Various real-world problems consist of partitioning a set of locations into disjoint subsets, each subset spread in a way that it covers the whole set with a certain radius. Given a finite set S, a metric d, and a radius r, define a subset (of S) S' to be an r-cover if and only if forall s in S there exists s' in S' such that d(s,s') is less or equal to r. We examine the problem of determining whether there exist m disjoint r-covers, naming it the Solidarity Cover Problem (SCP). We consider as well the related optimization problems of maximizing the number of r-covers, referred to as the partition size, and minimizing the radius. We analyze the relation between the SCP and a graph problem known as the Domatic Number Problem (DNP), both hard problems in the general case. We show that the SCP is hard already in the Euclidean 2D setting, implying hardness of the DNP already in the unit-disc-graph setting. As far as we know, the latter is a result yet to be shown. We use the tight approximation bound of (1-o(1))/ln(n) for the DNP's general case, shown by U.Feige, M.Halld'orsson, G.Kortsarz, and A.Srinivasan (SIAM Journal on computing, 2002), to deduce the same bound for partition-size approximation of the SCP in the Euclidean space setting. We show an upper bound of 3 and lower bounds of 2 and sqrt(2) for approximating the minimal radius in different settings of the SCP. Lastly, in the Euclidean 2D setting we provide a general bicriteria-approximation scheme which allows a range of possibilities for trading the optimality of the radius in return for better approximation of the partition size and vice versa. We demonstrate a usage of the scheme which achieves an approximation of (1/16,2) for the partition size and radius respectively.
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: …
We present a new generalization of the bin covering problem that is known to be a strongly NP-hard problem. In our generalization there is a positive constant $Δ$, and we are given a set of items each of which has a positive size. We would like to find a partition of the items into bins. We say that a bin is near exact covered if the total size of items packed into the bin is between $1$ and $1+Δ$. Our goal is to maximize the number of near exact covered bins. If $Δ=0$ or $Δ>0$ is given as part of the input, our problem is shown here to have no approximation algorithm with a bounded asymptotic…
Vertex Cover Problem Parameterized Above and Below Tight Bounds
我们研究了著名的顶点覆盖问题的参数上下紧界。我们证明了其中两个参数化（都是由Mahajan，Raman和Sikdar，J. Computer and System Sciences，75（2）：137- 153，2009提出的）是固定参数易处理的，另外两个参数化是W[1]-hard的（其中一个实际上是W[2]-hard的）。
We show, assuming the (randomized) Gap Exponential Time Hypothesis (Gap-ETH), that the following tasks cannot be done in $T(k) \cdot N^{o(k)}$-time for any function $T$ where $N$ denote the input size: - $\left(1 - \frac{1}{e} + ε\right)$-approximation for Max $k$-Coverage for any $ε> 0$, - $\left(1 + \frac{2}{e} - ε\right)$-approximation for $k$-Median (in general metrics) for any constant $ε> 0$. - $\left(1 + \frac{8}{e} - ε\right)$-approximation for $k$-Mean (in general metrics) for any constant $ε> 0$. - Any constant factor approximation for $k$-Unique Set Cover, $k$-Nearest Codeword Probl…
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…
We study fundamental point-line covering problems in computational geometry, in which the input is a set $S$ of points in the plane. The first is the Rich Lines problem, which asks for the set of all lines that each covers at least $λ$ points from $S$, for a given integer parameter $λ\geq 2$; this problem subsumes the 3-Points-on-Line problem and the Exact Fitting problem, which -- the latter -- asks for a line containing the maximum number of points. The second is the NP-hard problem Line Cover, which asks for a set of $k$ lines that cover the points of $S$, for a given parameter $k \in \math…
Obtaining strong linear relaxations of capacitated covering problems constitute a major technical challenge even for simple settings. For one of the most basic cases, the Knapsack-Cover (Min-Knapsack) problem, the relaxation based on knapsack-cover inequalities achieves an integrality gap of 2. These inequalities have been exploited in more general environments, many of which admit primal-dual approximation algorithms. Inspired by problems from power and transport systems, we introduce a new general setting in which items can be taken fractionally to cover a given demand. The cost incurred by …
我们进行了系统的研究，用于识别跨几个领域的自然覆盖问题，在参数化的复杂性领域。在{ \sc Test Cover}问题中，我们给出了一组项目$[n]=\{1,...,n\}$以及一个集合$\cal T$，这些项目的不同子集称为测试。我们假设$\cal T$是一个测试覆盖，即，对于每对项目，在$\cal T$中有一个测试，该测试恰好包含这些项目中的一个。目标是找到$\cal T$的最小大小子集，它仍然是一个测试封面。{ \sc Test Cover}的通用参数化版本由$p(k,n,|{\cal T}|)$ -{ \sc Test Cover}表示。在这里，我们给定$([n],\cal{T})$和正整数参数$k$作为输入，目标是决定是否存在大小至多为$p(k,n,|{\cal T}|)$的测试覆盖。我们研究了{ \sc Test Cover}的四种参数化，得到如下结论：（a）$k$ -{ \sc Test Cover}和$(n-k)$ -{ \sc Test Cover}是固定参数可处理的（FPT）。(b)$(|{\cal T}|-k)$ -{ \sc Test Cover}和$(\log n+k)$ -{ \sc Test Cover}为W[1]-hard。因此，这些问题不太可能是FPT。
