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1. Algorithms for covering multiple submodular constraints and applications

   https://doi.org/10.1007/s10878-022-00874-x  

   Chekuri, Chandra; Inamdar, Tanmay; Quanrud, Kent; Varadarajan, Kasturi; Zhang, Zhao (June 2022, Journal of Combinatorial Optimization)

   Abstract We consider the problem of covering multiple submodular constraints. Given a finite ground set N , a weight function $$w: N \rightarrow \mathbb {R}_+$$ w : N → R + , r monotone submodular functions $$f_1,f_2,\ldots ,f_r$$ f 1 , f 2 , … , f r over N and requirements $$k_1,k_2,\ldots ,k_r$$ k 1 , k 2 , … , k r the goal is to find a minimum weight subset $$S \subseteq N$$ S ⊆ N such that $$f_i(S) \ge k_i$$ f i ( S ) ≥ k i for $$1 \le i \le r$$ 1 ≤ i ≤ r . We refer to this problem as Multi-Submod-Cover and it was recently considered by Har-Peled and Jones (Few cuts meet many point sets. CoRR. arxiv:abs1808.03260 Har-Peled and Jones 2018) who were motivated by an application in geometry. Even with $$r=1$$ r = 1 Multi-Submod-Cover generalizes the well-known Submodular Set Cover problem ( Submod-SC ), and it can also be easily reduced to Submod-SC . A simple greedy algorithm gives an $$O(\log (kr))$$ O ( log ( k r ) ) approximation where $$k = \sum _i k_i$$ k = ∑ i k i and this ratio cannot be improved in the general case. In this paper, motivated by several concrete applications, we consider two ways to improve upon the approximation given by the greedy algorithm. First, we give a bicriteria approximation algorithm for Multi-Submod-Cover that covers each constraint to within a factor of $$(1-1/e-\varepsilon )$$ ( 1 - 1 / e - ε ) while incurring an approximation of $$O(\frac{1}{\epsilon }\log r)$$ O ( 1 ϵ log r ) in the cost. Second, we consider the special case when each $$f_i$$ f i is a obtained from a truncated coverage function and obtain an algorithm that generalizes previous work on partial set cover ( Partial-SC ), covering integer programs ( CIPs ) and multiple vertex cover constraints Bera et al. (Theoret Comput Sci 555:2–8 Bera et al. 2014). Both these algorithms are based on mathematical programming relaxations that avoid the limitations of the greedy algorithm. We demonstrate the implications of our algorithms and related ideas to several applications ranging from geometric covering problems to clustering with outliers. Our work highlights the utility of the high-level model and the lens of submodularity in addressing this class of covering problems. 

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2. Randomized Greedy Learning for Non-monotone Stochastic Submodular Maximization Under Full-bandit Feedback

   Fourati, Fares; Aggarwal, Vaneet; Quinn, Christopher John; Alouini, Mohamed-Slim (April 2023, Proceedings of the International Workshop on Artificial Intelligence and Statistics)

   We investigate the problem of unconstrained combinatorial multi-armed bandits with full-bandit feedback and stochastic rewards for submodular maximization. Previous works investigate the same problem assuming a submodular and monotone reward function. In this work, we study a more general problem, i.e., when the reward function is not necessarily monotone, and the submodularity is assumed only in expectation. We propose Randomized Greedy Learning (RGL) algorithm and theoretically prove that it achieves a $$\frac{1}{2}$$-regret upper bound of $$\Tilde{\mathcal{O}}(n T^{\frac{2}{3}})$$ for horizon $$T$$ and number of arms $$n$$. We also show in experiments that RGL empirically outperforms other full-bandit variants in submodular and non-submodular settings. 

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3. Randomized Greedy Learning for Non-monotone Stochastic Submodular Maximization Under Full-bandit Feedback

   Fares Fourati, Vaneet Aggarwal (January 2023, Proceedings of Machine Learning Research)

   We investigate the problem of unconstrained combinatorial multi-armed bandits with fullbandit feedback and stochastic rewards for submodular maximization. Previous works investigate the same problem assuming a submodular and monotone reward function. In this work, we study a more general problem, i.e., when the reward function is not necessarily monotone, and the submodularity is assumed only in expectation. We propose Randomized Greedy Learning (RGL) algorithm and theoretically prove that it achieves a 1 2 -regret upper bound of O˜(nT 2 3 ) for horizon T and number of arms n. We also show in experiments that RGL empirically outperforms other full-bandit variants in submodular and non-submodular settings. 

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4. Approximating Red-Blue Set Cover and Minimum Monotone Satisfying Assignment

   https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2023.11  

   Chlamtáč, Eden; Makarychev, Yury; Vakilian, Ali (September 2023, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023))

   Megow, Nicole; Smith, Adam (Ed.)

   We provide new approximation algorithms for the Red-Blue Set Cover and Circuit Minimum Monotone Satisfying Assignment (MMSA) problems. Our algorithm for Red-Blue Set Cover achieves Õ(m^{1/3})-approximation improving on the Õ(m^{1/2})-approximation due to Elkin and Peleg (where m is the number of sets). Our approximation algorithm for MMSA_t (for circuits of depth t) gives an Õ(N^{1-δ}) approximation for δ = 1/3 2^{3-⌈t/2⌉}, where N is the number of gates and variables. No non-trivial approximation algorithms for MMSA_t with t ≥ 4 were previously known. We complement these results with lower bounds for these problems: For Red-Blue Set Cover, we provide a nearly approximation preserving reduction from Min k-Union that gives an Ω(m^{1/4 - ε}) hardness under the Dense-vs-Random conjecture, while for MMSA we sketch a proof that an SDP relaxation strengthened by Sherali-Adams has an integrality gap of N^{1-ε} where ε → 0 as the circuit depth t → ∞. 

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5. On the Unreasonable Effectiveness of the Greedy Algorithm: Greedy Adapts to Sharpness

   Pokutta, S.; Singh, M.; Torrico, A. (June 2020, Thirty-seventh International Conference on Machine Learning)

   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. However, empirical studies show that its performance is substantially better in practice. This raises a natural question of explaining this improved performance of the greedy algorithm. In this work, we define sharpness for submodular functions as a candidate explanation for this phenomenon. We show that the greedy algorithm provably performs better as the sharpness of the submodular function increases. This improvement ties in closely with the faster convergence rates of first order methods for sharp functions in convex optimization. 

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