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In the fractional set cover problem, it is allowed to select fractions of sets, rather than entire sets. A fractional set cover is an assignment of a fraction (a number in [0,1]) to each set in , such that for each element x in the universe, the sum of fractions of sets that contain x is at least 1. The goal is to find a fractional set cover in ...
A subset Q of J is called a rainbow set if it contains at most a single interval of each color. A set of intervals J is called a covering of P if each point in P is contained in at least one interval of Q. The Rainbow covering problem is the problem of finding a rainbow set Q that is a covering of P. The problem is NP-hard (by reduction from ...
In the mathematical discipline of graph theory, a graph C is a covering graph of another graph G if there is a covering map from the vertex set of C to the vertex set of G. A covering map f is a surjection and a local isomorphism: the neighbourhood of a vertex v in C is mapped bijectively onto the neighbourhood of in G. The term lift is ...
A vertex-cover (aka hitting set or transversal) in H is set T ⊆ V such that, for all hyperedges e ∈ E, it holds that T ∩ e ≠ ∅. The vertex-cover number (aka transversal number) of a hypergraph H is the smallest size of a vertex cover in H. It is often denoted by τ(H). [1]: 466 For example, if H is this 3-uniform hypergraph:
The Hasse diagram depicting the covering relation of a Tamari lattice is the skeleton of an associahedron. The covering relation of any finite distributive lattice forms a median graph. On the real numbers with the usual total order ≤, the cover set is empty: no number covers another.
The maximum set packing need not cover every possible element. In the exact cover problem, every element of should be contained in exactly one of the subsets. Finding such an exact cover is an NP-complete problem, even in the special case in which the size of all sets is 3 (this special case is called exact 3 cover or X3C).
A set of cliques that cover all edges of a graph is called a clique edge cover [3] or edge clique cover, [4] or even just a clique cover, although the last term is ambiguous: a clique cover can also be a set of cliques that cover all vertices of a graph. [5] Sometimes "covering" is used in place of "cover". [6] As well as being called the ...
The free group G = π 1 (X) has n = 2 generators corresponding to loops a,b from the base point P in X.The subgroup H of even-length words, with index e = [G : H] = 2, corresponds to the covering graph Y with two vertices corresponding to the cosets H and H' = aH = bH = a −1 H = b − 1 H, and two lifted edges for each of the original loop-edges a,b.