Search for: All records

Graph compression or sparsification is a basic information-theoretic and computational question. A major open problem in this research area is whether $$(1+\epsilon)$$-approximate cut-preserving vertex sparsifiers with size close to the number of terminals exist. As a step towards this goal, we initiate the study of a thresholded version of the problem: for a given parameter $$c$$, find a smaller graph, which we call \emph{connectivity-$$c$$ mimicking network}, which preserves connectivity among $$k$$ terminals exactly up to the value of $$c$$. We show that connectivity-$$c$$ mimicking networks of size $O(kc^4)$ exist and can be found in time $$m(c\log n)^{O(c)}$$. We also give a separate algorithm that constructs such graphs of size $$k \cdot O(c)^{2c}$$ in time $$mc^{O(c)}\log^{O(1)}n$$. These results lead to the first offline data structures for answering fully dynamic $$c$$-edge-connectivity queries for $$c \ge 4$$ in polylogarithmic time per query as well as more efficient algorithms for survivable network design on bounded treewidth graphs.