
Vertex Sparsification for Edge Connectivity
Graph compression or sparsification is a basic informationtheoretic and...
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Faster Algorithms for Rooted Connectivity in Directed Graphs
We consider the fundamental problems of determining the rooted and globa...
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Mimicking Networks Parameterized by Connectivity
Given a graph G=(V,E), capacities w(e) on edges, and a subset of termina...
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A Faster Local Algorithm for Detecting BoundedSize Cuts with Applications to HigherConnectivity Problems
Consider the following "local" cutdetection problem in a directed graph...
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The starstructure connectivity and starsubstructure connectivity of hypercubes and folded hypercubes
As a generalization of vertex connectivity, for connected graphs G and T...
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Incomplete Directed Perfect Phylogeny in Linear Time
Reconstructing the evolutionary history of a set of species is a central...
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On Uniquely Registrable Networks
Consider a network with N nodes in ddimensional Euclidean space, and M ...
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Vertex Sparsification for Edge Connectivity in Polynomial Time
An important open question in the area of vertex sparsification is whether (1+ϵ)approximate cutpreserving vertex sparsifiers with size close to the number of terminals exist. The work Chalermsook et al. (SODA 2021) introduced a relaxation called connectivityc mimicking networks, which asks to construct a vertex sparsifier which preserves connectivity among k terminals exactly up to the value of c, and showed applications to dynamic connectivity data structures and survivable network design. We show that connectivityc mimicking networks with O(kc^3) edges exist and can be constructed in polynomial time in n and c, improving over the results of Chalermsook et al. (SODA 2021) for any c ≥log n, whose runtimes depended exponentially on c.
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