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A Refined Analysis of Submodular Greedy [article]

Ariel Kulik and Roy Schwartz and Hadas Shachnai
2021 arXiv   pre-print
We present a novel refined analysis of this greedy heuristic which enables us to: (1) reduce the enumeration in the tight (1-e^-1)-approximation of [Sviridenko 04] from subsets of size three to two; (2  ...  Many algorithms for maximizing a monotone submodular function subject to a knapsack constraint rely on the natural greedy heuristic.  ...  Based on this insight, we give in Section 2 a refined analysis of the greedy phase. The refined analysis is the key for the proof of Theorem 1.1, given in Section 3.  ... 
arXiv:2102.12879v2 fatcat:nybkydornrf53dhyugg3y2cgsy

On the Unreasonable Effectiveness of the Greedy Algorithm: Greedy Adapts to Sharpness [article]

Alfredo Torrico, Mohit Singh, Sebastian Pokutta
2020 arXiv   pre-print
It is well known that the standard greedy algorithm guarantees a worst-case approximation factor of 1-1/e when maximizing a monotone submodular function under a cardinality constraint.  ...  This raises a natural question of explaining this improved performance of the greedy algorithm.  ...  These results significantly improve the ones obtained by the curvature analysis and monotonic sharpness, providing evidence that more refined notions of sharpness can capture the behavior of the greedy  ... 
arXiv:2002.04063v1 fatcat:u5bypzyuxrbz5b2l27epyndfiy

Causal meets Submodular: Subset Selection with Directed Information

Yuxun Zhou, Costas J. Spanos
2016 Neural Information Processing Systems  
Moreover, we show that based on SmI, greedy algorithm has performance guarantee for the maximization of possibly non-monotonic and non-submodular functions, justifying its usage for a much broader class  ...  To substantiate the idea of approximate submodularity, we introduce a novel quantity, namely submodularity index (SmI), for general set functions.  ...  BEARS has been established by the University of California, Berkeley as a center for intellectual excellence in research and education in Singapore.  ... 
dblp:conf/nips/ZhouS16 fatcat:mdkovi2xvncuhpxvriwo3x7tki

A New Approximation Guarantee for Monotone Submodular Function Maximization via Discrete Convexity [article]

Tasuku Soma, Yuichi Yoshida
2017 arXiv   pre-print
In this paper, we provide a novel approximation guarantee by extracting an M^-concave function h:2^E → R_+, a notion in discrete convex analysis, from the objective function f:2^E → R_+.  ...  In monotone submodular function maximization, approximation guarantees based on the curvature of the objective function have been extensively studied in the literature.  ...  Acknowledgment The authors thank Yuni Iwamasa for pointing out a reference [8] on ultrametric fitting.  ... 
arXiv:1709.02910v1 fatcat:ayhp3fl6xbhudhtccxw6nob5ea

A New Approximation Guarantee for Monotone Submodular Function Maximization via Discrete Convexity

Tasuku Soma, Yuichi Yoshida, Michael Wagner
2018 International Colloquium on Automata, Languages and Programming  
In this paper, we provide a novel approximation guarantee by extracting an M -concave function h : 2 E → R + , a notion in discrete convex analysis, from the objective function f : 2 E → R + .  ...  In monotone submodular function maximization, approximation guarantees based on the curvature of the objective function have been extensively studied in the literature.  ...  Acknowledgements The authors thank Yuni Iwamasa for pointing out a reference [8] on ultrametric fitting. The authors also thank Jan Vondrák for telling us a reference [21].  ... 
doi:10.4230/lipics.icalp.2018.99 dblp:conf/icalp/SomaY18 fatcat:eiyaj5g3jjaojk7jzihhmbyuna

Maximizing Sequence-Submodular Functions and its Application to Online Advertising [article]

Saeed Alaei, Ali Makhdoumi, Azarakhsh Malekian
2019 arXiv   pre-print
We establish that if the objective function is sequence-submodular and sequence-non-decreasing, then there exists a greedy algorithm that achieves 1-1/e of the optimal solution.  ...  Motivated by applications in online advertising, we consider a class of maximization problems where the objective is a function of the sequence of actions as well as the running duration of each action  ...  We then find a greedy algorithm to maximize sequence submodular functions and establish its performance. Finally, we apply our algorithm and analysis to online ad allocation problem.  ... 
arXiv:1009.4153v4 fatcat:jhd52royobf7vc6dc5wi67qn7q

Weakly Submodular Function Maximization Using Local Submodularity Ratio [article]

Richard Santiago, Yuichi Yoshida
2020 arXiv   pre-print
We then provide a more refined analysis that takes into account that the weak submodularity parameter may change (sometimes improving) throughout the execution of the algorithm.  ...  Weak submodularity is a natural relaxation of the diminishing return property, which is equivalent to submodularity.  ...  Another important piece of our work is to provide a more refined analysis that allows for the submodularity ratio to change throughout the execution of the algorithm.  ... 
arXiv:2004.14650v2 fatcat:ym3qnirktbdnvaap34g4iq2egm

Maximizing Submodular or Monotone Functions under Partition Matroid Constraints by Multi-objective Evolutionary Algorithms [article]

