Covering and Packing in Linear Space
Given a family of subsets of an n-element universe, the k-cover problem asks whether there are k sets in the family whose union contains the universe; in the k-packing problem the sets are required to be pairwise disjoint and their union contained in the universe. When the size of the family is exponential in n, the fastest known algorithms for these problems use inclusion–exclusion and fast zeta transform, taking time and space 2n, up to a factor polynomial in n. Can one improve these bounds to only linear in the size of the family? Here, we answer the question in the affirmative regarding the space requirement, while not increasing the time requirement. Our key contribution is a new fast zeta transform that adapts its space usage to the support of the function to be transformed. Thus, for instance, the chromatic or domatic number of an n-vertex graph can be found in time within a polynomial factor of 2n and space proportional to the number of maximal independent sets, O(1.442n), or minimal dominating sets, O(1.716n), respectively. Moreover, by exploiting some properties of independent sets, we reduce the space requirement for computing the chromatic polynomial to O(1.292n). Our algorithms also parallelize efficiently.
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