Nearly tight approximation bounds for vertex cover on dense k-uniform k-partite hypergraphs

We establish almost tight upper and lower approximation bounds for the Vertex Cover problem on dense k-uniform k-partite hypergraphs.     

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.     

Polynomial-time perfect matchings in dense hypergraphs

Let H be a k-graph on n   vertices, with minimum codegree at least n/k+cn$n/k+cn$ for some fixed c>0$c>0$. In this paper we construct a polynomial-time algorithm which finds either a perfect matching in H   or a certificate that none exists. This essentially solves a problem of Karpiński, Ruciński and Szymańska; Szymańska previously showed that this problem is NP-hard for a minimum codegree of n/k−cn$n/k-cn$. Our algorithm relies on a theoretical result of independent interest, in which we characterise any such hypergraph with no perfect matching using a family of lattice-based constructions.     

Fractional decompositions of dense hypergraphs

A seminal result of Rödl (the Rödl nibble) assertsthat the edges of the complete r-uniform hypergraph can be packed, almost completely, with copiesof , where k is fixed.We prove that the same result holds in a dense hypergraph setting.It is shown that for every r-uniform hypergraph H₀, there existsa constant = (H₀) < 1 such that every r-uniform hypergraphH in which every (r – 1)-set is contained in at least n edges has an H₀-packing that covers |E(H)|(1 – o_n(1))edges. Our method of proof uses fractional decompositions andmakes extensive use of probabilistic arguments and additionalcombinatorial ideas.     

Some remarks on vertex Folkman numbers for hypergraphs

Let $F(r,G)$ be the least order of $H$ such that the clique numbers of $H$ and $G$ are equal and any $r$-coloring of the vertices of $H$ yields a monochromatic and induced copy of $G$. The problem of bounding of $F(r,G)$ was studied by several authors and it is well understood. In this note, we extend those results to uniform hypergraphs.     

Properties of vertex cover obstructions

We study properties of the sets of minimal forbidden minors for the families of graphs having a vertex cover of size at most k. We denote this set by O(k-VERTEX COVER) and call it the set of obstructions. Our main result is to give a tight vertex bound of O(k-VERTEX COVER), and then confirm a conjecture made by Liu Xiong that there is a unique connected obstruction with maximum number of vertices for k-VERTEX COVER and this graph is C_2k+1. We also find two iterative methods to generate graphs in O((k+1)-VERTEX COVER) from any graph in O(k-VERTEX COVER).     

Counting dense connected hypergraphs via the probabilistic method

In 1990 Bender, Canfield, and McKay gave an asymptotic formula for the number of connected graphs on with m edges, whenever and . We give an asymptotic formula for the number of connected r‐uniform hypergraphs on with m edges, whenever is fixed and with , that is, the average degree tends to infinity. This complements recent results of Behrisch, Coja‐Oghlan, and Kang (the case ) and the present authors (the case , ie, “nullity” or “excess” o(n)). The proof is based on probabilistic methods, and in particular on a bivariate local limit theorem for the number of vertices and edges in the largest component of a certain random hypergraph. The arguments are much simpler than in the sparse case; in particular, we can use “smoothing” techniques to directly prove the local limit theorem, without needing to first prove a central limit theorem.     

A list heuristic for vertex cover

We analyze a list heuristic for the vertex cover problem that handles the vertices in a given static order based on the degree sequence. We prove an approximation ratio of at most for a nonincreasing degree sequence, and show that no ordering can achieve an approximation ratio of less than .     

Improved approximation of maximum vertex cover

We provide a new LP relaxation of the maximum vertex cover problem and a polynomial-time algorithm that finds a solution within the approximation factor , where is the size of the smallest clique in a given clique-partition of the edge weighting of G.     

Minimum vertex degree conditions for loose Hamilton cycles in 3-uniform hypergraphs

We investigate minimum vertex degree conditions for 3-uniform hypergraphs which ensure the existence of loose Hamilton cycles. A loose Hamilton cycle is a spanning cycle in which only consecutive edges intersect and these intersections consist of precisely one vertex.     

Spanning Euler tours and spanning Euler families in hypergraphs with particular vertex cuts

An Euler tour in a hypergraph is a closed walk that traverses each edge of the hypergraph exactly once, while an Euler family, first defined by Bahmanian and ?ajna, is a family of closed walks that jointly traverse each edge exactly once and cannot be concatenated. In this paper, we study the notions of a spanning Euler tour and a spanning Euler family, that is, an Euler tour (family) that also traverses each vertex of the hypergraph at least once. We examine necessary and sufficient conditions for a hypergraph to admit a spanning Euler family, most notably when the hypergraph possesses a vertex cut consisting of vertices of degree two. Moreover, we characterise hypergraphs with a vertex cut of cardinality at most two that admit a spanning Euler tour (family). This result enables us to reduce the problem of existence of a spanning Euler tour (which is NP-complete), as well as the problem of a spanning Euler family, to smaller hypergraphs.     

On the powers of vertex cover ideals

Let $S=\mathbb{K}[x_1,\dots ,x_n]$ be a polynomial ring, where $\mathbb{K}$ is a field, and G be a simple graph on n vertices. Let $J(G)?S$ be the vertex cover ideal of G. Herzog, Hibi and Ohsugi have conjectured that all powers of vertex cover ideals of chordal graph are componentwise linear. Here we establish the conjecture for the special case of trees. We also show that if G is a unicyclic vertex decomposable graph, then symbolic powers of $J(G)$ are componentwise linear.     

Approximating the maximum vertex/edge weighted clique using local search

This paper extends the recently introduced Phased Local Search (PLS) algorithm to more difficult maximum clique problems and also adapts the algorithm to handle maximum vertex/edge weighted clique instances. PLS is a stochastic reactive dynamic local search algorithm that interleaves sub-algorithms which alternate between sequences of iterative improvement, during which suitable vertices are added to the current sub-graph, and plateau search, where vertices of the current sub-graph are swapped with vertices not contained in the current sub-graph. These sub-algorithms differ in firstly their vertex selection techniques in that selection can be solely based on randomly selecting a vertex, randomly selecting within highest vertex degree, or random selecting within vertex penalties that are dynamically adjusted during the search. Secondly, the perturbation mechanism used to overcome search stagnation differs between the sub-algorithms. PLS has no problem instance dependent parameters and achieves state-of-the-art performance for maximum clique and maximum vertex/edge weighted clique problems over a large range of the commonly used DIMACS benchmark instances.