Partial Vertex Cover Research Articles

Approximating power node-deletion problems

In the Power Vertex Cover(PVC) problem introduced in [1] as a generalization of the well-known Vertex Cover, we are allowed to specify costs for covering edges in a graph individually. Namely, two (nonnegative) weights, w(u,v) and w(v,u), are associated with each edge {u,v}∈E of an input graph G=(V,E), and to cover an edge {u,v}, it is required to assign “power” p∈R+V on vertices of G such that either p(u)≥w(u,v) or p(v)≥w(v,u). The objective is to minimize the total power assigned on V, ∑v∈Vp(v), while covering all the edges of G by p.The node-deletion problem for a graph property π is the problem of computing a minimum vertex subset C⊆V in a given graph G=(V,E), such that the graph satisfies π when all the vertices in C are removed from G. In this paper we consider node-deletion problems extended with the “covering-by-power” condition as in PVC, and present a unified approach for effectively approximating them. The node-deletion problems considered are Partial Vertex Cover (PartVC), Bounded Degree Deletion (BDD), and Feedback Vertex Set (FVS), each corresponding to graph properties π= “the graph has at most |E|−k edges”, π= “vertex degree of v is no larger than b(v)”, and π= “the graph is acyclic”, respectively. After reducing these problems to the Submodular Set Cover (SSC) problem, we conduct an extended analysis of the approximability of these problems in the new setting of power covering by applying some of the existing techniques for approximating SSC. It will be shown that 1) PPartVC can be approximated within a factor of 2, 2) PBDD for b∈Z+V within max⁡{2,1+bmax}, where bmax=maxv∈V⁡b(v), or within 2+log⁡bmax (for bmax≥1) by a combination of the greedy SSC algorithm and the local ratio method extended for power node-deletion problems, and 3) PFVS within 2, resulting in each of these bounds matching the best one known for the corresponding original problem.

Approximation Algorithms for Partial Vertex Covers in Trees

This paper is concerned with designing algorithms for and analyzing the computational complexity of the partial vertex cover problem in trees. Graphs (and trees) are frequently used to model risk management in various systems. In particular, Caskurlu et al. in [4] have considered a system which essentially represents a tripartite graph. The goal in this model is to reduce the risk in the system below a predefined risk threshold level. It can be shown that the main goal in this risk management system can be formulated as a Partial Vertex Cover problem on bipartite graphs. In this paper, we focus on a special case of the partial vertex cover problem, when the input graph is a tree. We consider four possible versions of this setting, depending on whether or not, the vertices and edges are weighted. Two of these versions, where edges are assumed to be unweighted, are known to be polynomial-time solvable. However, the computational complexity of this problem with weighted edges, and possibly with weighted vertices, remained open. The main contribution of this paper is to resolve these questions by fully characterizing which variants of partial vertex cover remain NP-hard in trees, and which can be solved in polynomial time. In the paper, we propose two pseudo-polynomial DP-based algorithms for the most general case in which weights are present on both the edges and the vertices of the tree. One of these algorithms leads to a polynomial-time procedure, when weights are confined to the edges of the tree. The insights used in this algorithm are combined with additional scaling ideas to derive an FPTAS for the general case. A secondary contribution of this work is to propose a novel way of using centroid decompositions in trees, which could be useful in other settings as well.

On the fixed-parameter tractability of the partial vertex cover problem with a matching constraint in edge-weighted bipartite graphs

