the eternal vertex cover number of cactus graphs. Unlike other graph classes for which polynomial time algorithms for eternal vertex cover number are based on efficient computability of a known lower bound directly derived from minimum vertex cover, we show that it is a certain substructure property that helps …
New Branching Rules: Improvements on Independent Set and Vertex Cover in Sparse Graphs
本文提出了一个在度为3的$n$ -顶点图中寻找最大独立集的$O^*(1.0919^n)$ -时间算法，改进了Bourgeois，Escoffier和Paschos [IWPEC 2008]提出的运行时间算法$O^*(1.0977^n)$.我们还提出了一个$O^*(1.1923^k)$ -时间算法，以决定是否有一个图的度有界为3的顶点覆盖的大小$k$，这改进了以前已知的算法运行时间$O^*(1.1939^k)$由陈，Kanj和夏[ISAAC 2003]。介绍了两种新的分支技术\emph{branching on a bottle}和\emph{branching on a 4-cycle}，它们帮助我们设计简单、快速的最大独立集和最小顶点覆盖问题算法，并避免繁琐的分支规则。
A famous conjecture of Tuza states that the minimum number of edges needed to cover all the triangles in a graph is at most twice the maximum number of edge-disjoint triangles. This conjecture was couched in a broader setting by Aharoni and Zerbib who proposed a hypergraph version of this conjecture, and also studied its implied fractional versions. We establish the fractional version of the Aharoni-Zerbib conjecture up to lower order terms. Specifically, we give a factor $t/2+ O(\sqrt{t \log t})$ approximation based on LP rounding for an algorithmic version of the hypergraph Turán problem (AH…
We give a parameterized algorithm for weighted vertex cover on graphs with maximum degree 3 whose time complexity is $O^*(1.402^t)$, where $t$ is the minimum size of a vertex cover of the input graph.
The Connected Vertex Cover problem, where the goal is to compute a minimum set of vertices in a given graph which forms a vertex cover and induces a connected subgraph, is a fundamental combinatorial problem and has received extensive attention in various subdomains of algorithmics. In the area of kernelization, it is known that this problem is unlikely to have efficient preprocessing algorithms, also known as polynomial kernelizations. However, it has been shown in a recent work of Lokshtanov et al. [STOC 2017] that if one considered an appropriate notion of approximate kernelization, then th…
Given a graph $G$, the Connected Vertex Cover problem (CVC) asks to find a minimum cardinality vertex cover of $G$ that induces a connected subgraph. In this paper we describe some approaches to solve the CVC problem exactly. First, we give compact mixed-integer extended formulations for CVC: these are the first formulations proposed for this problem, and can be easily adapted to variations of the problem such as Tree Cover. Second, we describe a simple branch and bound algorithm for the CVC problem. Finally, we implement our algorithm and compare its performance against our best formulation: …
For intractable problems on graphs of bounded treewidth, two graph parameters treedepth and vertex cover number have been used to obtain fine-grained complexity results. Although the studies in this direction are successful, we still need a systematic way for further investigations because the graphs of bounded vertex cover number form a rather small subclass of the graphs of bounded treedepth. To fill this gap, we use vertex integrity, which is placed between the two parameters mentioned above. For several graph problems, we generalize fixed-parameter tractability results parameterized by ver…
一个伪森林是一个图，它的连通分支最多有一个圈。设X是图G的伪森林调制器，i. e. G的一个顶点子集，使得G-X是一个伪森林。我们表明，顶点覆盖承认一个多项式核参数化的大小的pseudoforest调制器。换句话说，我们提供了一个多项式时间算法，对于输入图G和整数k，输出图G'和整数k'，使得G'具有O（|X| 12）个顶点，并且G具有大小为k的顶点覆盖当且仅当G'具有大小为k '的顶点覆盖。我们补充我们的研究结果证明，有没有多项式内核的顶点覆盖参数化的大小调制器到一个模拟森林（图中没有周期共享一个顶点），除非NP是一个子集的coNP/聚。特别是，这也排除了多项式内核时，参数化的调制器的大小外平面和仙人掌图。
In the 4-path vertex cover problem, the input is an undirected graph $G$ and an integer $k$. The goal is to decide whether there is a set of vertices $S$ of size at most $k$ such that every path with 4 vertices in $G$ contains at least one vertex of $S$. In this paper we give a parameterized algorithm for 4-path vertex cover whose time complexity is $O^*(2.619^k)$.
