Vertex Cover Problem In Graphs Research Articles

Some Algorithmic Results for Eternal Vertex Cover Problem in Graphs

The eternal vertex cover problem is a variant of the vertex cover problem. It is a two-player (attacker and defender) game in which, given a graph $G=(V,E)$, the defender needs to allocate guards at some vertices so that the allocated vertices form a vertex cover. The attacker can attack one edge at a time, and the defender needs to move the guards along the edges such that at least one guard moves through the attacked edge and the new configuration still remains a vertex cover. The attacker wins if no such move exists for the defender. The defender wins if there exists a strategy to defend the graph against infinite sequence of attacks. The minimum number of guards with which the defender can form a winning strategy is called the eternal vertex cover number of $G$, and is denoted by $evc(G)$. Given a graph $G$, the problem of finding the eternal vertex cover number is NP-hard for general graphs and remains NP-hard even for bipartite graphs. We have given a polynomial time algorithm to find the Eternal vertex cover number in chain graphs and $P_4$-sparse graphs. We have also given a linear-time algorithm to find the eternal vertex cover number for split graphs, an important subclass of chordal graphs.

The relationship between attribute reducts in rough sets and minimal vertex covers of graphs

The problems to find attribute reduction in rough sets and to obtain the minimal vertex cover for graphs are both NP-hard problems. This paper studies the relationship between the two problems. The vertex cover problem for graphs from the perspective of rough sets is first investigated. The attribute reduction of an information system is then studied in the framework of graph theory. The results in this paper show that finding the minimal vertex cover of a graph is equivalent to finding the attribute reduction of an information system induced from the graph. Conversely, the attribute reduction computation can be translated into the calculation of the minimal vertex cover of a derivative graph. Finally, a new algorithm for the vertex cover problem based on rough sets is presented. Furthermore, experiments are conducted to verify the effectiveness of the proposed method.

On the parameterized vertex cover problem for graphs with perfect matching

A vertex cover of an n-vertex graph with perfect matching contains at least n/2 vertices. In this paper, we study the parameterized complexity of the problem vc-pm* that decides if a given graph with perfect matching has a vertex cover of size bounded by n/2 +k. We first present an algorithm of running time O*(4 k ) for a variation of the vertex cover problem on Konig graphs with perfect matching. This algorithm combined with the iterative compression technique leads to an algorithm of running time O*(9 k ) for the problem vc-pm*. Our result improves the previous best algorithm of running time O*(15 k ) for the vc-pm* problem, which reduces the problem to the almost 2-sat problem and solves the latter by Razgon and O’Sullivan’s recent algorithm.

Vertex cover in graphs with locally few colors

Erdős et al. defined the local chromatic number of a graph as the minimum number of colors that must appear within distance 1 of a vertex. For any Δ⩾2, there are graphs with arbitrarily large chromatic number that can be colored so that (i) no vertex neighborhood contains more than Δ different colors (bounded local colorability), and (ii) adjacent vertices from two color classes induce a complete bipartite graph (biclique coloring).We investigate the weighted vertex cover problem in graphs when a locally bounded coloring is given. This generalizes the vertex cover problem in bounded degree graphs to a class of graphs with arbitrarily large chromatic number. Assuming the Unique Game Conjecture (UGC), we provide a tight characterization. We prove that it is UGC-hard to improve the approximation ratio of 2−2/(Δ+1) on (Δ+1)-locally (but not necessarily biclique) colorable graphs. A matching upper bound is also provided. Vice versa, when properties (i) and (ii) hold, we present a randomized algorithm with approximation ratio of 2−Ω(1)lnlnΔlnΔ. This matches known inapproximability results for the special case of bounded degree graphs.Moreover, we show that when both the above two properties (i) and (ii) hold, the obtained result finds a natural application in a classical scheduling problem, namely the precedence constrained single machine scheduling problem to minimize the total weighted completion time, denoted as 1|prec|∑wjCj in standard scheduling notation. In a series of recent papers it was established that this scheduling problem is a special case of the minimum weighted vertex cover in graphs GP of incomparable pairs defined in the dimension theory of partial orders. We show that GP satisfies properties (i) and (ii) where Δ−1 is the maximum number of predecessors (or successors) of each job.

Complexity and approximation results for the connected vertex cover problem in graphs and hypergraphs

We study a variation of the vertex cover problem where it is required that the graph induced by the vertex cover is connected. We prove that this problem is polynomial in chordal graphs, has a PTAS in planar graphs, is APX-hard in bipartite graphs and is 5/3-approximable in any class of graphs where the vertex cover problem is polynomial (in particular in bipartite graphs). Finally, dealing with hypergraphs, we study the complexity and the approximability of two natural generalizations.

Faster computation of maximum independent set and parameterized vertex cover for graphs with maximum degree 3

In this paper we propose an O ( 1.0892 n ) algorithm solving the Maximum Independent Set problem for graphs with maximum degree 3 improving the previously best upper bound of O ( 1.0977 n ) . A useful secondary effect of the proposed algorithm is that being applied to 2 k kernel, it improves the upper bound on the parameterized complexity of the Vertex Cover problem for graphs with maximum degree 3 (VC-3). In particular, the new upper bound for the VC-3 problem is O ( 1.1864 k + n ) , improving the previously best upper bound of O ( k 2 ∗ 1.194 k + n ) . The presented results have a methodological interest because, to the best of our knowledge, this is the first time when a new parameterized upper bound is obtained through design and analysis of an exact exponential algorithm.

Approximated Vertex Cover for Graphs with Perfect Matchings

Chen and Kanj considered the VERTEX COVER problem for graphs with perfect matchings (VC-PM). They showed that: (i) There is a reduction from general VERTEX COVER to VC-PM, which guarantees that if one can achieve an approximation factor of less than two for VC-PM, then one can do so for general VERTEX COVER as well. (ii) There is an algorithm for VC-PM whose approximation factor is given as 1.069 + 0.069d where d is the average degree of the given graph. In this paper we improve (ii). Namely we give a new VC-PM algorithm which greatly outperforms the above one and its approximation factor is roughly 2-6.74/d + 6.28. Our algorithm also works for graphs with large matchings, although its approximation factor is degenerated.

Improved Approximation Algorithms for the Vertex Cover Problem in Graphs and Hypergraphs

We obtain improved algorithms for finding small vertex covers in bounded degree graphs and hypergraphs. We use semidefinite programming to relax the problems and introduce new} rounding techniques for these relaxations. On graphs with maximum degree at most $\Delta$, the algorithm achieves a performance ratio of $2-(1-o(1))\frac{2 \ln \ln \Delta}{\ln \Delta}$ for large $\Delta$, which improves the previously known ratio of $2-\frac{\log \Delta + O(1)}{\Delta}$ obtained by Halld{órsson and Radhakrishnan. Using similar techniques, we also present improved approximations for the vertex cover problem in hypergraphs. For k-uniform hypergraphs with n vertices, we achieve a ratio of $k-(1-o(1))\frac{k\ln \ln n}{\ln n}$ for large n, and for k-uniform hypergraphs with maximum degree at most $\Delta$ the algorithm achieves a ratio of $k-(1-o(1))\frac{k(k-1)\ln \ln \Delta}{\ln \Delta}$ for large $\Delta$. These results considerably improve the previous best ratio of $k(1-c/\Delta^\frac{1}{k-1})$ for bounded degree k-uniform hypergraphs, and $k(1-c/n^\frac{k-1}{k})$ for general k-uniform hypergraphs, both obtained by Krivelevich. Using similar techniques, we also obtain an approximation algorithm for the weighted independent set problem, matching a recent result of Halldorsson.