In the classical partial vertex cover problem, we are given a graph $G$ and two positive integers $k_1$ and $k_2$. The goal is to check whether there is a subset $V'$ of $V$ of size at most $k_1$, such that $V'$ covers at least $k_2$ edges of $G$. The problem is NP-hard as it includes the Vertex Cover problem. Previous research has addressed the extension of this problem where one has weight-functions defined on sets of vertices and edges of $G$. In this paper, we consider the following version of the problem where as the input we are given an edge-weighted bipartite graph $G$ with weights from $\mathbb{N}$, and three positive integers $k_1$, $k_2$ and $k_3$. The goal is to check whether $G$ has a subset $V'$ of vertices of $G$ of size at most $k_1$, such that the edges of $G$ covered by $V'$ have weight at least $k_2$ and they include a matching of weight at least $k_3$. In the paper, we address this problem from the perspective of fixed-parameter tractability and algorithms. We present some W[1]-hardness, paraNP-hardness results for our problem. On the positive side, we show that the problem is fixed-parameter tractable with respect to certain parameters. One of our W[1]-hardness results is obtained via a reduction from the bi-objective knapsack problem, which we show to be W[1]-hard with respect to one of the parameters. We believe that this problem might be useful in obtaining similar results in other situations. Keywords: partial vertex cover; bipartite graph; fixed-parameter tractability; W[1]-hardness

Tight approximation for partial vertex cover with hard capacities

We consider the partial vertex cover problem with hard capacity constraints (Partial VC-HC) on hypergraphs. In this problem we are given a hypergraph G=(V,E) with a maximum edge size f and a covering requirement R. Each edge is associated with a demand, and each vertex is associated with a capacity and an (integral) available multiplicity. The objective is to compute a minimum vertex multiset such that at least R units of demand from the edges are covered by the capacities of the vertices in the multiset and the multiplicity of each vertex does not exceed its available multiplicity.In this paper we present an f-approximation for this problem, improving over a previous result of (2f+2)(1+ϵ) by Cheung et al. to the tight extent possible. Our new ingredient of this work is a generalized analysis on the extreme points of the natural LP, developed from previous works, and a strengthened LP lower-bound obtained for the optimal solutions.

An Improved Memetic Algorithm for the Partial Vertex Cover Problem

As a generalized form of the classical minimum vertex cover problem (MVCP), the minimum partial vertex cover problem not only inherits the value of MVCP in both theoretical and practical research but also extends the application scenarios under the current era of big data. In order to solve such a classical NP-hard combinatorial optimization problem, we propose an efficient memetic algorithm called MAPVC , the whose motivation behind the idea is a balance between exploitation and exploration. Specifically, the evolutionary process with a powerful backbone crossover and an adaptive mutation is adopted mainly for skipping the local optimum, and the local search with a three-layer encoding, promotive and inhibitory factor interaction is integrated into the framework to ameliorate the solutions. The experimental results on DIMACS and BHOSLIB benchmarks verify the efficiency of MAPVC through comparisons with an exact solver Gurobi and state-of-the-art local search GRASP-PVC . In addition, we also conduct the experiments on parameters setting and strategies validation to provide a comprehensive analysis of MAPVC .

Partial Vertex Cover and Budgeted Maximum Coverage in Bipartite Graphs

In this paper, we study two closely related problems on bipartite graphs, viz., the partial vertex cover problem and the budgeted maximum coverage problem. Both these problems arise in a number of ...

An efficient local search for partial vertex cover problem

In this paper, an efficient local search framework, namely GRASP-PVC, is proposed to solve the minimum partial vertex cover problem. In order to speed up the convergence, a novel least-cost vertex selecting strategy is applied into GRASP-PVC. As far as we know, no heuristic algorithms have ever been reported to solve this momentous problem and we compare GRASP-PVC with a commercial integer programming solver CPLEX as well as a 2-approximation algorithm on two standard benchmark libraries called DIMACS and BHOSLIB. Experimental results evince that GRASP-PVC finds much better partial vertex covers than CPLEX and the approximation algorithm on most instances. Additional experimental results also confirm the validity of the least-cost vertex selecting strategy.

An approximation algorithm for the partial vertex cover problem in hypergraphs

Let $$\mathcal {H}=(V,\mathcal {E})$$H=(V,E) be a hypergraph with set of vertices $$V, n:=|V|$$V,n:=|V| and set of (hyper-)edges $$\mathcal {E}, m:=|\mathcal {E}|$$E,m:=|E|. Let $$l$$l be the maxim...

