Fast and Deterministic Approximations for k-Cut

In an undirected graph, a k-cut is a set of edges whose removal breaks the graph into at least k connected components. The minimum weight k-cut can be computed in n^O(k) time, but when k is treated as part of the input, computing the minimum weight k-cut is NP-Hard [Goldschmidt and Hochbaum, 1994]. For poly(m,n,k)-time algorithms, the best possible approximation factor is essentially 2 under the small set expansion hypothesis [Manurangsi, 2017]. Saran and Vazirani [1995] showed that a (2 - 2/k)-approximately minimum weight k-cut can be computed via O(k) minimum cuts, which implies a O~(km) randomized running time via the nearly linear time randomized min-cut algorithm of Karger [2000]. Nagamochi and Kamidoi [2007] showed that a (2 - 2/k)-approximately minimum weight k-cut can be computed deterministically in O(mn + n^2 log n) time. These results prompt two basic questions. The first concerns the role of randomization. Is there a deterministic algorithm for 2-approximate k-cuts matching the randomized running time of O~(km)? The second question qualitatively compares minimum cut to 2-approximate minimum k-cut. Can 2-approximate k-cuts be computed as fast as the minimum cut - in O~(m) randomized time? We give a deterministic approximation algorithm that computes (2 + eps)-minimum k-cuts in O(m log^3 n / eps^2) time, via a (1 + eps)-approximation for an LP relaxation of k-cut

Some algorithms to solve a bi-objectives problem for team selection

In real life, many problems are instances of combinatorial optimization. Cross-functional team selection is one of the typical issues. The decision-maker has to select solutions among (kh) solutions in the decision space, where k is the number of all candidates, and h is the number of members in the selected team. This paper is our continuing work since 2018; here, we introduce the completed version of the Min Distance to the Boundary model (MDSB) that allows access to both the "deep" and "wide" aspects of the selected team. The compromise programming approach enables decision-makers to ignore the parameters in the decision-making process. Instead, they point to the one scenario they expect. The aim of model construction focuses on finding the solution that matched the most to the expectation. We develop two algorithms: one is the genetic algorithm and another based on the philosophy of DC programming (DC) and its algorithm (DCA) to find the optimal solution. We also compared the introduced algorithms with the MIQP-CPLEX search algorithm to show their effectiveness

A Fast Algorithm for a Weighted Low Rank Approximation

Matrix low rank approximation including the classical PCA and the robust PCA (RPCA) method have been applied to solve the background modeling problem in video analysis. Recently, it has been demonstrated that a special weighted low rank approximation of matrices can be made robust to the outliers similar to the $\ell_1$-norm in RPCA method. In this work, we propose a new algorithm that can speed up the existing algorithm for solving the special weighted low rank approximation and demonstrate its use in background estimation problem