The 3-Hitting Set problem, also called the Vertex Cover problem on 3-uniform hypergraphs, cannot be solved in polynomial time unless P=NP. However, this problem is fixed-parameter tractable in the parameterized complexity, which means that it has kernelizations. In this paper, we address such kernelizations of the Vertex Cover problem on 3-uniform hypergraphs. Moreover, we show that this proble...

For a class of graphs P, the Bounded P-Block Vertex Deletion problem asks, given a graph G on n vertices and positive integers k and d, whether there is a set S of at most k vertices such that each block of G−S has at most d vertices and is in P. We show that when P satisfies a natural hereditary property and is recognizable in polynomial time, Bounded P-Block Vertex Deletion can be solved in t...

A circular cover of a graph G is a cover {X0, · · · , Xn−1} of the topological space G by closed connected subsets, indexed over Zn, with the following properties: Each element in the cover contains a vertex of G, each vertex of G is contained in at most two elements of the cover, and Xa ∩ Xb 6= ∅ if and only if b − a ∈ {−1, 0, 1}. The circularity of G is the largest integer n for which there i...

A subset S of vertices of a graph G = (V,E) is called a k-path vertex cover if every path on k vertices in G contains at least one vertex from S. Denote by ψk(G) the minimum cardinality of a k-path vertex cover in G and form a sequence ψ(G) = (ψ1(G), ψ2(G), . . . , ψ|V |(G)), called the path sequence of G. In this paper we prove necessary and sufficient conditions for two integers to appear on ...

In this paper we take a closer look at the parameterized complexity of ∃∀SAT, the prototypical complete problem of the class Σp2, the second level of the polynomial hierarchy. We provide a number of tight fine-grained bounds on the complexity of this problem and its variants with respect to the most important structural graph parameters. Specifically, we show the following lower bounds (assumin...

The inapproximability of non NP-hard optimization problems is investigated. Techniques are given to show that problems Log Dominating Set and Log Hypergraph Vertex Cover cannot be approximated to a constant ratio in polynomial time unless the corresponding NP-hard versions are also approximable in deterministic subexponential time. A direct connection is established between non NP-hard problems...

Computing a minimum vertex cover in graphs and hypergraphs is a well-studied optimizaton problem. While intractable in general, it is well known that on bipartite graphs, vertex cover is polynomial time solvable. In this work, we study the natural extension of bipartite vertex cover to hypergraphs, namely finding a small vertex cover in kuniform k-partite hypergraphs, when the k-partition is gi...

Given a graph G = (V,E) with n vertices and an integer k 6 n, Max k-Vertex Cover consists of determining a subset K ⊆ V that covers the most of edges in E. We study the polynomial approximation of Max k-Vertex Cover in bipartite graphs by a purely combinatorial algorithm and present an experimental analysis of it that finds the worst case approximation guarantee that is bounded below by 0.794. ...

In this paper we focus on problems which do not admit a constant-factor approximation in polynomial time and explore how quickly their approximability improves as the allowed running time is gradually increased from polynomial to (sub-)exponential. We tackle a number of problems: For Min Independent Dominating Set, Max Induced Path, Forest and Tree, for any r(n), a simple, known scheme gives an...

The paper researched the DNA computing method of classical optimization problems on weighted graph, improved the method of Weight-coding that belongs to the original model of DNA, raised some new methods of DNA encoding and DNA computing. Generally speaking, through the Relative length graph design that belongs to an Undirected weighted graph , given DNA coding and DNA computing a new method by...