We study the fully dynamic AllPairs Shortest Paths (APSP) problem in undirected edgeweighted graphs. Given an nvertex graph G with nonnegative edge lengths, that undergoes an online sequence of edge insertions and deletions, the goal is to support approximate distance queries and shortestpath queries. We provide a deterministic algorithm for this problem, that, for a given precision parameter є, achieves approximation factor (loglogn)2O(1/є3), and has amortized update time O(nєlogL) per operation, where L is the ratio of longest to shortest edge length. Query time for distancequery is O(2O(1/є)· logn· loglogL), and query time for shortestpath query is O(E(P)+2O(1/є)· logn· loglogL), where P is the path that the algorithm returns. To the best of our knowledge, even allowing any o(n)approximation factor, no adaptiveupdate algorithms with better than Θ(m) amortized update time and better than Θ(n) query time were known prior to this work. We also note that our guarantees are stronger than the best current guarantees for APSP in decremental graphs in the adaptiveadversary setting.
In order to obtain these results, we consider an intermediate problem, called Recursive Dynamic Neighborhood Cover (RecDynNC), that was formally introduced in [Chuzhoy, STOC ’21]. At a high level, given an undirected edgeweighted graph G undergoing an online sequence of edge deletions, together with a distance parameter D, the goal is to maintain a sparse Dneighborhood cover of G, with some additional technical requirements. Our main technical contribution is twofolds. First, we provide a blackbox reduction from APSP in fully dynamic graphs to the RecDynNC problem. Second, we provide a new deterministic algorithm for the RecDynNC problem, that, for a given precision parameter є, achieves approximation factor (loglogm)2O(1/є2), with total update time O(m1+є), where m is the total number of edges ever present in G. This improves the previous algorithm of [Chuzhoy, STOC ’21], that achieved approximation factor (logm)2O(1/є) with similar total update time. Combining these two results immediately leads to the deterministic algorithm for fullydynamic APSP with the guarantees stated above.
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Stochastic Minimum Vertex Cover in General Graphs: A 3/2Approximation
We study the stochastic vertex cover problem. In this problem, G = (V, E) is an arbitrary known graph, and G⋆ is an unknown random subgraph of G where each edge e is realized independently with probability p. Edges of G⋆ can only be verified using edge queries. The goal in this problem is to find a minimum vertex cover of G⋆ using a small number of queries.
Our main result is designing an algorithm that returns a vertex cover of G⋆ with size at most (3/2+є) times the expected size of the minimum vertex cover, using only O(n/є p) nonadaptive queries. This improves over the bestknown 2approximation algorithm by Behnezhad, Blum and Derakhshan [SODA’22] who also show that Ω(n/p) queries are necessary to achieve any constant approximation.
Our guarantees also extend to instances where edge realizations are not fully independent. We complement this upperbound with a tight 3/2approximation lower bound for stochastic graphs whose edges realizations demonstrate mild correlations.
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 Award ID(s):
 2145898
 NSFPAR ID:
 10482192
 Publisher / Repository:
 ACM
 Date Published:
 Page Range / eLocation ID:
 242 to 253
 Format(s):
 Medium: X
 Location:
 Orlando FL USA
 Sponsoring Org:
 National Science Foundation
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