Vertex Cover

A Minimum Vertex Cover is the smallest set of vertices whose removal completely disconnects a graph. In this paper, we perform experiments on a number of graphs from standard complex networks databases addressing the problem of finding a "good" vertex cover (finding an optimum is a NP-Hard problem). In particular, we take advantage of the ubiquitous power law distribution present on many complex networks. In our experiments, we show that running a greedy algorithm in a power law graph we can obtain a very small vertex cover typically about 1.02 times the theoretical optimum. This is an interesting practical result since theoretically we know that: (1) In a general graph, on n vertices a greedy approach cannot guarantee a factor better than ln n; (2) The best approximation algorithm known at the moment is very involved and has a much larger factor of [Formula: see text]. In fact, in the context of approximation within a constant factor, it is conjectured that there is no (2...

We introduce a novel stochastic local search algorithm for the vertex cover problem. Compared to current exhaustive search techniques, our algorithm achieves excellent performance on a suite of problems drawn from the field of biology. We also evaluate our performance on the commonly used DIMACS benchmarks for the related clique problem, finding that our approach is competitive with the current best stochastic local search algorithm for finding cliques. On three very large problem instances, our algorithm establishes new records in solution quality.

Let C be a clutter and let I be its edge ideal. We present a combinatorial description of the minimal generators of the symbolic Rees algebra Rs(I) of I. It is shown that the minimal generators of Rs(I) are in one to one correspondence with the irreducible parallelizations of C. From our description some major results on symbolic Rees algebras of perfect graphs and clutters will follow. As a byproduct, we give a method, using Hilbert bases, to compute all irreducible parallelizations of C along with all the corresponding vertex covering numbers.

We give a new proof of K\"onig's theorem and generalize the Gallai-Edmonds decomposition to balanced hypergraphs in two different ways. Based on our decompositions we give two new characterizations of balanced hypergraphs and show some properties of matchings and vertex cover in balanced hypergraphs. Comment: 10 pages

We explore opportunities for parameterising constant factor approximation algorithms for vertex cover. We provide a simple algorithm that works on any approximation ratio of the form \frac 2l+1l+1\frac {2l+1}{l+1} and has complexity that outperforms an algorithm by Bourgeois et al. derived from a sophisticated exact parameterised algorithm. In particular, for l = 1 (factor 1.5 approximation) our algorithm runs in time O*(1.09k)\mathcal{O}^{*}(1.09^{k}).

The remarkable discovery of many large-scale real networks is the power-law distribution in degree sequence: the number of vertices with degree i is proportional to i− β for some constant β> 1. A lot of researchers believe that it may be easier to solve some optimization problems in power-law graphs. Unfortunately, many problems have been proved NP-hard even in power-law graphs. Intuitively, a theoretical question is raised: are these problems on power-law graphs still as hard as on general graphs? In this paper, we show that many ...

A function � : V ! {1,...,k} is a broadcast coloring of order k if �(u) = �(v) implies that the distance between u and v is more than �(u). The minimum order of a broadcast coloring is called the broadcast chromatic number of G, and is denotedb(G). In this pa- per we introduce this coloring and study its properties.

Multiple-interval graphs are a natural generalization of interval graphs where each vertex may have more then one interval associated with it. We initiate the study of optimization problems in multiple-interval graphs by considering three classical problems: Minimum Vertex Cover, Minimum Dominating Set, and Maximum Clique. We describe applications for each one of these problems, and then proceed to discuss approximation

