Paths, Flowers and Vertex Cover
It is well known that in a bipartite (and more generally in a König) graph, the size of the minimum vertex cover is equal to the size of the maximum matching. We first address the question whether (and if not when) this property still holds in a König graph if we insist on forcing one of the two vertices of some of the matching edges in the vertex cover solution. We characterize such graphs using the classical notions of augmenting paths and flowers used in Edmonds’ matching algorithm. Using this characterization, we develop an O*(9k) algorithm for the question of whether a general graph has a vertex cover of size at most m + k where m is the size of the maximum matching. Our algorithm for this well studied Above Guarantee Vertex Cover problem uses the technique of iterative compression and the notion of important separators, and improves the runtime of the previous best algorithm that took O*(15k) time. As a consequence of this result we get that well known problems like Almost 2 SAT (deleting at most k clauses to get a satisfying 2 SAT formula) and König Vertex Deletion (deleting at most k vertices to get a König graph) also have an algorithm with O*(9k) running time, improving on the previous bound of O*(15k).
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