Skip to main content
`Ctrl`+`K`

# Source code for networkx.algorithms.approximation.vertex_cover

    """Functions for computing an approximate minimum weight vertex cover.

    A |vertex cover|_ is a subset of nodes such that each edge in the graph
    is incident to at least one node in the subset.

    .. _vertex cover: https://en.wikipedia.org/wiki/Vertex_cover
    .. |vertex cover| replace:: *vertex cover*

    """

    import networkx as nx

    __all__ = ["min_weighted_vertex_cover"]

    @nx._dispatchable(node_attrs="weight")
    def min_weighted_vertex_cover(G, weight=None):
        r"""Returns an approximate minimum weighted vertex cover.

        The set of nodes returned by this function is guaranteed to be a
        vertex cover, and the total weight of the set is guaranteed to be at
        most twice the total weight of the minimum weight vertex cover. In
        other words,

        .. math::

           w(S) \leq 2 * w(S^*),

        where $S$ is the vertex cover returned by this function,
        $S^*$ is the vertex cover of minimum weight out of all vertex
        covers of the graph, and $w$ is the function that computes the
        sum of the weights of each node in that given set.

        Parameters
        ----------
        G : NetworkX graph

        weight : string, optional (default = None)
            If None, every node has weight 1. If a string, use this node
            attribute as the node weight. A node without this attribute is
            assumed to have weight 1.

        Returns
        -------
        min_weighted_cover : set
            Returns a set of nodes whose weight sum is no more than twice
            the weight sum of the minimum weight vertex cover.

        Notes
        -----
        For a directed graph, a vertex cover has the same definition: a set
        of nodes such that each edge in the graph is incident to at least
        one node in the set. Whether the node is the head or tail of the
        directed edge is ignored.

        This is the local-ratio algorithm for computing an approximate
        vertex cover. The algorithm greedily reduces the costs over edges,
        iteratively building a cover. The worst-case runtime of this
        implementation is $O(m \log n)$, where $n$ is the number
        of nodes and $m$ the number of edges in the graph.

        References
        ----------
        .. [1] Bar-Yehuda, R., and Even, S. (1985). "A local-ratio theorem for
           approximating the weighted vertex cover problem."
           *Annals of Discrete Mathematics*, 25, 27–46
           <http://www.cs.technion.ac.il/~reuven/PDF/vc_lr.pdf>

        """
        cost = dict(G.nodes(data=weight, default=1))
        # While there are uncovered edges, choose an uncovered and update
        # the cost of the remaining edges.
        cover = set()
        for u, v in G.edges():
            if u in cover or v in cover:
                continue
            if cost[u] <= cost[v]:
                cover.add(u)
                cost[v] -= cost[u]
            else:
                cover.add(v)
                cost[u] -= cost[v]
        return cover