The minimum vertex cover (MVC) and maximum independent set (MIS) problems are to be determined in terms of a graph
of the small set of vertices, which cover all the edges, and a large set of vertices, no two of which are adjacent. MVC and MIS are
notable for its capability of modelling other combinatorial problems and real-world applications. The aim of this paper is twofold: first
to investigate failures of the state-of- the-art algorithms for MVC problem on small graphs and second to propose a simple and efficient
approximation algorithm for the minimum vertex cover problem. Mostly the state of art approximation algorithms for the MVC problem
are based on greedy approaches, or inspired from MIS approaches. Hence, these approaches regularly fail to provide optimal results on
specific graph instances. These problems motivated to propose Max Degres Around (MDA) approximation algorithm for the MVC
problem. The proposed algorithm is simple and efficient than the other heuristic algorithms for the MVC problem. In this paper, we have
introduced small benchmark instances and some state of the art algorithms (MDG, VSA, and MVSA) have been tested along with the
proposed algorithm. The proposed algorithm performed well as compared to counterpart algorithms tested on graphs with up to 1000
vertices and 150,000 vertices for Minimum Vertex Cover (MVC) and Maximum Independent Set (MIS).

The MINIMUM 2SAT-DELETION problem is to delete the minimum number of clauses in a 2SAT instance to make it satisfiable. It is one of the prototypes in the approximability hierarchy of minimization problems Khanna et al. [Constraint satisfaction: the approximability of minimization problems, Proceedings of the 12th Annual IEEE Conference on Computational Complexity, Ulm, Germany, 24–27 June, 1997, pp. 282–296], and its approximability is largely open. We prove a lower approximation bound of 85-15≈2.88854, improving the previous bound of 105-21≈1.36067 by Dinur and Safra [The importance of being biased, Proceedings of the 34th Annual ACM Symposium on Theory of Computing (STOC), May 2002, pp. 33–42, also ECCC Report TR01-104, 2001]. For highly restricted instances with exactly four occurrences of every variable we provide a lower bound of 32. Both inapproximability results apply to instances with no mixed clauses (the literals in every clause are both either negated, or unnegated).We further prove that any k-approximation algorithm for the MINIMUM 2SAT-DELETION problem polynomially reduces to a (2-2/(k+1))(2-2/(k+1))-approximation algorithm for the MINIMUM VERTEX COVER problem.One ingredient of these improvements is our proof that the MINIMUM VERTEX COVER problem is hardest to approximate on graphs with perfect matching. More precisely, the problem to design a ρρ-approximation algorithm for the MINIMUM VERTEX COVER on general graphs polynomially reduces to the same problem on graphs with perfect matching. This improves also on the results by Chen and Kanj [On approximating minimum vertex cover for graphs with perfect matching, Proceedings of the 11st ISAAC, Taipei, Taiwan, Lecture Notes in Computer Science, vol. 1969, Springer, Berlin, 2000, pp. 132–143].

An odd graceful labeling of a graph G = (V ,E ) is a function f :V (G)® {0,1,2, . . .2 E(G) −1}such that |f (u)−f (v)| is odd value less than or equal to 2 E(G) −1 for any u ,v ÎV (G) . In spite of the large number of papers published on the subject of graph labeling, there are few algorithms to be used by researchers to gracefully label graphs. This work provides generalized odd graceful solutions to all the vertices and edges for the tensor product of the two paths n P and m P denoted n m P Ù P . Firstly, we describe an algorithm to
label the vertices and the edges of the vertex set ( ) n m V P ÙP and the edge set ( ) n m E P Ù P respectively.
Finally, we prove that the graph n m P ÙP is odd graceful for all integers n and m .

In this paper, we show that the number of edges for any odd harmonious Eulerian graph is congruent to 0 or 2 (mod 4), and we found a counter example for the inverse of this statement is not true. We also proved that, the graphs which are constructed by two copies of even cycle Cn sharing a common edge are odd  harmonious. In addition, we obtained an odd harmonious labeling for the graphs which are constructed by two copies of cycle Cn sharing a common vertex when n is congruent to 0 (mod 4). Moreover, we show
that, the Cartesian product of cycle graph Cm and path Pn for each n ≥ ,2 m ≡ 0 (mod )4 are odd harmonious graphs. Finally many new families of odd harmonious graphs are introduced.

A variety of efficient kernelization strategies for the classic vertex cover problem are developed, implemented and compared experimentally. A new technique, termed crown reduction, is introduced and analyzed. Applications to computational biology are ...