Several algorithms with an approximation guarantee of $O(\log n)$ are known for the Set Cover problem, where $n$ is the number of elements. We study a generalization of the Set Cover problem, called the Partition Set Cover problem. Here, the elements are partitioned into $r$ \emph{color classes}, and we are required to cover at least $k_t$ elements from each color class $\mathcal{C}_t$, using the minimum number of sets. We give a randomized LP-rounding algorithm that is an $O(β+ \log r)$ approximation for the Partition Set Cover problem. Here $β$ denotes the approximation guarantee for a relat…
Considering the set cover problem, by modifying the approach that gives a logarithmic approximation guarantee for the greedy algorithm, we obtain an estimation of the greedy algorithm's accuracy for a particular input. We compare the presented estimation to another estimations of this type. We give such examples of the set cover problem instances that the presented estimation sagnificantly improves over linear programming relaxation based estimation.
Partial set cover problem and set multi-cover problem are two generalizations of set cover problem. In this paper, we consider the partial set multi-cover problem which is a combination of them: given an element set $E$, a collection of sets $\mathcal S\subseteq 2^E$, a total covering ratio $q$ which is a constant between 0 and 1, each set $S\in\mathcal S$ is associated with a cost $c_S$, each element $e\in E$ is associated with a covering requirement $r_e$, the goal is to find a minimum cost sub-collection $\mathcal S'\subseteq\mathcal S$ to fully cover at least $q|E|$ elements, where element…
We consider single-machine scheduling problems that are natural generalizations or variations of the min-sum set cover problem and the min-sum vertex cover problem. For each of these problems, we give new approximation algorithms. Some of these algorithms rely on time-indexed LP relaxations. We show how a variant of alpha-point scheduling leads to the best-known approximation ratios, including a guarantee of 4 for an interesting special case of the so-called generalized min-sum set cover problem. We also make explicit the connection between the greedy algorithm for min-sum set cover and the co…
In this paper, we perform theoretical analyses on the behaviour of an evolutionary algorithm and a randomised search algorithm for the dynamic vertex cover problem based on its dual formulation. The dynamic vertex cover problem has already been theoretically investigated to some extent and it has been shown that using its dual formulation to represent possible solutions can lead to a better approximation behaviour. We improve some of the existing results, i.e. we find a linear expected re-optimization time for a (1+1) EA to re-discover a 2-approximation when edges are dynamically deleted from …
We study the minimum connected sensor cover problem (MIN-CSC) and the budgeted connected sensor cover (Budgeted-CSC) problem, both motivated by important applications (e.g., reduce the communication cost among sensors) in wireless sensor networks. In both problems, we are given a set of sensors and a set of target points in the Euclidean plane. In MIN-CSC, our goal is to find a set of sensors of minimum cardinality, such that all target points are covered, and all sensors can communicate with each other (i.e., the communication graph is connected). We obtain a constant factor approximation alg…
We consider the minimum vertex cover problem having applications in e.g. biochemistry and network security. Quantum annealers can find the optimum solution of such NP-hard problems, given they can be embedded on the hardware. This is often infeasible due to limitations of the hardware connectivity structure. This paper presents a decomposition algorithm for the minimum vertex cover problem: The algorithm recursively divides an arbitrary problem until the generated subproblems can be embedded and solved on the annealer. To speed up the decomposition, we propose several pruning and reduction tec…
Given a metric space $(F \cup C, d)$, we consider star covers of $C$ with balanced loads. A star is a pair $(f, C_f)$ where $f \in F$ and $C_f \subseteq C$, and the load of a star is $\sum_{c \in C_f} d(f, c)$. In minimum load $k$-star cover problem $(\mathrm{MLkSC})$, one tries to cover the set of clients $C$ using $k$ stars that minimize the maximum load of a star, and in minimum size star cover $(\mathrm{MSSC})$ one aims to find the minimum number of stars of load at most $T$ needed to cover $C$, where $T$ is a given parameter. We obtain new bicriteria approximations for the two problems us…
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.
