We introduce a notion of universality in the context of optimization problems with partial information. Universality is a framework for dealing with uncertainty by guaranteeing a certain quality of goodness for all possible completions of the partial information set. Universal variants of optimization problems can be defined that are both natural and well-motivated. We consider universal versions of three classical problems: TSP, Steiner Tree and Set Cover. We present a polynomial-time algorithm to find a universal tour on a given metric space over Ò vertices such that for any subset of the vertices, the sub-tour induced by the subset is within Ç´ÐÓ Ò ÐÓ ÐÓ Òµ of an optimal tour for the subset. Similarly, we show that given a metric space over Ò vertices and a root vertex, we can find a universal spanning tree such that for any subset of vertices containing the root, the sub-tree induced by the subset is within Ç´ÐÓ Ò ÐÓ ÐÓ Òµ of an optimal Steiner tree for the subset. Our algorithms rely on a new notion of sparse partitions, that may be of independent interest. For the special case of doubling metrics, which includes both constant-dimensional Euclidean and growth-restricted metrics, our algorithms achieve an Ç´ÐÓ Òµ upper bound. We complement our results for the universal Steiner tree problem with a lower bound of ª´ÐÓ Ò ÐÓ ÐÓ Òµ that holds even for Ò vertices on the plane. We also show that a slight generalization of the universal Steiner Tree problem is coNP-hard and present nearly tight upper and lower bounds for a universal version of Set Cover.
Jia et al. REF first proposed the Universal approximation model for the TSP, Steiner tree and Set Cover problems.
Universal approximations for TSP, Steiner tree, and set cover
STOC '05
Mathematics Computer Science