Approximation algorithms for the partition vertex cover problem

We consider a natural generalization of the Partial Vertex Cover problem. Here an instance consists of a graph G=(V,E), a cost function c:V→Z+, a partition P1,…,Pr of the edge set E, and a parameter ki for each partition Pi. The objective is to find a minimum cost set of vertices which cover at least ki edges from the partition Pi. We call this the Partition-VC problem. In this paper, we give matching upper and lower bound on the approximability of this problem. Our algorithm is based on a novel LP relaxation for this problem. This LP relaxation is obtained by adding knapsack cover inequalities to a natural LP relaxation of the problem. We show that this LP has integrality gap of O(log⁡r), where r is the number of sets in the partition of the edge set. We also extend our result to more general settings. For example we consider a problem where additionally edges have profits, and we would like to pick a minimum cost set of vertices which cover edges of total profit at least Πi for each partition Pi. We call this the Knapsack Partition Vertex Cover problem. We give an O(log⁡r) approximation algorithm for this problem as well.

On the [formula omitted]-edge-incident subgraph problem and its variants

Given a node-weighted graph G=(V,E) and an integer k, the k-edge-incident subgraph problem requires one to find a vertex set S⊆V of maximum weight that covers at most k edges, and the minimum partial vertex cover problem requires one to find a set of k vertices that covers the minimum number of edges. These two problems are closely related to the well-studied densestk-subgraph problem, and are interesting on their own. In this paper, we study these two problems from an approximation point of view. We obtain the following results. 1.For the k-edge-incident subgraph problem, we present a (2+ε) approximation algorithm for any fixed ε>0, which improves the previous best approximation ratio of 3 and matches that of its unweighted version [O. Goldschmidt and D.S. Hochbaum, k-edge subgraph problems, Discrete Appl. Math. 74 (2) (1997) 159–169].2.For the minimum partial vertex cover problem, we give a 2-approximation algorithm. We then propose a polynomial-time approximation scheme (PTAS) for it on the class of everywhere-c-dense graphs on which many well-studied combinatorial problems have been investigated.

Analytical models for risk-based intrusion response

Risk analysis has been used to manage the security of systems for several decades. However, its use has been limited to offline risk computation and manual response. In contrast, we use risk computation to drive changes in an operating system’s security configuration. This allows risk management to occur in real time and reduces the window of exposure to attack. We posit that it is possible to protect a system by reducing its functionality temporarily when it is under siege. Our goal is to minimize the tension between security and usability by trading them dynamically. Instead of statically configuring a system, we aim to monitor the risk level, using it to drive the tradeoff between security and utility. The advantage of this approach is that it provides users with the maximum possible functionality for any predefined level of risk tolerance.Risk management can be framed as an exercise in managing the constraints on edge and vertex weights of a tripartite graph, with the partitions corresponding to the threats, vulnerabilities, and assets in the system. If a threat requires a specific permission and affects a particular asset, an edge is added between the threat and the permission that mediates access to the vulnerable resource. Another edge is added between the permission and the asset. The presence of a path from a threat, through a permission check, to an asset contributes an element of risk. Risk can be reduced by denying access to a resource that contains a vulnerability or activating data protection measures. We first show that algorithmic underpinnings of optimal risk management can be formulated as the Partial Vertex Cover (PVC) problem in bipartite graphs. We then experimentally compare several heuristics and a 1+22+∊-approximation algorithm we designed for the problem.

