Approximation Algorithm For Vertex Cover Research Articles

Approximation Algorithm for Vertex Cover with Multiple Covering Constraints

We consider the vertex cover problem with multiple coverage constraints in hypergraphs. In this problem, we are given a hypergraph \(G=(V,E)\) with a maximum edge size f, a cost function \(w: V\rightarrow {\mathbb {Z}}^+\), and edge subsets \(P_1,P_2,\ldots ,P_r\) of E along with covering requirements \(k_1,k_2,\ldots ,k_r\) for each subset. The objective is to find a minimum cost subset S of V such that, for each edge subset \(P_i\), at least \(k_i\) edges of it are covered by S. This problem is a basic yet general form of classical vertex cover problem and the edge-partitioned vertex cover problem considered by Bera et al. We present a primal-dual algorithm yielding an \(\left( f \cdot H_r + H_r\right) \)-approximation for this problem, where \(H_r\) is the \(r^{th}\) harmonic number. This improves over the previous ratio of \((3cf\log r)\), where c is a large constant used to ensure a low failure probability for Monte-Carlo randomized algorithms. Compared to the previous result, our algorithm is deterministic and pure combinatorial, meaning that no Ellipsoid solver is required for this basic problem. Our result can be seen as a novel reinterpretation of a few classical tight results using the language of LP primal-duality.

A fault-containing self-stabilizing [formula omitted]-approximation algorithm for vertex cover in anonymous networks

The non-computability of many distributed tasks in anonymous networks is well known. This paper presents a deterministic self-stabilizing algorithm to compute a ( 3 − 2 Δ + 1 ) -approximation of a minimum vertex cover in anonymous networks. The algorithm operates under the distributed unfair scheduler, stabilizes after O ( n + m ) moves respectively O ( Δ ) rounds, and requires O ( log n ) storage per node. Recovery from a single fault is reached within a constant time and the contamination number is O ( Δ ) . For trees the algorithm computes a 2 -approximation of a minimum vertex cover.

Divide-and-Conquer Approximation Algorithm for Vertex Cover

The vertex cover problem is a classical NP-complete problem for which the best worst-case approximation ratio is $2-o(1)$. In this paper, we use a collection of simple graph transformations, each of which guarantees an approximation ratio of $\frac{3}{2}$, to find approximate vertex covers for a large collection of randomly generated graphs and test graphs from various sources. The graph reductions are extremely fast, and even though they by themselves are not guaranteed to find a vertex cover, we manage to find a $\frac{3}{2}$-approximate vertex cover for almost every single graph in our collection. The few graphs that we cannot solve have specific structure: they are triangle-free, with a minimum degree of at least 3, a lower bound of $\frac{n}{2}$ on the optimal vertex cover, and are unlikely to have a large bipartite subgraph.

On hard instances of approximate vertex cover

We show that if there is a 2 - ϵ approximation algorithm for vertex cover on graphs with vector chromatic number at most 2 + δ, then there is a 2 - f (ϵ, δ) approximation algorithm for vertex cover for all graphs.

An improved approximation algorithm for vertex cover with hard capacities

We study the capacitated vertex cover problem, a generalization of the well-known vertex-cover problem. Given a graph G = ( V , E ) , the goal is to cover all the edges by picking a minimum cover using the vertices. When we pick a vertex, we can cover up to a pre-specified number of edges incident on this vertex (its capacity). The problem is clearly NP-hard as it generalizes the well-known vertex-cover problem. Previously, approximation algorithms with an approximation factor of 2 were developed with the assumption that an arbitrary number of copies of a vertex may be chosen in the cover. If we are allowed to pick at most a fixed number of copies of each vertex, the approximation algorithm becomes much more complex. Chuzhoy and Naor (FOCS, 2002) have shown that the weighted version of this problem is at least as hard as set cover; in addition, they developed a 3-approximation algorithm for the unweighted version. We give a 2-approximation algorithm for the unweighted version, improving the Chuzhoy–Naor bound of three and matching (up to lower-order terms) the best approximation ratio known for the vertex-cover problem.