Incremental flow

Hartline, Jeff; Sharp, Alexa

doi:10.1002/net.20168

Cited by 10 publications

(11 citation statements)

References 8 publications

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“…Hartline and Sharp [18] considered an incremental variant of the maximum flow problem where capacities increase over time. This is in contrast to our framework where the cardinality of the solution increases.…”

Section: Theorem 2 For Every Cardinality Constrained Problem With a Mmentioning

confidence: 99%

General bounds for incremental maximization

Bernstein¹,

Disser

Groß

2020

Math. Program.

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We propose a theoretical framework to capture incremental solutions to cardinality constrained maximization problems. The defining characteristic of our framework is that the cardinality/support of the solution is bounded by a value $$k\in {\mathbb {N}}$$ k ∈ N that grows over time, and we allow the solution to be extended one element at a time. We investigate the best-possible competitive ratio of such an incremental solution, i.e., the worst ratio over all k between the incremental solution after k steps and an optimum solution of cardinality k. We consider a large class of problems that contains many important cardinality constrained maximization problems like maximum matching, knapsack, and packing/covering problems. We provide a general 2.618-competitive incremental algorithm for this class of problems, and we show that no algorithm can have competitive ratio below 2.18 in general. In the second part of the paper, we focus on the inherently incremental greedy algorithm that increases the objective value as much as possible in each step. This algorithm is known to be 1.58-competitive for submodular objective functions, but it has unbounded competitive ratio for the class of incremental problems mentioned above. We define a relaxed submodularity condition for the objective function, capturing problems like maximum (weighted) d-dimensional matching, maximum (weighted) (b-)matching and a variant of the maximum flow problem. We show a general bound for the competitive ratio of the greedy algorithm on the class of problems that satisfy this relaxed submodularity condition. Our bound generalizes the (tight) bound of 1.58 slightly beyond sub-modular functions and yields a tight bound of 2.313 for maximum (weighted) (b-)matching. Our bound is also tight for a more general class of functions as the relevant parameter goes to infinity. Note that our upper bounds on the competitive ratios translate to approximation ratios for the underlying cardinality constrained problems, and our bounds for the greedy algorithm carry over both.

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Section: Theorem 2 For Every Cardinality Constrained Problem With a Mmentioning

confidence: 99%

General bounds for incremental maximization

Bernstein¹,

Disser

Groß

2020

Math. Program.

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“…The session "Flows, Cuts, and Connectivity" contained one paper describing the problem of finding two-connected orientations of graphs [64], another describing two problems related to q-route flows [65], and a third one on multilateral networks [120]. A fourth session "Flows and Cuts" contained papers on the maximum congested cut problem [104], on the quickest flow problem [81], and on a hierarchical version of the maximum flow problem [56].…”

Section: The Session "Designing Network With Connectivitymentioning

confidence: 99%

Further contributions to network optimization

Ben‐Ameur¹,

Gouveia

2006

Networks

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“…We executed GRASP on the unweighted dataset, for a varying number of iterations per edge weight update (more iterations mean higher runtime, and a higher likelihood of identifying more dense subgraphs), and measured its runtime, and recall (fraction of output-dense subgraphs that it identified, excluding disconnected subgraphs, which it does not produce). We limited GRASP to searching for subgraphs of cardinalities up to Nmax = 5, and normalized the runtime of GRASP to the runtime of DYNDENS for the same parameters 18 (i.e. the normalized runtime of DYNDENS is 1).…”

Section: Comparison With Other Techniquesmentioning

confidence: 99%

“…17 The average (over the values of all other parameters tested) standard deviation of varying α ∈ (0, 1) was 4%, and the median standard deviation was 1%. 18 For DYNDENS we selected a reasonable value of δit, given the values of the rest of the parameters. the number of recomputations that BASELINE was able to perform, given the same time as DYNDENS took for the entire dataset Even given the above restricted problem setting, we observed that BASELINE was generally not up to the task of realtime story identification.…”

Section: Comparison With Other Techniquesmentioning

confidence: 99%

“…While max-flow algorithms can be dynamized [22], [18], these algorithms can only identify and maintain clusters containing user-specified nodes. In a related vein, [14] uses max-flow to find the top-1 dense subgraph (for AVGDE-GREE); however their techniques cannot be efficiently applied to a top-k or threshold variant, nor can they be efficiently dynamized.…”

Section: Related Workmentioning

confidence: 99%

See 1 more Smart Citation

Dense subgraph maintenance under streaming edge weight updates for real-time story identification

et al. 2013

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Recent years have witnessed an unprecedented proliferation of social media. People around the globe author, every day, millions of blog posts, micro-blog posts, social network status updates, etc. This rich stream of information can be used to identify, on an ongoing basis, emerging stories, and events that capture popular attention. Stories can be identified via groups of tightly-coupled realworld entities, namely the people, locations, products, etc., that are involved in the story. The sheer scale, and rapid evolution of the data involved necessitate highly efficient techniques for identifying important stories at every point of time.The main challenge in real-time story identification is the maintenance of dense subgraphs (corresponding to groups of tightlycoupled entities) under streaming edge weight updates (resulting from a stream of user-generated content). This is the first work to study the efficient maintenance of dense subgraphs under such streaming edge weight updates. For a wide range of definitions of density, we derive theoretical results regarding the magnitude of change that a single edge weight update can cause. Based on these, we propose a novel algorithm, DYNDENS, which outperforms adaptations of existing techniques to this setting, and yields meaningful results. Our approach is validated by a thorough experimental evaluation on large-scale real and synthetic datasets.

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Incremental flow

Cited by 10 publications

References 8 publications

General bounds for incremental maximization

General bounds for incremental maximization

Further contributions to network optimization

Dense subgraph maintenance under streaming edge weight updates for real-time story identification

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