Subexponential algorithms for partial cover problems

Partial Cover problems are optimization versions of fundamental and well-studied problems like Vertex Cover and Dominating Set. Here one is interested in covering (or dominating) the maximum number of edges (or vertices) using a given number k of vertices, rather than covering all edges (or vertices). In general graphs, these problems are hard for parameterized complexity classes when parameterized by k. It was recently shown by Amini et al. (2008) [1] that Partial Vertex Cover and Partial Dominating Set are fixed parameter tractable on large classes of sparse graphs, namely H-minor-free graphs, which include planar graphs and graphs of bounded genus. In particular, it was shown that on planar graphs both problems can be solved in time 2 O ( k ) n O ( 1 ) . During the last decade there has been an extensive study on parameterized subexponential algorithms. In particular, it was shown that the classical Vertex Cover and Dominating Set problems can be solved in subexponential time on H-minor-free graphs. The techniques developed to obtain subexponential algorithms for classical problems do not apply to partial cover problems. It was left as an open problem by Amini et al. (2008) [1] whether there is a subexponential algorithm for Partial Vertex Cover and Partial Dominating Set. In this paper, we answer the question affirmatively by solving both problems in time 2 O ( k ) n O ( 1 ) not only on planar graphs but also on much larger classes of graphs, namely, apex-minor-free graphs. Compared to previously known algorithms for these problems our algorithms are significantly faster and simpler.

Local search with edge weighting and configuration checking heuristics for minimum vertex cover

The Minimum Vertex Cover (MVC) problem is a well-known combinatorial optimization problem of great importance in theory and applications. In recent years, local search has been shown to be an effective and promising approach to solve hard problems, such as MVC. In this paper, we introduce two new local search algorithms for MVC, called EWLS (Edge Weighting Local Search) and EWCC (Edge Weighting Configuration Checking). The first algorithm EWLS is an iterated local search algorithm that works with a partial vertex cover, and utilizes an edge weighting scheme which updates edge weights when getting stuck in local optima. Nevertheless, EWLS has an instance-dependent parameter. Further, we propose a strategy called Configuration Checking for handling the cycling problem in local search. This is used in designing a more efficient algorithm that has no instance-dependent parameters, which is referred to as EWCC. Unlike previous vertex-based heuristics, the configuration checking strategy considers the induced subgraph configurations when selecting a vertex to add into the current candidate solution. A detailed experimental study is carried out using the well-known DIMACS and BHOSLIB benchmarks. The experimental results conclude that EWLS and EWCC are largely competitive on DIMACS benchmarks, where they outperform other current best heuristic algorithms on most hard instances, and dominate on the hard random BHOSLIB benchmarks. Moreover, EWCC makes a significant improvement over EWLS, while both EWLS and EWCC set a new record on a twenty-year challenge instance. Further, EWCC performs quite well even on structured instances in comparison to the best exact algorithm we know. We also study the run-time behavior of EWLS and EWCC which shows interesting properties of both algorithms.

EWLS: A New Local Search for Minimum Vertex Cover

A number of algorithms have been proposed for the Minimum Vertex Cover problem. However, they are far from satisfactory, especially on hard instances. In this paper, we introduce Edge Weighting Local Search (EWLS), a new local search algorithm for the Minimum Vertex Cover problem. EWLS is based on the idea of extending a partial vertex cover into a vertex cover. A key point of EWLS is to find a vertex set that provides a tight upper bound on the size of the minimum vertex cover. To this purpose, EWLS employs an iterated local search procedure, using an edge weighting scheme which updates edge weights when stuck in local optima. Moreover, some sophisticated search strategies have been taken to improve the quality of local optima. Experimental results on the broadly used DIMACS benchmark show that EWLS is competitive with the current best heuristic algorithms, and outperforms them on hard instances. Furthermore, on a suite of difficult benchmarks, EWLS delivers the best results and sets a new record on the largest instance.

