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Submodular maximization is one of the central topics in combinatorial optimization. It has found numerous applications in the real world. In the past decades, a series of algorithms have been proposed for this problem. However, most of the state-of-the-art algorithms are randomized. There remain non-negligible gaps with respect to approximation ratios between deterministic and randomized algorithms in submodular maximization. In this paper, we propose deterministic algorithms with improved approximation ratios for non-monotone submodular maximization. Specifically, for the matroid constraint, we provide a deterministic 0.283 − o (1) approximation algorithm, while the previous best deterministic algorithm only achieves a 1/4 approximation ratio. For the knapsack constraint, we provide a deterministic 1/4 approximation algorithm, while the previous best deterministic algorithm only achieves a 1/6 approximation ratio. For the linear packing constraints with large widths, we provide a deterministic 1 / 6 − ϵ approximation algorithm. To the best of our knowledge, there is currently no deterministic approximation algorithm for the constraints. [ABSTRACT FROM AUTHOR]

In this paper we study the problem of selfish bin packing with punishment. Different from the selfish bin packing problem proposed by Bilò in 2006, where the items are astute to minimize their shared proportional cost in a bin, we consider the case that items are into the utility defined as the load of bin it is packed in and, a unilateral moving may incur punishment. We define and consider several kinds of punishment, which can be roughly classified into two types: the punishment according to the behavior and the punishment based on the result. For both types we present the tight bound for Price of Anarchy (PoA) under the objective of minimizing the number of bins used. The results proved show that punishment does not always work, a punishment based on the result can be viewed as a delicate designed threat in advance which could perform much better, with an approximate 1.48 upper bound comparing to the optimal solution. • Initiate the consideration of punishment into the bin packing game. • Define three versions of punishment for the selfish bin packing problem. • Prove that punishment may not help to get better performance ratio. • Propose a dissuasive mechanism which could encourage agents to form a better equilibrium profile. [ABSTRACT FROM AUTHOR]