Wireless sensor and actor networks (WSANs) consist of powerful actors and resource constraint sensors that are linked together in wireless networks. They mostly rely on actors to make proper decisions and perform desired coordination to achieve the goals of the entire network. They are usually deployed in critical applications and actor-actor network connectivity is thus vital to their effective utilization. Since WSAN applications are mostly deployed in harsh environments, actor nodes may fail and so partition their network. We propose a comparatively more efficient distributed approach, nicknamed AOM, to restore actor-actor connectivity upon the failure of any actor. We identify critical actors by combining the result of determining critical actors using the Stojmenovich's method with the connectivity dominating set (CDS) of the network. This hybrid method of detecting critical actors helps in detecting critical nodes and candidate replacement actors more precisely while minimizing the total number of required messages for network restoration. The failure handling of actors is done in a proactive manner. Our proposed method minimizes both the restoration time of network and the total number of actor movements. When a failed actor is a critical node, actors in its neighborhood are relocated in a coordinated way to reconnect the actor network. The superiority of our approach compared to other works is shown by simulative experiments measuring two important parameters to WSANS, namely, the total number of transmitted messages and the total number of actor movements during actor-actor network reconnection process.

Given a graph and an integer k, the biclique cover problem asks whether the edge-set of the given graph can be covered with at most k bicliques (complete bipartite subgraphs); the biclique vertex-cover problem asks whether the vertex-set of the given graph can be covered with at most k bicliques. Both problems are known to be NP-complete even if the given graph is bipartite. In this paper we investigate these two problems in the framework of parameterized complexity: do the problems become easier if k is assumed to be small? We show that, considering k as the parameter, the first problem is fixed-parameter tractable, while the second one is not fixed-parameter tractable unless P=NP.

Broadcasting in processor networks means disseminating a single piece of information, which is originally known only at some nodes, to all members of the network. The goal is to inform everybody using as few rounds as possible, that is to minimize the broadcasting time. Given a graph and a subset of nodes, the sources, the problem to determine its specific broadcast time, or more general to find a broadcast schedule of minimal length has been shown to be NP-complete. In contrast to other optimization problems for graphs, like vertex cover or traveling salesman, little was known about restricted graph classes for which polynomial time algorithms exist, for example for graphs of bounded treewidth. The broadcasting problem is harder in this respect because it does not have the finite state property. Here, we will investigate this problem in detail and prove that it remains hard even if one restricts to planar graphs of bounded degree or constant broadcasting time. A simple consequence is that the minimal broadcasting time cannot even be approximated with an error less than 1/8, unless P=NP. On the other hand, we will investigate for which classes of graphs this problem can be solved efficiently and show that broadcasting and even a more general version of this problem becomes easy for graphs with good decomposition properties. The solution strategy can efficiently be parallelized, too. Combining the negative and the positive results reveals the parameters that make broadcasting difficult. Depending on simple graph properties the complexity jumps from NC or P to NP.

In this paper we present a detailed study of the hitting set (HS) problem. This problem is a generalization of the standard vertex cover to hypergraphs: one seeks a configuration of particles with minimal density such that every hyperedge of the hypergraph contains at least one particle. It can also be used in important practical tasks, such as the group testing procedures where one wants to detect defective items in a large group by pool testing. Using a statistical mechanics approach based on the cavity method, we study the phase diagram of the HS problem, in the case of random regular hypergraphs. Depending on the values of the variables and tests degrees different situations can occur: The HS problem can be either in a replica symmetric phase, or in a one-step replica symmetry breaking one. In these two cases, we give explicit results on the minimal density of particles, and the structure of the phase space. These problems are thus in some sense simpler than the original vertex ...

Abstract: The framework of Bodlaender et al.(ICALP 2008) and Fortnow and Santhanam (STOC 2008) allows us to exclude the existence of polynomial kernels for a range of problems under reasonable complexity-theoretical assumptions. However, there are also some issues that are not addressed by this framework, including the existence of Turing kernels such as the" kernelization" of Leaf Out Branching (k) into a disjunction over n instances of size poly (k). Observing that Turing kernels are preserved by polynomial ...