A Primal-Dual Approximation Algorithm for Partial Vertex Cover: Making Educated Guesses

We study the partial vertex cover problem. Given a graph G=(V,E), a weight function w:V→R +, and an integer s, our goal is to cover all but s edges, by picking a set of vertices with minimum weight. The problem is clearly NP-hard as it generalizes the well-known vertex cover problem. We provide a primal-dual 2-approximation algorithm which runs in O(nlog n+m) time. This represents an improvement in running time from the previously known fastest algorithm. Our technique can also be used to get a 2-approximation for a more general version of the problem. In the partial capacitated vertex cover problem each vertex u comes with a capacity k u . A solution consists of a function x:V→ℕ0 and an orientation of all but s edges, such that the number of edges oriented toward vertex u is at most x u k u . Our objective is to find a cover that minimizes ∑v∈V x v w v . This is the first 2-approximation for the problem and also runs in O(nlog n+m) time.

Parameterized Complexity of Vertex Cover Variants

Important variants of theVERTEX COVER problem (among others, CONNECTED VERTEX COVER, CAPACITATED VERTEX COVER, and MAXIMUM PARTIAL VERTEX COVER) have been intensively studied in terms of polynomial-time approximability. By way of contrast, their parameterized complexity has so far been completely open. We close this gap here by showing that, with the size of the desired vertex cover as the parameter, CONNECTED VERTEX COVER and CAPACITATED VERTEX COVER are both fixed-parameter tractable while MAXIMUM PARTIAL VERTEX COVER is W[1]-complete. This answers two open questions from the literature. The results extend to several closely related problems. Interestingly, although the considered variants of VERTEX COVER behave very similar in terms of constant factor approximability, they display a wide range of different characteristics when investigating their parameterized complexities.

On Combinatorial Approximation of Covering 0-1 Integer Programs and Partial Set Cover

The problems dealt with in this paper are generalizations of the set cover problem, min{cx | Ax ≥ b, x ∈{0,1} n }, where c ∈ Q+ n , A ∈ {0,1}m × n, b ∈ 1. The covering 0-1 integer program is the one, in this formulation, with arbitrary nonnegative entries of A and b, while the partial set cover problem requires only m–K constrains (or more) in Ax ≥ b to be satisfied when integer K is additionall specified. While many approximation algorithms have been recently developed for these problems and their special cases, using computationally rather expensive (albeit polynomial) LP-rounding (or SDP-rounding), we present a more efficient purely combinatorial algorithm and investigate its approximation capability for them. It will be shown that, when compared with the best performance known today and obtained by rounding methods, although its performance comes short in some special cases, it is at least equally good in general, extends for partial vertex cover, and improves for weighted multicover, partial set cover, and further generalizations.

Using Homogeneous Weights for Approximating the Partial Cover Problem

In this paper we consider the natural generalizations of two fundamental problems, the Set-Cover problem and the Min-Knapsack problem. We are given a hypergraph, each vertex of which has a nonnegative weight, and each edge of which has a nonnegative length. For a given threshold ℓ̂, our objective is to find a subset of the vertices with minimum total cost, such that at least a length of ℓ̂ of the edges is covered. This problem is called the partial set cover problem. We present an O(|V|2+|H|)-time, ΔE-approximation algorithm for this problem, where ΔE≥2 is an upper bound on the edge cardinality of the hypergraph and |H| is the size of the hypergraph (i.e., the sum of all its edges cardinalities). The special case where ΔE=2 is called the partial vertex cover problem. For this problem a 2-approximation was previously known, however, the time complexity of our solution, i.e., O(|V|2), is a dramatic improvement.We show that if the weights are homogeneous (i.e., proportional to the potential coverage of the sets) then any minimal cover is a good approximation. Now, using the local-ratio technique, it is sufficient to repeatedly subtract a homogeneous weight function from the given weight function.

On approximation of the submodular set cover problem

We design a primal-dural heuristic for the submodular set cover problem and analyze its performance giving an approximation bound as a generalization of the one for the set cover problem. As an application, a capacitated version of the partial vertex cover problem on hypergraphs with edge size at most d is considered. It will be shown that the problem can be approximated in polynomial time within a factor of d of the optimum, generalizing some existing results.