8 research outputs found
Dynamic Maxflow via Dynamic Interior Point Methods
In this paper we provide an algorithm for maintaining a
-approximate maximum flow in a dynamic, capacitated graph
undergoing edge additions. Over a sequence of -additions to an -node
graph where every edge has capacity our algorithm runs in
time . To obtain this result we
design dynamic data structures for the more general problem of detecting when
the value of the minimum cost circulation in a dynamic graph undergoing edge
additions obtains value at most (exactly) for a given threshold . Over a
sequence -additions to an -node graph where every edge has capacity
and cost we solve this thresholded
minimum cost flow problem in . Both of our algorithms
succeed with high probability against an adaptive adversary. We obtain these
results by dynamizing the recent interior point method used to obtain an almost
linear time algorithm for minimum cost flow (Chen, Kyng, Liu, Peng, Probst
Gutenberg, Sachdeva 2022), and introducing a new dynamic data structure for
maintaining minimum ratio cycles in an undirected graph that succeeds with high
probability against adaptive adversaries.Comment: 30 page
A Deterministic Almost-Linear Time Algorithm for Minimum-Cost Flow
We give a deterministic time algorithm that computes exact
maximum flows and minimum-cost flows on directed graphs with edges and
polynomially bounded integral demands, costs, and capacities. As a consequence,
we obtain the first running time improvement for deterministic algorithms that
compute maximum-flow in graphs with polynomial bounded capacities since the
work of Goldberg-Rao [J.ACM '98].
Our algorithm builds on the framework of
Chen-Kyng-Liu-Peng-Gutenberg-Sachdeva [FOCS '22] that computes an optimal flow
by computing a sequence of -approximate undirected minimum-ratio
cycles. We develop a deterministic dynamic graph data-structure to compute such
a sequence of minimum-ratio cycles in an amortized time per edge
update. Our key technical contributions are deterministic analogues of the
vertex sparsification and edge sparsification components of the data-structure
from Chen et al. For the vertex sparsification component, we give a method to
avoid the randomness in Chen et al. which involved sampling random trees to
recurse on. For the edge sparsification component, we design a deterministic
algorithm that maintains an embedding of a dynamic graph into a sparse spanner.
We also show how our dynamic spanner can be applied to give a deterministic
data structure that maintains a fully dynamic low-stretch spanning tree on
graphs with polynomially bounded edge lengths, with subpolynomial average
stretch and subpolynomial amortized time per edge update.Comment: Accepted to FOCS 202
Fully-Dynamic Graph Sparsifiers Against an Adaptive Adversary
Designing efficient dynamic graph algorithms against an adaptive adversary is a major goal in the field of dynamic graph algorithms and has witnessed many exciting recent developments in, e.g., dynamic matching (Wajc STOC'20) and decremental shortest paths (Chuzhoy and Khanna STOC'19). Compared to other graph primitives (e.g. spanning trees and matchings), designing such algorithms for graph spanners and (more broadly) graph sparsifiers poses a unique challenge since there is no fast deterministic algorithm known for static computation and the lack of a way to adjust the output slowly (known as “small recourse/replacements”). This paper presents the first non-trivial efficient adaptive algorithms for maintaining many sparsifiers against an adaptive adversary. Specifically, we present algorithms that maintain 1. a polylog(n)-spanner of size Õ(n) in polylog(n) amortized update time, 2. an O(k)-approximate cut sparsifier of size Õ(n) in Õ(n1/k) amortized update time, and 3. a polylog(n)-approximate spectral sparsifier in polylog(n) amortized update time. Our bounds are the first non-trivial ones even when only the recourse is concerned. Our results hold even against a stronger adversary, who can access the random bits previously used by the algorithms and the amortized update time of all algorithms can be made worst-case by paying sub-polynomial factors. Our spanner result resolves an open question by Ahmed et al. (2019) and our results and techniques imply additional improvements over existing results, including (i) answering open questions about decremental single-source shortest paths by Chuzhoy and Khanna (STOC'19) and Gutenberg and Wulff-Nilsen (SODA'20), implying a nearly-quadratic time algorithm for approximating minimum-cost unit-capacity flow and (ii) de-amortizing a result of Abraham et al. (FOCS'16) for dynamic spectral sparsifiers. Our results are based on two novel techniques. The first technique is a generic black-box reduction that allows us to assume that the graph is initially an expander with almost uniform-degree and, more importantly, stays as an almost uniform-degree expander while undergoing only edge deletions. The second technique is called proactive resampling: here we constantly re-sample parts of the input graph so that, independent of an adversary's computational power, a desired structure of the underlying graph can be always maintained. Despite its simplicity, the analysis of this sampling scheme is far from trivial, because the adversary can potentially create dependencies between the random choices used by the algorithm. We believe these two techniques could be useful for developing other adaptive algorithms.ISSN:1868-896
Evaluation of a mobile screening service for abdominal aortic aneurysm in Broken Hill, a remote regional centre in far western NSW
Objectives: To evaluate the feasibility of a mobile screening service model for abdominal aortic aneurysm (AAA) in a remote population centre in Australia. Design: Screening test evaluation. Setting: A remote regional centre (population: 20 000) in far western NSW. Participants: Men aged 65-74 years, identified from the Australian Electoral roll. Interventions: A mobile screening service using directed ultrasonography, a basic health check and post-screening consultation. Main outcome measures: Attendance at the screening program, occurrence of AAA in the target population and effectiveness of screening processes. Results: A total of 516 men without a previous diagnosis of AAA were screened, an estimated response rate of 60%. Of these, 463 (89.7%) had a normal aortic diameter, 28 (5.4%) ectatic and 25 (4.9%) a small, moderate or significant aneurysm. Two men with AAA were recommended for surgery. Feedback from participants indicated that the use of a personalised letter of invitation helped with recruitment, that the screening process was acceptable and the service valued. Conclusions: It is feasible to organise and operate a mobile AAA screening service from moderate sized rural and remote population centres. This model could be scaled up to provide national coverage for rural and remote residents. © 2010 The Authors. Journal compilatio