Anh Viet Do, Frank Neumann
2020 arXiv   pre-print
While there have been many studies on the subject, most of existing run-time analyses for GSEMO assume a single cardinality constraint.  ...  A simple multi-objective evolutionary algorithm called GSEMO has been shown to achieve good approximation for submodular functions efficiently.  ...  Acknowledgements The experiments were run using the HPC service provided by the University of Adelaide.  ... 
arXiv:2006.12773v2 fatcat:cz2zqk3rv5g5nkc2gtmvjgjwli

A Unified Continuous Greedy Algorithm for Submodular Maximization

M. Feldman, Joseph Naor, R. Schwartz
2011 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science  
The study of combinatorial problems with a submodular objective function has attracted much attention in recent years, and is partly motivated by the importance of such problems to economics, algorithmic  ...  The main bottleneck of such continuous techniques is how to approximately solve a non-convex relaxation for the submodular problem at hand.  ...  We analyze it for general non-monotone submodular functions and then refine the analysis to get improved results for monotone submodular functions.  ... 
doi:10.1109/focs.2011.46 dblp:conf/focs/FeldmanNS11 fatcat:ajjddbnuwzcjxd23rcwup2ptne

Using Partial Monotonicity in Submodular Maximization [article]

Loay Mualem, Moran Feldman
2022 arXiv   pre-print
The monotonicity property of set functions plays a central role in submodular maximization.  ...  Traditionally, the study of submodular functions was based on binary function properties.  ...  Analysis of the Greedy algorithm A version of the greedy algorithm designed for matroid constraints appears as Algorithm 3.  ... 
arXiv:2202.03051v2 fatcat:ftvtlw67abcfvemhu5p4esigcy

On Submodular Prophet Inequalities and Correlation Gap [article]

Chandra Chekuri, Vasilis Livanos
2021 arXiv   pre-print
Along the way they showed a variant of the correlation gap for non-negative submodular functions.  ...  Rubinstein and Singla developed a notion of combinatorial prophet inequalities in order to generalize the standard prophet inequality setting to combinatorial valuation functions such as submodular and  ...  Moreover, we presented a refined analysis of the Measured Continuous Greedy algorithm for polytopes with small coordinates and general non-negative submodular functions, showing that, for these cases,  ... 
arXiv:2107.03662v1 fatcat:r6eazxwdtbannmlkz5gxvvk5ta

A Parallel Double Greedy Algorithm for Submodular Maximization [article]

Alina Ene, Huy L. Nguyen, Adrian Vladu
2018 arXiv   pre-print
We study parallel algorithms for the problem of maximizing a non-negative submodular function.  ...  Our algorithm is based on a continuous variant of the double greedy algorithm of Buchbinder et al. that achieves the optimal 1/2 approximation in the sequential setting.  ...  This allows us to extend the analysis at a loss in the approximation of O(ǫ 3 M ) per iteration, and thus O(ǫM ) overall. Analysis of the approximation guarantee.  ... 
arXiv:1812.01591v1 fatcat:qjgbyzcmezhztjsvkkeovvw2eq

Batch greedy maximization of non-submodular functions: Guarantees and applications to experimental design [article]

Jayanth Jagalur-Mohan, Youssef Marzouk
2021 arXiv   pre-print
We propose and analyze batch greedy heuristics for cardinality constrained maximization of non-submodular non-decreasing set functions.  ...  Based on that analogy, we propose a new class of methods exploiting any plausible modular bound.  ...  for a helpful correspondence clarifying a detail about operator concave inequalities; and to Arvind Saibaba for help in simplifying some linear algebra arguments.  ... 
arXiv:2006.04554v3 fatcat:6bxokkvm6jgktpv3bcznlmfpnu

Online Submodular Welfare Maximization

Nitish Korula, Vahab Mirrokni, Morteza Zadimoghaddam
2015 Proceedings of the Forty-Seventh Annual ACM on Symposium on Theory of Computing - STOC '15  
In the Submodular Welfare Maximization (SWM) problem, the input consists of a set of n items, each of which must be allocated to one of m agents.  ...  This problem is motivated by applications to Internet advertising, where user ad impressions must be allocated to advertisers whose value is a submodular function of the set of users / impressions they  ...  Further, for a broad class of submodular functions, we can strengthen our analysis.  ... 
doi:10.1145/2746539.2746626 dblp:conf/stoc/KorulaMZ15 fatcat:s2ea4ksdr5aprenrtnk4562fai

Scaling Submodular Optimization Approaches for Control Applications in Networked Systems [article]

Arun V Sathanur
2018 arXiv   pre-print
Often times, in many design problems, there is a need to select a small set of informative or representative elements from a large ground set of entities in an optimal fashion.  ...  In this work, we explore a well-known paradigm, namely leader-selection in a multi-agent networked environment to illustrate strategies for scalable submodular optimization.  ...  A quick analysis shows a speedup of the order of O(n 1.7 ) and this is confirmed by regression analysis on the wall-times for each of the algorithms which shows a speedup order of magnitude in the range  ... 
arXiv:1810.02837v1 fatcat:k4oma735gfdmtapjljjx2rwuzq
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