Fully-Dynamic Submodular Cover with Bounded Recourse

Gupta, Anupam; Levin, Roie

doi:10.1109/focs46700.2020.00110

Cited by 6 publications

(4 citation statements)

References 51 publications

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“…In addition, there has been extensive work on online algorithms in the recourse model for a variety of different problems. For amortized recourse, studied problems include online bipartite matching [6], graph coloring [9], minimum spanning tree and traveling salesperson [21], Steiner tree [12], online facility location [11], bin packing [13], submodular covering [14], and constrained optimization [3].…”

Section: Related Workmentioning

confidence: 99%

The Power of Amortized Recourse for Online Graph Problems

Liu¹,

Jonathan²

2022

Preprint

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In this work, we study graph problems with monotone-sum objectives. We propose a general two-fold greedy algorithm that references α-approximation algorithms (where α ≥ 1) to achieve (t • α)competitiveness while incurring at most wmax•(t+1) min{1,wmin}•(t−1) amortized recourse, where w max and w min are the largest value and the smallest positive value that can be assigned to an element in the sum. We further refine this trade-off between competitive ratio and amortized recourse for three classical graph problems. For IndependentSet, we refine the analysis of our general algorithm and show that t-competitiveness can be achieved with t t−1 amortized recourse. For MaximumMatching, we use an existing algorithm with limited greed to show that t-competitiveness can be achieved with3−t * amortized recourse, where t * is the largest number such that t * = 1 + 1 j ≤ t for some integer j. For VertexCover, we introduce a polynomial-time algorithm that further limits greed to show that (2 − 2 OPT )-competitiveness, where OPT is the size of the optimal vertex cover, can be achieved with at most 10 3 amortized recourse by a potential function argument. We remark that this online result can be used as an offline approximation result (without violating the unique games conjecture [19]) to improve upon that of Monien and Speckenmeyer [22] for graphs containing odd cycles of length no less than 2k + 3, using an algorithm that is also constructive.

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Section: Related Workmentioning

confidence: 99%

The Power of Amortized Recourse for Online Graph Problems

Liu¹,

Jonathan²

2022

Preprint

View full text Add to dashboard Cite

show abstract

“…Finally, we are ready to write our LP, which is is the intersection of the submodular cover LPs of (2.1) for the functions , across all 5 This formulation is reminiscent of online and dynamic submodular cover problems [22,23] in which the goal is also to maintain a feasible submodular cover while the underlying submodular function changes over time. However the cost models in these other works are different.…”

Section: Submodular Cover Lp Formulationmentioning

confidence: 99%

Competitive Algorithms for Block-Aware Caching

Coester

Levin

Naor

et al. 2022

Proceedings of the 34th ACM Symposium on Parallelism in Algorithms and Architectures

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Motivated by the design of real system storage hierarchies, we study the block-aware caching problem, a generalization of classic caching in which fetching (or evicting) pages from the same block incurs the same cost as fetching (or evicting) just one page from the block. Given a cache of size , and a sequence of requests from pages partitioned into given blocks of size ≤ , the goal is to minimize the total cost of fetching to (or evicting from) cache. This problem captures generalized caching as a special case, which is already NP-hard offline. We show the following suite of results:• For the eviction cost model, we show an (log )-approximate offline algorithm, a -competitive deterministic online algorithm, and an (log 2 )-competitive randomized online algorithm.

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“…In addition, there has been extensive work on online algorithms in the recourse model for a variety of different problems. For amortized recourse, studied problems include online bipartite matching [6], graph coloring [9], minimum spanning tree and traveling salesperson [22], Steiner tree [12], online facility location [11], bin packing [13], submodular covering [14], and constrained optimization [3].…”

Section: Minimum Vertex Covermentioning

confidence: 99%

The Power of Amortized Recourse for Online Graph Problems

Liu

Jonathan²

2022

Approximation and Online Algorithms

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In this work, we study online graph problems with monotonesum objectives, where the vertices or edges of the graph are revealed one by one and need to be assigned to a value such that certain properties of the solution hold. We propose a general two-fold greedy algorithm that augments its current solution greedily and references yardstick algorithms. The algorithm maintains competitiveness by strategically aligning to the yardstick solution and incurring recourse. We show that our general algorithm achieves t-competitiveness while incurring at most wmax •(t+1) t−1 amortized recourse for any monotone-sum problems with integral solution, where wmax is the largest value that can be assigned to a vertex or an edge. For fractional monotone-sum problems where each of the assigned values is between [0, 1], our general algorithm incurs at most t+1 w min •(t−1) amortized recourse, where wmin is the smallest non-negative value that can be assigned. We further show that the general algorithm can be improved for three classical graph problems. For Independent Set, we refine the analysis of our general algorithm and show that t-competitiveness can be achieved with t t−1 amortized recourse. For Maximum Cardinality Matching, we limit our algorithm's greed to show that t-competitiveness can be achieved with3−t * amortized recourse, where t * is the largest number such that t * = 1+ 1 j ≤ t for some integer j. For Vertex Cover, we show that our algorithm guarantees a competitive ratio strictly smaller than 2 for any finite instance in polynomial time while incurring at most 3.33 amortized recourse. We beat the almost unbreakable 2-approximation in polynomial time by using the optimal solution as the reference without computing it. We remark that this online result can be used as an offline approximation result (without violating the unique games conjecture [20]) to partially improve upon the constructive algorithm of Monien and Speckenmeyer [23].

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Fully-Dynamic Submodular Cover with Bounded Recourse

Cited by 6 publications

References 51 publications

The Power of Amortized Recourse for Online Graph Problems

The Power of Amortized Recourse for Online Graph Problems

Competitive Algorithms for Block-Aware Caching

The Power of Amortized Recourse for Online Graph Problems

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