The Closest String Problem is an NP-hard problem that aims to find a string that has the minimum distance from all sequences that belong to the given set of strings. Its applications can be found in coding theory, computational biology, and designing degenerated primers, among others. There are efficient exact algorithms that have reached high-quality solutions for binary sequences. However, there is still room for improvement concerning the quality of solutions over DNA and protein sequences. In this paper, we introduce a three-stage algorithm that comprises the following process: first, we apply a novel alphabet pruning method to reduce the search space for effectively finding promising search regions. Second, a variant of beam search to find a heuristic solution is employed. This method utilizes a newly developed guiding function based on an expected distance heuristic score of partial solutions. Last, we introduce a local search to improve the quality of the solution obtained from the beam search. Furthermore, due to the lack of real-world benchmarks, two real-world datasets are introduced to verify the robustness of the method. The extensive experimental results show that the proposed method outperforms the previous approaches from the literature.

Compressed sensing (CS) is an important theory for sub-Nyquist sampling and recovery of compressible data. Recently, it has been extended by Pham and Venkatesh to cope with the case where corruption to the CS data is modeled as impulsive noise. The new formulation, termed as robust CS, combines robust statistics and CS into a single framework to suppress outliers in the CS recovery. To solve the newly formulated robust CS problem, Pham and Venkatesh suggested a scheme that iteratively solves a number of CS problems, the solutions from which converge to the true robust compressed sensing solution. However, this scheme is rather inefficient as it has to use existing CS solvers as a proxy. To overcome limitation with the original robust CS algorithm, we propose to solve the robust CS problem directly in this paper and drive more computationally efficient algorithms by following latest advances in large-scale convex optimization for non-smooth regularization. Furthermore, we also extend the robust CS formulation to various settings, including additional affine constraints, $\ell_1$-norm loss function, mixed-norm regularization, and multi-tasking, so as to further improve robust CS. We also derive simple but effective algorithms to solve these extensions. We demonstrate that the new algorithms provide much better computational advantage over the original robust CS formulation, and effectively solve more sophisticated extensions where the original methods simply cannot. We demonstrate the usefulness of the extensions on several CS imaging tasks.