Let $vc(G)$, $fvs(G)$ and $oct(G)$, respectively, denote the size of a minimum vertex cover, minimum feedback vertex set and minimum odd cycle transversal in a graph $G$. One can ask, when looking for these sets in a graph, how much bigger might they be if we require that they are independent; that is, what is the price of independence? If $G$ has a vertex cover, feedback vertex set or odd cycle transversal that is an independent set, then we let $ivc(G)$, $ifvs(G)$ or $ioct(G)$, respectively, denote the minimum size of such a set. Similar to a recent study on the price of connectivity (Hartin…
An $O(1)$-Approximation Algorithm for Dynamic Weighted Vertex Cover with Soft Capacity
本研究考虑动态环境中的（软）容量限制顶点覆盖问题。该问题推广了近年来得到广泛研究的顶点覆盖问题的动态模型。给定一个动态变化的顶点加权图$G=(V,E)$，它允许边插入和边删除，目标是设计一个数据结构，保持一个近似的最小顶点覆盖，同时满足每个顶点的容量约束。也就是说，当拾取覆盖中的顶点$v$的副本时，由副本覆盖的$v$的关联边的数量达到给定容量的$v$。我们扩展Bhattacharya等人。的工作[SODA'15和ICALP'15]，以获得一个确定性的原始-对偶算法，用于保持一个常数因子近似最小容量顶点覆盖与$O(\log n / ε)$摊销更新时间，其中$n$是在图中的顶点数。该算法可以扩展到（1）一个更一般的模型，其中每个边与一个非均匀和不可分裂的需求，（2）更一般的容量限制集覆盖问题。
In the vertex cover problem, the input is a graph $G$ and an integer $k$, and the goal is to decide whether there is a set of vertices $S$ of size at most $k$ such that every edge of $G$ is incident on at least one vertex in $S$. We study the vertex cover problem on graphs with maximum degree 4 and minimum degree at least 2, parameterized by $r = k-n/3$. We give an algorithm for this problem whose running time is $O^*(1.6253^r)$. As a corollary, we obtain an $O^*(1.2403^k)$-time algorithm for vertex cover on graphs with maximum degree 4.
A vertex subset $I$ of a graph $G$ is called a $k$-path vertex cover if every path on $k$ vertices in $G$ contains at least one vertex from $I$. The \textsc{$k$-Path Vertex Cover Reconfiguration ($k$-PVCR)} problem asks if one can transform one $k$-path vertex cover into another via a sequence of $k$-path vertex covers where each intermediate member is obtained from its predecessor by applying a given reconfiguration rule exactly once. We investigate the computational complexity of \textsc{$k$-PVCR} from the viewpoint of graph classes under the well-known reconfiguration rules: $\mathsf{TS}$, …
It is a celebrated result in early combinatorics that, in bipartite graphs, the size of maximum matching is equal to the size of a minimum vertex cover. Kőnig's proof of this fact gave an algorithm for finding a minimum vertex cover from a maximum matching. In this paper, we revisit the connection this algorithm induces between the two types of structures. We find that all minimum vertex covers can be found by applying this algorithm to some matching and then classify which matchings give minimum vertex covers when this algorithm is applied to them.
\textproc{Weighted Vertex Cover} is a variation of an extensively studied NP-complete problem, \textproc{Vertex Cover}, in which we are given a graph, $G = (V,E,w)$, where function $w:V \rightarrow \mathbb{Q}^{+}$ and a parameter $k$. The objective is to determine if there exists a vertex cover, $S$, such that $\sum_{v \in S}w(v) \leq k$. In our work, we first study the hardness of \textproc{Weighted Vertex Cover} and then examine this problem under parameterization by $l$ and $k$, where $l$ is the number of vertices with fractional weights. Then, we study the \textproc{Red-Blue Weighted Verte…
In the $l$-path vertex cover problem the input is an undirected graph $G$ and an integer $k$. The goal is to decide whether there is a set of vertices $S$ of size at most $k$ such that $G-S$ does not contain a path with $l$ vertices. In this paper we give parameterized algorithms for $l$-path vertex cover for $l = 5,6,7$, whose time complexities are $O^*(3.945^k)$, $O^*(4.947^k)$, and $O^*(5.951^k)$, respectively.
Graduate School of Information Sciences, Tohoku University（东北大学大学院信息科学研究科）；Faculty of Science and Technology, Keio University（庆应义塾大学理工学部）；Graduate School of Informatics, Kyoto University（京都大学情报学府）；Research Institute for Mathematical Sciences, Kyoto University（京都大学数理物理学研究所）；Graduate School of Informatics and Engineering, The University of Electro-Communications（电气通信大学情报工学研究科）
Our main result is designing an algorithm that returns a vertex cover of $\mathcal{G}^\star$ with size at most $(3/2+ε)$ times the expected size of the minimum vertex cover, using only $O(n/εp)$ non-adaptive queries. This improves over the best-known 2-approximation algorithm by Behnezhad, Blum, and Derakhshan [SODA'22], who also show that $Ω(n/p)$ queries are necessary to achieve any constant approximation. Our guarantees also extend to instances where edge realizations are not fully independent. We complement this upper bound with a tight $3/2$-approximation lower bound for stochastic graphs…
We study the Minimum Sum Vertex Cover problem, which asks for an ordering of vertices in a graph that minimizes the total cover time of edges. In particular, n vertices of the graph are visited according to an ordering, and for each edge this induces the first time it is covered. The goal of the problem is to find the ordering which minimizes the sum of the cover times over all edges in the graph. In this work we give the first explicit hardness of approximation result for Minimum Sum Vertex Cover. In particular, assuming the Unique Games Conjecture, we show that the Minimum Sum Vertex Cover p…
Given a graph and two integers $k$ and $\ell$, Partial Vertex Cover asks for a set of at most $k$ vertices whose deletion results in a graph with at most $\ell$ edges. Based on the expansion lemma, we provide a problem kernel with $(\ell + 2)(k + \ell)$ vertices. We then introduce a new, additive version of the expansion lemma and show it can be used to prove a kernel with $(\ell + 1)(k + \ell)$ vertices for $\ell \ge 1$.
